JEE Magnetic Effects of Current Questions

Given: point charge $$q = +10^{-8}\,\text{C}$$ at a distance $$d = 20\,\text{cm} = 0.20\,\text{m}$$ from the centre $$O$$ of a conducting sphere of radius $$R = 10\,\text{cm} = 0.10\,\text{m}$$.
Throughout we use $$k = \dfrac{1}{4\pi\varepsilon_0}=9\times10^9\;\text{N m}^2\!\text{/C}^2$$.

Step 1 : Image-charge data for a charge outside a sphere
For a point charge $$q$$ kept at distance $$d\,(d\gt R)$$ from the centre of a conducting sphere of radius $$R$$:

• An image charge $$q' = -\dfrac{R}{d}\,q$$ is located on the same line at a distance $$b = \dfrac{R^{2}}{d}$$ from the centre.
• The combination $$(q,q')$$ alone makes the potential of the sphere zero (grounded condition).
• If the sphere is isolated (not grounded) and must keep a specified net charge $$Q_{\text{sphere}}$$, an additional charge $$q_c$$ is placed at the centre. $$q_c$$ is fixed by the condition $$q_c + q' = Q_{\text{sphere}}$$. The potential of the sphere then becomes a constant $$V_{\text{sphere}} = k\dfrac{q_c}{R}$$.

Part (A) : Potential before grounding
Initially the sphere is neutral $$(Q_{\text{sphere}}=0)$$ and isolated.

Image charge: $$q' = -\dfrac{R}{d}\,q = -\dfrac{0.10}{0.20}\,10^{-8} = -0.5\times10^{-8} = -5\times10^{-9}\,\text{C}$$.
To keep the sphere neutral, add a central charge $$q_c = -q' = +5\times10^{-9}\,\text{C}$$.

The potential of the sphere is therefore $$V_{\text{before}} = k\dfrac{q_c}{R} = 9\times10^9 \times\dfrac{5\times10^{-9}}{0.10} = 9\times10^9 \times 5\times10^{-8} = 45\times10^{1} = 450\;\text{V}.$$

Hence Option A is correct.

Part (B) : Charge that flows when the sphere is grounded
On connecting the sphere to earth, its potential must become zero. That is achieved by removing the central charge $$q_c$$ (ground provides/absorbs charge) so that only $$q'$$ remains on the sphere.

Charge leaving the sphere to the ground $$\;\Delta Q = q_c = +5\times10^{-9}\,\text{C}.$$

Thus $$5\times10^{-9}\,\text{C}$$ of positive charge flows from the sphere to the earth. Option B is correct.

Part (C) : Charge on the isolated sphere after the grounding wire is removed
With the earth connection opened, the induced charge that remains on the sphere equals $$q' = -5\times10^{-9}\,\text{C}$$.

Option C is correct.

Part (D) : Potential after the external charge is shifted to 30 cm
The grounding has already been removed, so the total charge on the sphere stays fixed at $$Q_{\text{sphere}} = -5\times10^{-9}\,\text{C}.$$ Now the point charge is shifted to $$d_2 = 30\,\text{cm} = 0.30\,\text{m}$$.

New image charge for the new position:
$$q'_2 = -\dfrac{R}{d_2}\,q = -\dfrac{0.10}{0.30}\,10^{-8} = -\dfrac{1}{3}\times10^{-8} = -3.33\times10^{-9}\,\text{C}.$$

To keep the sphere’s total charge fixed, the central charge becomes $$q_{c2} = Q_{\text{sphere}} - q'_2 = \bigl(-5\times10^{-9}\bigr) - \bigl(-3.33\times10^{-9}\bigr) = -1.67\times10^{-9}\,\text{C}.$$

The potential of the sphere is therefore $$V_{\text{final}} = k\dfrac{q_{c2}}{R} = 9\times10^9 \times\dfrac{-1.67\times10^{-9}}{0.10} \approx -150\;\text{V}.$$

This is neither 300 V nor positive. Hence Option D is incorrect.

Result
Correct statements: Option A, Option B, Option C.

Voltage sensitivity $$S_V$$ of a moving-coil galvanometer is defined as the deflection produced per unit applied voltage.
If the torsional constant of the suspension spring is $$k$$, then

Current sensitivity $$S_I$$ is $$\displaystyle S_I = \frac{NAB}{k} \quad -(1)$$
where
$$N$$ = number of turns, $$A$$ = area of each turn, $$B$$ = magnetic field.

Voltage sensitivity is obtained by dividing current sensitivity by the total resistance $$R$$ of the coil:

$$\displaystyle S_V = \frac{S_I}{R} = \frac{NAB}{kR} \quad -(2)$$

Because both coils have the same spring, the torsional constant $$k$$ is identical for them and cancels in the ratio.

For coil $$M_1$$:
$$N_1 = 15$$, $$A_1 = 3.6 \times 10^{-3}\,\text{m}^2$$, $$B_1 = 0.25\,\text{T}$$, $$R_1 = 5\,\Omega$$

For coil $$M_2$$:
$$N_2 = 21$$, $$A_2 = 1.8 \times 10^{-3}\,\text{m}^2$$, $$B_2 = 0.50\,\text{T}$$, $$R_2 = 7\,\Omega$$

Using $$-(2)$$ for each coil and taking the ratio:

$$\frac{S_{V1}}{S_{V2}} = \frac{\dfrac{N_1A_1B_1}{kR_1}}{\dfrac{N_2A_2B_2}{kR_2}} = \frac{N_1A_1B_1R_2}{N_2A_2B_2R_1}$$

Substituting the given values:

$$N_1A_1B_1R_2 = 15 \times 3.6\times10^{-3} \times 0.25 \times 7$$
$$= 15 \times 0.0036 \times 0.25 \times 7$$
$$= 0.0945$$

$$N_2A_2B_2R_1 = 21 \times 1.8\times10^{-3} \times 0.50 \times 5$$
$$= 21 \times 0.0018 \times 0.50 \times 5$$
$$= 0.0945$$

Therefore

$$\frac{S_{V1}}{S_{V2}} = \frac{0.0945}{0.0945} = 1$$

Thus, the voltage sensitivities of $$M_1$$ and $$M_2$$ are equal, giving the ratio

$$S_{V1} : S_{V2} = 1 : 1$$

Option A is correct.

When an electron is projected perpendicular to a uniform magnetic field $$B$$, it moves in a circular path due to the magnetic force acting as the centripetal force. The force balance equation is given by:

$$ e v B = \frac{m v^2}{r} $$

where $$e$$ is the charge of the electron, $$v$$ is its speed, $$m$$ is its mass, and $$r$$ is the radius of the circular path. Simplifying this equation:

$$ e B = \frac{m v}{r} $$

Rearranging, we get:

$$ m v = e B r \quad \text{(1)} $$

Bohr's quantization condition states that the angular momentum is quantized and given by:

$$ m v r = n \frac{h}{2\pi} \quad \text{(2)} $$

where $$n$$ is the principal quantum number and $$h$$ is Planck's constant. The first excited state corresponds to $$n = 2$$ (since the ground state is $$n = 1$$). Substituting $$n = 2$$ into equation (2):

$$ m v r = 2 \times \frac{h}{2\pi} = \frac{h}{\pi} \quad \text{(3)} $$

We now have two equations: equation (1) $$m v = e B r$$ and equation (3) $$m v r = \frac{h}{\pi}$$. Substituting equation (1) into equation (3):

$$ (e B r) \times r = \frac{h}{\pi} $$

Simplifying:

$$ e B r^2 = \frac{h}{\pi} $$

Solving for $$r^2$$:

$$ r^2 = \frac{h}{\pi e B} $$

Taking the square root of both sides:

$$ r = \sqrt{\frac{h}{\pi e B}} $$

This is the radius of the electronic orbit in the first excited state. Comparing with the given options, this expression matches option A, $$\sqrt{\frac{h}{\pi eeB}}$$, when interpreted as $$\sqrt{\frac{h}{\pi e B}}$$ (considering "ee" as a typographical error for "e").

Thus, the correct option is A.

For an electromagnetic wave propagating in the +x direction:

- The electric and magnetic fields must be perpendicular to the direction of propagation (transverse wave)

- $$\vec{E}$$, $$\vec{B}$$, and the propagation direction must form a right-handed coordinate system

Since the wave propagates along +x, neither $$\vec{E}$$ nor $$\vec{B}$$ can have an x-component.

This eliminates Options 1 (has $$E_x$$), 3 (has $$B_y$$ but we need to check), and 4 (has $$B_x$$).

For $$\hat{E} \times \hat{B} = \hat{k}_{propagation} = \hat{x}$$:

$$\hat{y} \times \hat{z} = \hat{x}$$ ✓

So $$E_y, B_z$$ is consistent with propagation along +x.

Check Option 3: $$E_z, B_y$$: $$\hat{z} \times \hat{y} = -\hat{x}$$ (propagation in -x direction). Incorrect.

The correct answer is Option 2: $$E_y, B_z$$.

For a circular coil of radius $$R$$ carrying current $$I$$, the standard results for magnetic field are:

• At the centre of the coil (on its plane) $$B_1 = \frac{\mu_0 I}{2R} \quad -(1)$$

• On the axis of the coil at a distance $$x$$ from the centre $$B_2 = \frac{\mu_0 I R^{2}}{2\left(R^{2}+x^{2}\right)^{3/2}} \quad -(2)$$

To obtain the required ratio, divide $$(2)$$ by $$(1)$$:

$$\frac{B_2}{B_1} = \frac{\mu_0 I R^{2}}{2\left(R^{2}+x^{2}\right)^{3/2}} \times \frac{2R}{\mu_0 I} = \frac{R^{3}}{\left(R^{2}+x^{2}\right)^{3/2}} \quad -(3)$$

Rewrite $$(3)$$ more compactly:

$$\frac{B_2}{B_1} = \left(\frac{R^{2}}{R^{2}+x^{2}}\right)^{3/2} \quad -(4)$$

The problem specifies $$x : R = 3 : 4$$, so choose $$x = 3k$$ and $$R = 4k$$ for some constant $$k$$.

Substitute these into $$(4)$$:

$$\frac{B_2}{B_1} = \left( \frac{(4k)^{2}}{(4k)^{2} + (3k)^{2}} \right)^{3/2} = \left( \frac{16k^{2}}{16k^{2} + 9k^{2}} \right)^{3/2} = \left( \frac{16}{25} \right)^{3/2} \quad -(5)$$

Evaluate the power:

$$\left( \frac{16}{25} \right)^{3/2} = \frac{16^{3/2}}{25^{3/2}} = \frac{( \sqrt{16} )^{3}}{( \sqrt{25} )^{3}} = \frac{4^{3}}{5^{3}} = \frac{64}{125}$$

Hence, $$\displaystyle \frac{B_2}{B_1} = \frac{64}{125}$$, or in ratio form $$64 : 125$$.

Therefore, the correct option is Option C (64 : 125).

Assertion (A): An electron moves with constant velocity in a straight line in a uniform magnetic field.

The magnetic force on a moving charge is $$\vec{F} = q\vec{v} \times \vec{B}$$. If $$\vec{v} \parallel \vec{B}$$, then $$\vec{v} \times \vec{B} = 0$$ and the force is zero, so the electron moves in a straight line with constant velocity. TRUE.

Reason (R): The magnetic field is along the direction of velocity of the electron.

This is exactly the condition that makes the cross product zero. TRUE.

R correctly explains A: when $$\vec{B} \parallel \vec{v}$$, there is no magnetic force, so the electron travels in a straight line.

The correct answer is Option C: Both (A) and (R) are true and (R) is the correct explanation of (A).

When a charged particle of charge $$q$$ and linear momentum $$p$$ enters a uniform magnetic field $$\mathbf{B}$$ perpendicular to the field, the magnetic force $$|q|vB$$ provides the required centripetal force $$\dfrac{mv^{2}}{r}$$.

Substituting $$p = mv$$ and $$v = \dfrac{p}{m}$$, the radius (called radius of curvature) is obtained as
$$r = \dfrac{p}{|q|B}$$ $$-(1)$$

Case 1: Oxide ion $$O^{2-}$$ versus proton $$H^{+}$$
For $$O^{2-}: |q| = 2e \quad(\text{two electronic charges})$$
For $$H^{+}: |q| = e$$
Both ions are given to have the same momentum $$p$$ and the same field $$B$$, so using $$(1)$$:

$$r_{O^{2-}} = \dfrac{p}{2eB}, \qquad r_{H^{+}} = \dfrac{p}{eB} = 2\,r_{O^{2-}}$$

Thus $$r_{O^{2-}} \lt r_{H^{+}}$$, i.e. the oxide ion describes a circle of smaller radius (larger bending) than the proton. Consequently its curvature radius is smaller. Hence Assertion A is correct.

Case 2: Proton versus electron with the same linear momentum
For a proton $$|q_p| = e$$ and for an electron $$|q_e| = e$$. With equal momenta $$p$$, equation $$(1)$$ gives

$$r_p = \dfrac{p}{eB}, \qquad r_e = \dfrac{p}{eB}$$

The radii are identical; the proton does not form a path of smaller radius than the electron. Therefore Reason R is false.

Conclusion: Assertion A is true, Reason R is false. The correct choice is Option A.

The magnetic field at the centre of a square loop carrying current can be found by applying the Biot-Savart law to one side and then using symmetry. Given that I = 5 A, a = 1/√2 m and μ₀ = 4π×10⁻⁷ T m A⁻¹, consider a single straight segment of the loop. For a finite straight wire the field at a perpendicular distance d, subtending angles θ₁ and θ₂ at the observation point, is $$B_{\text{wire}} = \frac{\mu_0 I}{4\pi d}(\sin\theta_1 + \sin\theta_2).$$ In our square loop the distance from the centre to each side is d = a/2, and each side subtends equal angles θ₁ = θ₂ = 45° at the centre. Substituting these values gives $$B_1 = \frac{\mu_0 I}{4\pi\,(a/2)}\bigl(\sin45°+\sin45°\bigr) = \frac{\mu_0 I}{2\pi a}\times\frac{2}{\sqrt2} = \frac{\sqrt2\,\mu_0 I}{2\pi a}.$$ By symmetry all four sides contribute equally and their fields add in the same direction, so the total field is $$B = 4B_1 = 4\;\frac{\sqrt2\,\mu_0 I}{2\pi a} = \frac{2\sqrt2\,\mu_0 I}{\pi a}.$$ Substituting I = 5 A, a = 1/√2 m and μ₀ = 4π×10⁻⁷ T m A⁻¹ yields $$B = \frac{2\sqrt2\times4\pi\times10^{-7}\times5}{\pi\times(1/\sqrt2)}.$$ The π factors cancel, giving $$B = \frac{2\sqrt2\times4\times5\times10^{-7}}{1/\sqrt2} = 40\sqrt2\times10^{-7}\times\sqrt2 = 40\times2\times10^{-7} = 8\times10^{-6}\text{ T}.$$ Therefore, the magnetic field at the centre of the square loop is $$B = 8\times10^{-6}\text{ T},$$ and the required value of p is 8.

When a charged particle is fired perpendicular to a uniform magnetic field, it traces a circular path. The time required for the particle to return to its original location for the first time is exactly the time period of this circular motion.

The magnetic force provides the centripetal force: $$qvB = \frac{mv^{2}}{r}$$

This gives the radius $$r = \frac{mv}{qB}$$.

The time period is $$T = \frac{2\pi r}{v} = \frac{2\pi m}{qB}$$

The time period does not depend on the speed of the particle.

Substituting into the time period formula:

$$T = \frac{2 \times 3.14 \times 16 \times 10^{-6}}{1.6 \times 10^{-6} \times 6.28}$$

$$T = \frac{100.48 \times 10^{-6}}{10.048 \times 10^{-6}} = \frac{100.48}{10.048} = 10 \text{ s}$$

So, the answer is $$10$$.

We are told a proton moves undeflected in crossed electric and magnetic fields at $$v = 2 \times 10^5$$ m/s. When the electric field is switched off, the proton moves in a circle of radius $$r = 2$$ cm. We need to find the electric field.

Condition for undeflected motion in crossed fields

When a charged particle moves undeflected through crossed electric and magnetic fields, the electric force and magnetic force balance each other:

$$qE = qvB$$

$$E = vB \quad \ldots (1)$$

Find $$B$$ from the circular motion when $$E$$ is switched off

When the electric field is switched off, only the magnetic force acts, providing the centripetal force for circular motion:

$$qvB = \frac{mv^2}{r}$$

$$B = \frac{mv}{qr} \quad \ldots (2)$$

Substituting given values ($$m = 1.6 \times 10^{-27}$$ kg, $$v = 2 \times 10^5$$ m/s, $$q = 1.6 \times 10^{-19}$$ C, $$r = 2$$ cm $$= 0.02$$ m):

$$B = \frac{1.6 \times 10^{-27} \times 2 \times 10^5}{1.6 \times 10^{-19} \times 0.02}$$

$$B = \frac{3.2 \times 10^{-22}}{3.2 \times 10^{-21}} = \frac{1}{10} = 0.1 \text{ T}$$

Calculate the electric field using equation (1)

$$E = vB = 2 \times 10^5 \times 0.1 = 2 \times 10^4 \text{ N/C}$$

Therefore, $$x = 2$$.

The answer is 2.

The magnetic field inside a solenoid is given by:

$$ B = \mu_0 n I = \mu_0 \frac{N}{L} I $$

We are given that $$B = 2.9 \times 10^{-4}$$ T, $$N = 200$$, $$I = 0.29$$ A, and $$\mu_0 = 4\pi \times 10^{-7}$$ T·m/A.

We rearrange this formula to solve for $$L$$:

$$ L = \frac{\mu_0 N I}{B} = \frac{4\pi \times 10^{-7} \times 200 \times 0.29}{2.9 \times 10^{-4}} $$

It follows that

$$ L = \frac{4\pi \times 10^{-7} \times 58}{2.9 \times 10^{-4}} = \frac{4\pi \times 58 \times 10^{-7}}{2.9 \times 10^{-4}} $$

Simplifying further gives

$$ = \frac{4\pi \times 20 \times 10^{-7}}{10^{-4}} = 4\pi \times 20 \times 10^{-3} = 80\pi \times 10^{-3} \text{ m} $$

Finally,

$$ = 8\pi \text{ cm} $$

Therefore, the answer is 8.

The magnetic force $$F$$ on a straight conductor of length $$l$$ carrying current $$I$$ in a uniform magnetic field of magnitude $$B$$ is given by the formula

$$F = I\,l\,B\,\sin\theta$$,

where $$\theta$$ is the angle between the direction of current and the magnetic field.

Here the wire is placed perpendicular to the field, so $$\theta = 90^{\circ}$$ and $$\sin 90^{\circ} = 1$$. Thus,

$$F = I\,l\,B$$ $$-(1)$$

Convert the length from centimetres to metres:
$$l = 4.0\ \text{cm} = 4.0 \times 10^{-2}\ \text{m} = 0.04\ \text{m}$$

Substitute $$I = 8\ \text{A}$$, $$l = 0.04\ \text{m}$$ and $$B = 0.15\ \text{T}$$ into $$(1)$$:

$$F = 8 \times 0.04 \times 0.15$$

First multiply $$8 \times 0.04 = 0.32$$.
Then multiply $$0.32 \times 0.15 = 0.048$$.

So, $$F = 0.048\ \text{N}$$.

To express the force in milli-newtons (1 mN = $$10^{-3}$$ N):
$$0.048\ \text{N} = 0.048 \times 10^{3}\ \text{mN} = 48\ \text{mN}$$.

Therefore, the magnetic force acting on the wire is $$\mathbf{48\ mN}$$.

$$I_X = 5 \text{ A}$$

$$I_Y = 4 \text{ A}$$

Distance between wires X and Y is $$6 \text{ cm}$$

Point P is at a distance of $$4 \text{ cm}$$ to the right of wire Y.

The total distance of point P from wire X is:

$$ r_X = 6 \text{ cm} + 4 \text{ cm} = 10 \text{ cm} = 0.1 \text{ m} $$

The distance of point P from wire Y is:

$$ r_Y = 4 \text{ cm} = 0.04 \text{ m} $$

The magnetic field due to a long straight wire is given by the formula:

$$ B = \frac{\mu_0 I}{2\pi r} $$

Calculate the magnetic field at point P due to wire X ($$B_X$$):

$$ B_X = \frac{4\pi \times 10^{-7} \times 5}{2\pi \times 0.1} $$

$$ B_X = \frac{2 \times 10^{-7} \times 5}{0.1} $$

$$ B_X = 10 \times 10^{-6} \text{ T} = 1 \times 10^{-5} \text{ T} $$

By the right-hand grip rule, with current flowing upwards, the direction of $$B_X$$ at point P is into the plane of the paper.

Calculate the magnetic field at point P due to wire Y ($$B_Y$$):

$$ B_Y = \frac{4\pi \times 10^{-7} \times 4}{2\pi \times 0.04} $$

$$ B_Y = \frac{2 \times 10^{-7} \times 4}{0.04} $$

$$ B_Y = 2 \times 10^{-5} \text{ T} $$

By the right-hand grip rule, with current flowing downwards, the direction of $$B_Y$$ at point P is out of the plane of the paper.

Since the magnetic fields are in opposite directions, the resultant magnetic field ($$B_{net}$$) is the difference between their magnitudes:

$$ B_{net} = B_Y - B_X $$

$$ B_{net} = 2 \times 10^{-5} \text{ T} - 1 \times 10^{-5} \text{ T} $$

$$ B_{net} = 1 \times 10^{-5} \text{ T} $$

Comparing this with the given expression $$x \times 10^{-5} \text{ T}$$, we get the value of $$x$$:

$$ x = 1 $$

Find percentage error in spring constant $$k$$ given 2% error in time and 1% error in mass.

Relation between k, m, and T.

$$T = 2\pi\sqrt{\frac{m}{k}}$$, so $$k = \frac{4\pi^2 m}{T^2}$$.

Error propagation.

$$\frac{\Delta k}{k} = \frac{\Delta m}{m} + 2\frac{\Delta T}{T} = 1\% + 2(2\%) = 1\% + 4\% = 5\%$$

Note: we take absolute values of errors for maximum percentage error regardless of sign.

The correct answer is Option A: 5%.

A proton and a deuteron with the same kinetic energy enter a uniform magnetic field perpendicular to $$\vec{B}$$. We need the ratio of their radii of circular paths.

When a charged particle moves perpendicular to a magnetic field, it follows a circular path with radius:

$$ r = \frac{mv}{qB} $$

Since $$KE = \frac{1}{2}mv^2$$, we get $$mv = \sqrt{2m \cdot KE}$$. Therefore:

$$ r = \frac{\sqrt{2m \cdot KE}}{qB} $$

For the proton: mass = $$m_p$$, charge = $$e$$, so $$r_p = \frac{\sqrt{2m_p \cdot KE}}{eB}$$

For the deuteron: mass = $$2m_p$$ (approximately twice the proton mass), charge = $$e$$ (same charge as proton), so:

$$ r_d = \frac{\sqrt{2 \cdot 2m_p \cdot KE}}{eB} = \frac{\sqrt{2} \cdot \sqrt{2m_p \cdot KE}}{eB} = \sqrt{2} \cdot r_p $$

The ratio is:

$$ \frac{r_d}{r_p} = \sqrt{2} : 1 $$

The correct answer is Option (1): $$\sqrt{2} : 1$$.

{image}

Side of square loop,

$$a=15\ \text{cm}$$

Speed of loop,

$$v=2\ \text{cm s}^{-1}$$

Time taken by the complete loop to enter the magnetic field:

$$t=\frac{15}{2}=7.5\ \text{s}$$

Since,

$$10\ \text{s} > 7.5\ \text{s}$$

the complete loop is inside the magnetic field at t=10 s.

Therefore, magnetic flux linked with the loop does not change.

$$e=\frac{d\phi}{dt}$$

Since,

$$\frac{d\phi}{dt}=0$$

Hence,

$$\boxed{e=0}$$

Convert a galvanometer (resistance 50 $$\Omega$$, max current 5 mA) into a voltmeter reading up to 100 V.

A galvanometer measures current. To use it as a voltmeter, we connect a high resistance in series so that when the full voltage is applied, only the maximum safe current flows through the galvanometer.

By Ohm’s law, when $$V = 100$$ V is applied and maximum current $$I_g = 5$$ mA = $$5 \times 10^{-3}$$ A flows:

$$ R_{\text{total}} = \frac{V}{I_g} = \frac{100}{5 \times 10^{-3}} = \frac{100}{0.005} = 20000 \text{ } \Omega $$

Subtracting the galvanometer resistance of 50 $$\Omega$$ gives the required series resistance:

$$ R_{\text{series}} = R_{\text{total}} - R_g = 20000 - 50 = 19950 \text{ } \Omega $$

The correct answer is Option C: 19950 $$\Omega$$.

We need to find the ratio of the magnetic field at distances $$\frac{a}{2}$$ and $$2a$$ from the axis of a long straight wire of radius $$a$$ carrying a steady, uniformly distributed current $$I$$.

Recall that Ampere's law states: $$\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$$ where $$I_{enc}$$ is the current enclosed by the Amperian loop.

Inside the wire at a distance $$r$$ from the axis, the current is uniformly distributed over the cross-section of radius $$a$$. The enclosed current is proportional to the enclosed area: $$I_{enc} = I \cdot \frac{\pi r^2}{\pi a^2} = I \cdot \frac{r^2}{a^2}$$.

Applying Ampere's law with a circular Amperian loop of radius $$r$$ gives $$B \cdot 2\pi r = \mu_0 \cdot I \cdot \frac{r^2}{a^2}$$, so $$B_{inside} = \frac{\mu_0 I r}{2\pi a^2}$$. At $$r = \frac{a}{2}$$, this becomes $$B_1 = \frac{\mu_0 I \cdot \frac{a}{2}}{2\pi a^2} = \frac{\mu_0 I}{4\pi a}$$.

Outside the wire at a distance $$r$$ from the axis, the entire current $$I$$ is enclosed, so Ampere's law gives $$B \cdot 2\pi r = \mu_0 I$$ and hence $$B_{outside} = \frac{\mu_0 I}{2\pi r}$$. At $$r = 2a$$, this becomes $$B_2 = \frac{\mu_0 I}{2\pi \cdot 2a} = \frac{\mu_0 I}{4\pi a}$$.

The ratio of these fields is $$\frac{B_1}{B_2} = \frac{\frac{\mu_0 I}{4\pi a}}{\frac{\mu_0 I}{4\pi a}} = \frac{1}{1}$$, so $$B_1 : B_2 = 1 : 1$$.

The correct answer is Option (2): 1 : 1.

A proton moves with constant velocity, meaning the net force on it is zero.

(A) $$E = 0, B = 0$$: No force, proton moves at constant velocity. Possible.

(B) $$E = 0, B \neq 0$$: If the proton moves parallel to $$\vec{B}$$, the magnetic force $$q\vec{v} \times \vec{B} = 0$$. Possible.

(C) $$E \neq 0, B = 0$$: The electric force $$q\vec{E} \neq 0$$, so the proton would accelerate. Not possible.

(D) $$E \neq 0, B \neq 0$$: If $$q\vec{E} + q\vec{v} \times \vec{B} = 0$$, the electric and magnetic forces can cancel. Possible.

The correct options are (A), (B) and (D) only, which corresponds to Option (3).

Magnetic field $$B = 2 \times 10^{-3}$$ T along +Y. Rectangular loop 20 cm × 10 cm in Y-Z plane, current 5 A anticlockwise w.r.t. negative X-axis.

The area vector of the loop in Y-Z plane points along $$\pm\hat{x}$$. Since current is anticlockwise when viewed from negative X-axis, by the right-hand rule, the area vector points along $$-\hat{x}$$.

$$\vec{m} = I\vec{A} = 5 \times (20 \times 10 \times 10^{-4})(-\hat{x}) = 5 \times 0.02(-\hat{x}) = -0.1\hat{x}$$ A m$$^2$$.

Torque: $$\vec{\tau} = \vec{m} \times \vec{B} = (-0.1\hat{x}) \times (2 \times 10^{-3}\hat{y})$$

$$= -0.1 \times 2 \times 10^{-3}(\hat{x} \times \hat{y}) = -2 \times 10^{-4}\hat{z}$$

Magnitude: $$2 \times 10^{-4}$$ N m, direction: along negative Z-direction.

The correct answer is Option B: $$2 \times 10^{-4}$$ N m along negative Z-direction.

This question asks for the correct expressions for the electrostatic force and magnetic force acting on a charge $$q$$ moving with velocity $$\vec{v}$$.

Recall that the electrostatic force on a charge $$q$$ in an electric field $$\vec{E}$$ is given by:

$$\vec{F_1} = q\vec{E}$$

This force acts in the direction of $$\vec{E}$$ for a positive charge and opposite to $$\vec{E}$$ for a negative charge. Importantly, this force does not depend on the velocity of the charge.

The magnetic force on a charge $$q$$ moving with velocity $$\vec{v}$$ in a magnetic field $$\vec{B}$$ is given by:

$$\vec{F_2} = q(\vec{v} \times \vec{B})$$

Key features of this force:

- It involves the cross product of velocity and magnetic field, not a dot product.

- The order matters: it is $$\vec{v} \times \vec{B}$$, not $$\vec{B} \times \vec{v}$$ (they differ by a negative sign).

- This force is always perpendicular to both $$\vec{v}$$ and $$\vec{B}$$, so it does no work on the charge.

- If the charge is stationary ($$\vec{v} = 0$$), the magnetic force is zero.

Comparing with the given options shows that option (1) gives $$\vec{F_1} = q\vec{E}$$ and $$\vec{F_2} = q(\vec{V} \times \vec{B})$$, which matches both formulas correctly.

Option (2) incorrectly uses $$\vec{B}$$ for the electrostatic force.

Option (3) has the cross product in the wrong order ($$\vec{B} \times \vec{V}$$).

Option (4) incorrectly uses dot products instead of cross products.

The correct answer is Option (1): $$\vec{F_1} = q\vec{E}$$, $$\vec{F_2} = q(\vec{V} \times \vec{B})$$.

We need to find the mass ratio $$\frac{m_X}{m_Y}$$ of two particles with equal charges accelerated through the same potential difference, making circular paths of radii $$R_1$$ and $$R_2$$.

When a particle of mass m and charge q is accelerated through a potential $$V$$, its kinetic energy satisfies $$\frac{1}{2}mv^2 = qV \Rightarrow v = \sqrt{\frac{2qV}{m}}$$. In a magnetic field, the circular radius is given by $$R = \frac{mv}{qB} = \frac{m}{qB}\sqrt{\frac{2qV}{m}} = \frac{\sqrt{2mV/q}}{B} = \frac{1}{B}\sqrt{\frac{2mV}{q}}$$, so that $$R \propto \sqrt{m}$$ when q, V, and B are constant.

Hence, $$\frac{R_1}{R_2} = \sqrt{\frac{m_X}{m_Y}}$$ and thus $$\frac{m_X}{m_Y} = \left(\frac{R_1}{R_2}\right)^2$$.

Therefore, the answer is Option B.

The series circuit consists of a pure resistor $$R = 90\,\Omega$$ and a coil whose resistance is negligible (so it behaves as a pure inductor with inductive reactance $$X_L$$). The applied rms supply is $$V = 120\,\text{V}$$ at frequency $$f = 60\,\text{Hz}$$.

The voltmeter shows the rms voltage across the resistor to be $$V_R = 36\,\text{V}$$. For a pure resistor, voltage and current are in phase, so the rms current in the entire series circuit is

$$I = \frac{V_R}{R} = \frac{36}{90} = 0.40\,\text{A}$$

In an $$R$$-$$L$$ series combination, the resistor voltage $$V_R$$ is in phase with the current while the inductor voltage $$V_L$$ leads the current by $$90^{\circ}$$. Therefore, the supply voltage is the vector (phasor) sum of $$V_R$$ and $$V_L$$, giving the rms relation

$$V = \sqrt{V_R^{2} + V_L^{2}} \quad -(1)$$

Rearranging $$(1)$$ for $$V_L$$:

$$V_L = \sqrt{V^{2} - V_R^{2}} = \sqrt{(120)^{2} - (36)^{2}}$$

$$V_L = \sqrt{14400 - 1296} = \sqrt{13104} \approx 1.145 \times 10^{2}\,\text{V}$$

Thus, $$V_L \approx 114\,\text{V}$$.

The inductive reactance is then obtained from Ohm’s law for the inductor:

$$X_L = \frac{V_L}{I} = \frac{114}{0.40} \approx 285\,\Omega$$

For a pure inductor, $$X_L = 2\pi f L$$. Therefore, the inductance $$L$$ is

$$L = \frac{X_L}{2\pi f} = \frac{285}{2\pi \times 60}$$

$$L = \frac{285}{120\pi} \approx \frac{285}{376.99} \approx 0.756\,\text{H}$$

Rounding to two significant figures, $$L \approx 0.76\,\text{H}$$.

Answer: Option B (0.76 H)

Statement I: "When currents vary with time, Newton's third law is valid only if momentum carried by the electromagnetic field is taken into account."

This is true. For time-varying currents, the electromagnetic field itself carries momentum. The mechanical momentum of the charges alone is not conserved; one must include the electromagnetic field momentum (given by $$\frac{1}{c^2}\vec{E} \times \vec{B}$$ per unit volume) for Newton's third law (conservation of momentum) to hold. This is a fundamental result in classical electrodynamics.

Statement II: "Ampere's circuital law does not depend on Biot-Savart's law."

This is false. For steady (magnetostatic) currents, Ampere's circuital law can be derived from the Biot-Savart law. The Biot-Savart law gives the magnetic field due to a current element, and integrating it appropriately yields Ampere's law. They are mathematically equivalent descriptions of magnetostatics, so Ampere's law does depend on (and is derivable from) the Biot-Savart law.

The answer is Option B: Statement I is true but Statement II is false.

The electric field is $$\vec{E} = \hat{i}40\cos\omega(t - z/c)$$ NC⁻¹.

The wave propagates in the +z direction (from the argument $$t - z/c$$).

Direction of E: $$\hat{i}$$ (x-direction)

For EM wave: $$\vec{B} = \frac{\hat{k} \times \vec{E}}{c}$$ where $$\hat{k} = \hat{z}$$

$$\hat{z} \times \hat{i} = \hat{j}$$

$$\vec{B} = \hat{j}\frac{40}{c}\cos\omega(t - z/c)$$

The correct answer is Option 4.

The z-axis carries an infinitely long straight wire with current $$I$$ along $$+z$$. At a point with position vector $$\vec{r}=x\,\hat{i}+y\,\hat{j}$$ in the $$xy$$-plane its magnetic field is

$$\vec{B}=\frac{\mu_0 I}{2\pi r}\,\hat{\phi} ,\qquad r=\sqrt{x^{2}+y^{2}}$$

For current in $$+z$$, $$\hat{\phi}= -\sin\phi\,\hat{i}+\cos\phi\,\hat{j}$$, where $$\tan\phi=\dfrac{y}{x}$$. With $$x$$ variable and constant $$y=a$$ (our path), we have

$$\sin\phi=\frac{a}{r},\qquad \cos\phi=\frac{x}{r}$$

Hence

$$\vec{B}= \frac{\mu_0 I}{2\pi r}\Bigl(-\frac{a}{r}\,\hat{i} +\frac{x}{r}\,\hat{j}\Bigr) =\frac{\mu_0 I}{2\pi r^{2}}\bigl(-a\,\hat{i}+x\,\hat{j}\bigr)$$

The required line integral is taken along the straight segment from $$P_1(-\sqrt{3}a,\,a,\,0)$$ to $$P_2(a,\,a,\,0)$$. Along this segment $$y=a,\;z=0$$ and $$x$$ varies, so

$$d\vec{l}=dx\,\hat{i}$$

Therefore

$$\vec{B}\cdot d\vec{l}= \frac{\mu_0 I}{2\pi r^{2}}\bigl(-a\,\hat{i}+x\,\hat{j}\bigr)\!\cdot\!(dx\,\hat{i}) =-\frac{\mu_0 I\,a}{2\pi (x^{2}+a^{2})}\,dx$$

Integrate from $$x=-\sqrt{3}a$$ to $$x=a$$:

$$\int_{P_1}^{P_2}\vec{B}\cdot d\vec{l} =-\frac{\mu_0 I\,a}{2\pi}\int_{-\sqrt{3}a}^{a}\frac{dx}{x^{2}+a^{2}}$$

Use the standard integral $$\int\frac{dx}{x^{2}+a^{2}}=\frac{1}{a}\tan^{-1}\!\Bigl(\frac{x}{a}\Bigr)$$:

$$\int_{P_1}^{P_2}\vec{B}\cdot d\vec{l} =-\frac{\mu_0 I}{2\pi}\Bigl[\tan^{-1}\!\Bigl(\frac{x}{a}\Bigr)\Bigr]_{x=-\sqrt{3}a}^{x=a}$$

Evaluate the limits:

$$\tan^{-1}(1)=\frac{\pi}{4},\qquad \tan^{-1}(-\sqrt{3})=-\frac{\pi}{3}$$

Thus

$$\int_{P_1}^{P_2}\vec{B}\cdot d\vec{l} =-\frac{\mu_0 I}{2\pi}\!\Bigl(\frac{\pi}{4}-\bigl(-\frac{\pi}{3}\bigr)\Bigr) =-\frac{\mu_0 I}{2\pi}\!\Bigl(\frac{7\pi}{12}\Bigr) =-\frac{7\mu_0 I}{24}$$

The negative sign only indicates that $$\vec{B}$$ and $$d\vec{l}$$ are oppositely directed. The question asks for the magnitude, hence

$$\Bigl|\int_{P_1}^{P_2}\vec{B}\cdot d\vec{l}\Bigr| =\frac{7\mu_0 I}{24}$$

Option A which is: $$\dfrac{7\mu_0 I}{24}$$

A charge $$q = 4.0\;\mu\text{C} = 4.0 \times 10^{-6}$$ C moves with velocity $$\vec{v} = 4.0 \times 10^6\;\hat{j}$$ m/s in a magnetic field $$\vec{B} = 2\hat{k}$$ T. The magnetic force on a moving charge is given by $$\vec{F} = q(\vec{v} \times \vec{B}).$$

To compute the cross product $$\vec{v} \times \vec{B},$$ observe that $$\vec{v} \times \vec{B} = (4.0 \times 10^6\;\hat{j}) \times (2\hat{k}) = 4.0 \times 10^6 \times 2\;(\hat{j} \times \hat{k}).$$ Using the rule $$\hat{j} \times \hat{k} = \hat{i},$$ it follows that $$\vec{v} \times \vec{B} = 8.0 \times 10^6\;\hat{i}.$$

Multiplying by the charge yields $$\vec{F} = q(\vec{v} \times \vec{B}) = 4.0 \times 10^{-6} \times 8.0 \times 10^6\;\hat{i},$$ so $$\vec{F} = 32\;\hat{i}\text{ N}.$$

The answer is $$x = 32$$.

We need to find the work done in turning a current-carrying coil through 90° in a uniform magnetic field. First, recall that the magnetic moment of a coil is $$M = NIA$$, where $$N$$ is the number of turns, $$I$$ is the current, and $$A$$ is the area. The potential energy of a magnetic dipole in a field is $$U = -MB\cos\theta$$, and the work done is the change in potential energy, given by $$W = U_f - U_i$$.

Using the specified values, the magnetic moment is $$M = NIA = 100 \times 1 \times 10^{-3} \times 5 \times 10^{-3}$$, which simplifies to $$M = 100 \times 5 \times 10^{-6} = 5 \times 10^{-4}\,\text{A·m}^2$$.

The plane of the coil is initially perpendicular to the magnetic field, so the magnetic moment, which is normal to the coil’s plane, is parallel to the field. Therefore, $$\theta_i = 0°$$. After rotating through 90°, the angle becomes $$\theta_f = 90°$$.

Applying the expression for work, we have $$W = U_f - U_i = (-MB\cos 90°) - (-MB\cos 0°) = 0 - (-MB) = MB$$.

Substituting $$M = 5 \times 10^{-4}\,\text{A·m}^2$$ and $$B = 0.20\,\text{T}$$ yields $$W = 5 \times 10^{-4} \times 0.20 = 1 \times 10^{-4}\,\text{J} = 100\,\mu\text{J}$$.

The correct answer is 100 $$\mu$$J.

Find the torsional constant $$k = x \times 10^{-5}$$ N-m/rad of a galvanometer.

For a moving coil galvanometer, at equilibrium the magnetic torque equals the restoring torque:

$$ NBIA = k\theta $$

where $$N$$ = number of turns, $$B$$ = magnetic field, $$I$$ = current, $$A$$ = area of each turn, $$k$$ = torsional constant, $$\theta$$ = deflection angle.

$$N = 100$$, $$B = 0.01$$ T, $$I = 10$$ mA $$= 10 \times 10^{-3}$$ A, $$A = 2.0$$ cm$$^2$$ $$= 2.0 \times 10^{-4}$$ m$$^2$$, $$\theta = 0.05$$ rad.

$$ k = \frac{NBIA}{\theta} = \frac{100 \times 0.01 \times 10 \times 10^{-3} \times 2.0 \times 10^{-4}}{0.05} $$

Numerator: $$100 \times 0.01 = 1$$

$$1 \times 10 \times 10^{-3} = 10^{-2}$$

$$10^{-2} \times 2.0 \times 10^{-4} = 2.0 \times 10^{-6}$$

$$ k = \frac{2.0 \times 10^{-6}}{0.05} = \frac{2.0 \times 10^{-6}}{5 \times 10^{-2}} = 0.4 \times 10^{-4} = 4 \times 10^{-5} \text{ N-m/rad} $$

So $$x = 4$$.

The answer is 4.

A regular hexagon is formed by bending a wire of length $$4\pi$$ m. A current $$I = 4\pi\sqrt{3}$$ A flows through it, and we wish to find the magnetic field at the centre.

A regular hexagon has 6 sides, so its side length is $$a = \frac{4\pi}{6} = \frac{2\pi}{3}$$ m. The perpendicular distance from the centre to each side (the apothem) is $$d = \frac{a\sqrt{3}}{2} = \frac{2\pi}{3} \cdot \frac{\sqrt{3}}{2} = \frac{\pi\sqrt{3}}{3}$$ m.

Each side subtends an angle of $$60°$$ at the centre, giving a half-angle of $$30°$$. The magnetic field due to a finite straight conductor at perpendicular distance $$d$$ is given by $$B_1 = \frac{\mu_0 I}{4\pi d}(\sin\alpha_1 + \sin\alpha_2)$$, and since $$\alpha_1 = \alpha_2 = 30°$$, this becomes $$B_1 = \frac{\mu_0 I}{4\pi d}(2\sin 30°) = \frac{\mu_0 I}{4\pi d}$$.

The total magnetic field from all six sides is $$B = 6 \times \frac{\mu_0 I}{4\pi d} = \frac{6\mu_0 I}{4\pi d} = \frac{6 \times 4\pi \times 10^{-7} \times 4\pi\sqrt{3}}{4\pi \times \frac{\pi\sqrt{3}}{3}} = \frac{6 \times 4\pi\sqrt{3} \times 10^{-7}}{\frac{\pi\sqrt{3}}{3}} = \frac{6 \times 4\pi\sqrt{3} \times 3}{\pi\sqrt{3}} \times 10^{-7} = 72 \times 10^{-7}$$ T.

The value of $$x$$ is 72.

We need to find the number of turns $$m$$ in a solenoid, given its dimensions and the magnetic field it produces.

The magnetic field inside a long solenoid is given by $$B = \mu_0 n I = \mu_0 \frac{m}{L} I$$ where $$\mu_0 = 4\pi \times 10^{-7}$$ T m/A is the permeability of free space, $$n = m/L$$ is the number of turns per unit length, $$m$$ is the total number of turns, $$L$$ is the length of the solenoid, and $$I$$ is the current.

Substituting the given values $$B = 6.28 \times 10^{-3}$$ T, $$L = 0.5$$ m, and $$I = 5$$ A into this formula gives $$6.28 \times 10^{-3} = 4\pi \times 10^{-7} \times \frac{m}{0.5} \times 5$$.

This simplifies to $$6.28 \times 10^{-3} = 4\pi \times 10^{-7} \times 10m$$ and hence $$6.28 \times 10^{-3} = 4\pi \times 10^{-6} \times m$$. Using $$\pi \approx 3.14$$, we have $$6.28 \times 10^{-3} = 4 \times 3.14 \times 10^{-6} \times m = 12.56 \times 10^{-6} \times m$$. From this, $$m = \frac{6.28 \times 10^{-3}}{12.56 \times 10^{-6}} = \frac{6.28}{12.56} \times 10^{3} = 0.5 \times 10^{3} = 500$$.

The answer is 500.

A circular coil of $$N$$ turns and radius $$R$$ produces, at its centre, a magnetic field along the axis of the coil given by
$$B = \frac{\mu_0 N I}{2R} \quad -(1)$$
where $$I$$ is the current through the coil.

For coil $$P$$:
Number of turns, $$N_P = 100$$
Current, $$I_P = 1 \, \text{A}$$
Radius, $$R = \pi \, \text{cm} = \pi \times 10^{-2} \, \text{m}$$

Substituting in $$(1)$$,
$$B_P = \frac{\mu_0 N_P I_P}{2R}$$
$$= \frac{(4\pi \times 10^{-7}) (100)(1)}{2(\pi \times 10^{-2})}$$
$$= \frac{4\pi \times 10^{-5}}{2\pi \times 10^{-2}}$$
$$= 2 \times 10^{-3} \, \text{T}$$
$$= 2 \, \text{mT} \quad -(2)$$

For coil $$Q$$:
Number of turns, $$N_Q = 100$$ (same as $$P$$)
Current, $$I_Q = 2 \, \text{A}$$

Again using $$(1)$$,
$$B_Q = \frac{\mu_0 N_Q I_Q}{2R}$$
$$= \frac{(4\pi \times 10^{-7}) (100)(2)}{2(\pi \times 10^{-2})}$$
$$= \frac{8\pi \times 10^{-5}}{2\pi \times 10^{-2}}$$
$$= 4 \times 10^{-3} \, \text{T}$$
$$= 4 \, \text{mT} \quad -(3)$$

The planes of the two coils are mutually perpendicular, so the fields $$B_P$$ and $$B_Q$$ are perpendicular to each other. Hence, the resultant field at the common centre is obtained by vector addition using the Pythagoras theorem:

$$B_{\text{res}} = \sqrt{B_P^{\,2} + B_Q^{\,2}}$$
$$= \sqrt{(2 \, \text{mT})^{2} + (4 \, \text{mT})^{2}}$$
$$= \sqrt{4 + 16} \, \text{mT}$$
$$= \sqrt{20} \, \text{mT}$$

The question states $$B_{\text{res}} = \sqrt{x} \, \text{mT}$$. Comparing, $$x = 20$$.

Hence, the required value is 20.

Square loop in east-west plane, B in NE direction. B makes 45° with the plane of the loop.

$$\Phi = BA\cos 45° = 0.20 \times 0.01 \times \frac{1}{\sqrt{2}} = \frac{0.002}{\sqrt{2}}$$

$$\varepsilon = \frac{\Delta\Phi}{\Delta t} = \frac{0.002/\sqrt{2}}{1} = \frac{0.002}{\sqrt{2}} = \sqrt{2} \times 10^{-3}$$ V.

$$\sqrt{x} \times 10^{-3} = \sqrt{2} \times 10^{-3}$$. $$x = 2$$.

The answer is $$\boxed{2}$$.

Magnetic field at center of a square loop of side $$a$$ with current $$I$$:

Each side contributes $$B_1 = \frac{\mu_0 I}{4\pi d}(\sin\alpha_1 + \sin\alpha_2)$$ where $$d = a/2$$ and $$\alpha_1 = \alpha_2 = 45°$$.

$$B_1 = \frac{\mu_0 I}{4\pi (a/2)} \times 2\sin 45° = \frac{\mu_0 I}{2\pi a} \times \sqrt{2} = \frac{\sqrt{2}\mu_0 I}{2\pi a}$$

Total for 4 sides: $$B = 4B_1 = \frac{4\sqrt{2}\mu_0 I}{2\pi a} = \frac{2\sqrt{2}\mu_0 I}{\pi a}$$

With $$I = 5$$A, $$a = 1$$m:

$$B = \frac{2\sqrt{2} \times 4\pi \times 10^{-7} \times 5}{\pi \times 1} = 2\sqrt{2} \times 4 \times 5 \times 10^{-7} = 40\sqrt{2} \times 10^{-7}$$

So $$X = 40$$.

Therefore, the answer is $$\boxed{40}$$.

The force per unit length on a current-carrying conductor in a magnetic field is given by $$\frac{F}{L} = I \times B \times \sin\theta$$, where $$\theta$$ is the angle between the direction of current and the magnetic field. The horizontal component of Earth’s magnetic field is $$B_H = 3.5 \times 10^{-5} \text{ T}$$ directed towards geographic North, and the current is $$I = \sqrt{2} \text{ A}$$ flowing from South-East to North-West. The angle between the North-West direction and the North direction is $$45°$$, so the angle between the current direction (SE to NW) and $$B_H$$ (pointing North) is $$\theta = 45°$$.

Substituting into the formula gives

$$\frac{F}{L} = I \times B_H \times \sin 45°$$

$$\frac{F}{L} = \sqrt{2} \times 3.5 \times 10^{-5} \times \frac{1}{\sqrt{2}}$$

$$\frac{F}{L} = 3.5 \times 10^{-5} = 35 \times 10^{-6} \text{ N/m}$$

Therefore, the correct answer is $$35$$.

Two long parallel wires $$P$$ and $$Q$$ carry equal current $$I = 10$$ A, separated by distance $$d = 5$$ cm. We need to find the new force when the distance is halved and the currents are doubled. The magnetic force per unit length between two parallel wires is $$\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}$$ and for a length $$L = 10$$ cm $$= 0.1$$ m of wire $$P$$ this becomes $$F = \frac{\mu_0 I_1 I_2 L}{2\pi d}$$.

For the original configuration, substituting $$I_1 = I_2 = 10$$ A and $$d = 0.05$$ m gives $$F_1 = \frac{\mu_0 \times 10 \times 10 \times 0.1}{2\pi \times 0.05} = \frac{10\mu_0}{\pi \times 0.1} = \frac{100\mu_0}{\pi}.$$

In the new configuration where the distance is halved ($$d' = 2.5$$ cm) and currents are doubled ($$I' = 20$$ A), we have $$F_2 = \frac{\mu_0 \times 20 \times 20 \times 0.1}{2\pi \times 0.025} = \frac{40\mu_0}{\pi \times 0.05} = \frac{800\mu_0}{\pi}.$$

The ratio of the forces is $$\frac{F_2}{F_1} = \frac{(I')^2 / d'}{I^2 / d} = \frac{(20)^2 \times 0.05}{(10)^2 \times 0.025} = \frac{400 \times 0.05}{100 \times 0.025} = \frac{20}{2.5} = 8.$$ Alternatively, doubling each current increases the force by a factor of $$2 \times 2 = 4$$ and halving the distance increases it by a factor of $$2$$, giving a total factor of $$4 \times 2 = 8$$. Hence $$F_2 = 8F_1.$$

Answer: Option (1): $$\boxed{8F_1}$$.

We need to find the ratio of the magnetic field at the centre of a circular loop to the field at a point on the axis.

The magnetic field at the centre is given by $$B_{\text{centre}} = \frac{\mu_0 i}{2R}$$, and the field on the axis at a distance $$x$$ from the centre is $$B_{\text{axis}} = \frac{\mu_0 i R^2}{2(R^2 + x^2)^{3/2}}$$.

The ratio is $$\frac{B_{\text{centre}}}{B_{\text{axis}}} = \frac{(R^2 + x^2)^{3/2}}{R^3}$$, and at $$x = R$$ this becomes $$\frac{(2R^2)^{3/2}}{R^3} = \frac{2\sqrt{2}R^3}{R^3} = 2\sqrt{2}$$, so the ratio is $$2\sqrt{2}:1$$.

This matches Option C: $$2\sqrt{2}:1$$.

The magnetic field inside a solenoid is given by:

$$B = \mu_0 n I$$

where $$n$$ is the number of turns per unit length and $$I$$ is the current.

Given: $$n = 70$$ turns/cm $$= 7000$$ turns/m, $$I = 2.0$$ A, $$\mu_0 = 4\pi \times 10^{-7}$$ T m A$$^{-1}$$.

$$B = 4\pi \times 10^{-7} \times 7000 \times 2.0$$

$$= 4\pi \times 14000 \times 10^{-7}$$

$$= 56000\pi \times 10^{-7}$$

$$= 56\pi \times 10^{-4}$$

$$= 56 \times 3.14159 \times 10^{-4}$$

$$\approx 175.93 \times 10^{-4}$$

$$\approx 176 \times 10^{-4}$$ T

The correct answer is Option 2: $$176 \times 10^{-4}$$ T.

Find the magnetic intensity at the centre of a solenoid with 1200 turns, length 2 m, diameter 0.2 m, carrying a current of 2 A.

Calculate the number of turns per unit length.

$$n = \frac{N}{L} = \frac{1200}{2} = 600 \text{ turns/m}$$

Apply the formula for magnetic intensity (H).

The magnetic intensity (magnetic field strength H) inside a solenoid is:

$$H = nI = 600 \times 2 = 1200 \text{ A m}^{-1} = 1.2 \times 10^3 \text{ A m}^{-1}$$

Note: The diameter of the solenoid does not affect the magnetic intensity H inside it.

The correct answer is $$1.2 \times 10^3$$ A m$$^{-1}$$.

Statement I: If the number of turns N is doubled, current sensitivity = $$\frac{NBA}{k}$$. Doubling N doubles current sensitivity. This is true.

Statement II: Voltage sensitivity = Current sensitivity / Resistance = $$\frac{NBA}{kR}$$. When N doubles, resistance R also doubles (since $$R \propto N$$). So voltage sensitivity = $$\frac{2NBA}{k \cdot 2R} = \frac{NBA}{kR}$$, which remains the same. This is false.

The correct answer is Option 1: Statement I is true but Statement II is false.

Given: Cross-section area $$A = 2$$ cm$$^2$$ $$= 2 \times 10^{-4}$$ m$$^2$$, length $$L = 40$$ cm $$= 0.4$$ m, $$N = 400$$ turns, $$I = 0.4$$ A, total magnetic flux $$\Phi = 4\pi \times 10^{-6}$$ Wb.

The magnetic flux inside the solenoid is:

$$ \Phi = B \times A = \mu_r \mu_0 n I \times A $$

where $$n = N/L$$ is the number of turns per unit length.

$$ n = \frac{400}{0.4} = 1000 \text{ turns/m} $$

$$ \Phi = \mu_r \times 4\pi \times 10^{-7} \times 1000 \times 0.4 \times 2 \times 10^{-4} $$

$$ \Phi = \mu_r \times 4\pi \times 10^{-7} \times 400 \times 2 \times 10^{-4} $$

$$ \Phi = \mu_r \times 4\pi \times 10^{-7} \times 8 \times 10^{-2} $$

$$ \Phi = \mu_r \times 32\pi \times 10^{-9} = \mu_r \times 3.2\pi \times 10^{-8} $$

Setting this equal to the given flux:

$$ \mu_r \times 3.2\pi \times 10^{-8} = 4\pi \times 10^{-6} $$

$$ \mu_r = \frac{4 \times 10^{-6}}{3.2 \times 10^{-8}} = \frac{4}{0.032} = 125 $$

Inside a solenoid, the magnetic field is uniform and directed along the axis.

When an electron moves with constant velocity along the axis of the solenoid, its velocity $$\vec{v}$$ is parallel to the magnetic field $$\vec{B}$$.

The magnetic force on the electron is:

$$\vec{F} = q(\vec{v} \times \vec{B})$$

Since $$\vec{v} \parallel \vec{B}$$, the cross product $$\vec{v} \times \vec{B} = 0$$.

Therefore:

(B) The electron will not experience magnetic force — True

(C) The electron will continue to move along the axis — True (no force to change its motion)

(A) Force along axis — False (no force at all)

(D) Accelerated along axis — False

(E) Parabolic path — False

The correct answer is B and C only.

In a moving coil galvanometer, a fixed core made of non-magnetic metallic material (such as soft iron) serves an important purpose.

When the coil rotates in the magnetic field, the metallic core has eddy currents induced in it. These eddy currents produce a damping effect (electromagnetic damping) that opposes the motion of the coil, bringing it to rest quickly without prolonged oscillation. This is known as "dead-beat" behavior.

This quick damping allows the galvanometer needle to settle rapidly to the correct reading, rather than oscillating back and forth around the equilibrium position.

The answer is Option B: to bring the coil to rest quickly.

We need to find the area of each turn of a moving coil galvanometer.

Deflection $$\theta = 0.05$$ rad, current $$I = 10$$ mA = 0.01 A, torsional constant $$C = 4.0 \times 10^{-5}$$ N m/rad, magnetic field $$B = 0.01$$ T, number of turns $$N = 200$$.

We start by noting that

In a moving coil galvanometer, the magnetic torque on the coil ($$\tau_m = NBIA$$) is balanced by the restoring torque of the suspension wire ($$\tau_r = C\theta$$):

$$ NBIA = C\theta $$

where $$A$$ is the area of each turn.

Next,

$$ A = \frac{C\theta}{NBI} $$

$$ A = \frac{4.0 \times 10^{-5} \times 0.05}{200 \times 0.01 \times 0.01} $$

$$ A = \frac{2.0 \times 10^{-6}}{0.02} = 1.0 \times 10^{-4}\;\text{m}^2 $$

Converting to cm$$^2$$: $$1.0 \times 10^{-4}$$ m$$^2$$ = $$1.0$$ cm$$^2$$.

The area of each turn is 1.0 cm$$^2$$.

The correct answer is Option 2: 1.0.

We need to find the condition for equal magnetic moments of two circular coils A and B.

Magnetic moment of a coil:

$$M = NIA = NI\pi r^2$$

For coil A: $$M_A = N_A I_A \pi r_A^2 = N_A I_A \pi (0.1)^2 = 0.01\pi N_A I_A$$

For coil B: $$M_B = N_B I_B \pi r_B^2 = N_B I_B \pi (0.2)^2 = 0.04\pi N_B I_B$$

Setting $$M_A = M_B$$:

$$0.01\pi N_A I_A = 0.04\pi N_B I_B$$

$$N_A I_A = 4 N_B I_B$$

The correct answer is Option 3: $$N_A I_A = 4N_B I_B$$.

We are given that a charged particle moving in a magnetic field $$\vec{B}$$ has velocity components both along $$\vec{B}$$ and perpendicular to $$\vec{B}$$.

Decompose the velocity. Let the velocity be split as $$\vec{v} = v_{\parallel}\hat{B} + v_{\perp}\hat{n}$$, where $$v_{\parallel}$$ is along $$\vec{B}$$ and $$v_{\perp}$$ is perpendicular to $$\vec{B}$$.

Effect of the perpendicular component. The magnetic force $$\vec{F} = q(\vec{v} \times \vec{B})$$ acts only on $$v_{\perp}$$. Since this force is always perpendicular to $$v_{\perp}$$, it causes the particle to move in a circular path in the plane perpendicular to $$\vec{B}$$.

Effect of the parallel component. Since $$\vec{v}_{\parallel} \times \vec{B} = 0$$, there is no magnetic force along $$\vec{B}$$. The particle moves with constant velocity $$v_{\parallel}$$ along the field direction.

Combined motion. The superposition of circular motion (perpendicular to $$\vec{B}$$) and uniform linear motion (along $$\vec{B}$$) gives a helical (spiral) path with its axis along the direction of $$\vec{B}$$.

The correct answer is Option B: helical path with the axis along magnetic field B.

Let the long straight wire coincide with the $$x$$-axis and carry a steady current $$I = 10 \, \text{A}$$ in the $$+\hat{x}$$-direction.
Hence the magnetic field produced by the wire at a point at distance $$r$$ from the wire (in the $$y$$-direction) is

$$B = \frac{\mu_0 I}{2\pi r}$$
and it is perpendicular to the $$xy$$-plane (along $$\pm \hat{z}$$).
Therefore the magnetic flux through a rectangular loop lying in the $$xy$$-plane depends only on its distance from the wire.

The rectangle has
length (side along $$y$$) : $$l = 4 \, \text{cm} = 0.04 \, \text{m}$$
width (side parallel to the wire, along $$x$$) : $$w = 2 \, \text{cm} = 0.02 \, \text{m}$$.

When the nearer edge of the loop is at a distance $$d = 4 \, \text{cm} = 0.04 \, \text{m}$$ from the wire, the farther edge is at $$d + l = 0.04 + 0.04 = 0.08 \, \text{m}$$.

The magnetic flux through the loop is

$$\Phi = \int_{x = 0}^{w} \int_{y = d}^{d+l} B(y)\, dy \, dx = w \frac{\mu_0 I}{2\pi}\int_{d}^{d+l}\frac{dy}{y} = w \frac{\mu_0 I}{2\pi}\ln\!\left(\frac{d+l}{d}\right).$$

The loop is pulled away from the wire with constant velocity
$$\vec{v}=v\left(\frac{\sqrt{3}}{2}\hat{x}+\frac{1}{2}\hat{y}\right).$$
Only the $$y$$-component changes the distance $$d$$, so

$$\frac{dd}{dt}=v_y=\frac{v}{2}.$$

Induced emf (magnitude) is

$$\mathcal{E}= \left|\frac{d\Phi}{dt}\right| = w\frac{\mu_0 I}{2\pi}\left|\frac{-l}{d(d+l)}\right|\frac{dd}{dt} = w\frac{\mu_0 I l}{2\pi d(d+l)}\frac{v}{2}.$$

Using Ohm’s law, $$\mathcal{E}=IR_{\text{loop}}$$, where
induced current $$I_{\text{loop}} = 10\,\mu\text{A}=1.0\times10^{-5}\,\text{A}$$ and
resistance $$R_{\text{loop}} = 0.1\,\Omega$$.

Hence
$$\mathcal{E}=I_{\text{loop}}R_{\text{loop}} =(1.0\times10^{-5})(0.1) =1.0\times10^{-6}\,\text{V}.$$

Equating the two expressions for $$\mathcal{E}$$:

$$1.0\times10^{-6} = w\frac{\mu_0 I l}{2\pi d(d+l)}\frac{v}{2}.$$

Insert the numerical values:
$$w = 0.02,\; l = 0.04,\; d = 0.04,\; I = 10,\; \mu_0 = 4\pi\times10^{-7}.$$

$$1.0\times10^{-6} = 0.02\frac{(4\pi\times10^{-7})(10)(0.04)}{2\pi(0.04)(0.08)}\frac{v}{2}.$$

Simplify:

$$1.0\times10^{-6} = 0.02\frac{(4\times10^{-7})(10)(0.04)}{2(0.04)(0.08)}\frac{v}{2}$$

$$1.0\times10^{-6} = \frac{0.02\times4\times10^{-7}\times10\times0.04}{2\times0.04\times0.08}\,\frac{v}{2}$$

Evaluate the constant factor:
Numerator $$=0.02\times4\times10^{-7}\times10\times0.04=3.2\times10^{-9}$$
Denominator $$=2\times0.04\times0.08=6.4\times10^{-3}$$

Thus
$$1.0\times10^{-6} =\left(\frac{3.2\times10^{-9}}{6.4\times10^{-3}}\right)\frac{v}{2} =5.0\times10^{-7}\,\frac{v}{2}.$$

Hence $$1.0\times10^{-6}=2.5\times10^{-7}v$$
so $$v = \frac{1.0\times10^{-6}}{2.5\times10^{-7}} = 4 \, \text{m\,s}^{-1}.$$

Therefore, the speed of the loop is
4 m s−1.

We have a charge $$q = 2 \mu C = 2 \times 10^{-6}$$ C accelerated through a potential difference $$V = 100$$ V, entering a magnetic field $$B = 4$$ mT $$= 4 \times 10^{-3}$$ T where it traces a semicircle of radius $$r = 3$$ cm $$= 0.03$$ m.

The kinetic energy gained by the charge after acceleration is

Now, the radius of circular motion in a magnetic field gives us

So $$mv = qBr$$. Substituting into the kinetic energy equation,

Solving for mass,

Hence, the mass of the charged particle is $$144 \times 10^{-18}$$ kg. So, the answer is $$144$$.

Magnetic force on a current carrying conductor is given by:

F=BILsin⁡θ

where
$$B=0.75T$$
$$I=2A$$
L= length of the conductor
θ=angle between current direction and magnetic field.

The magnetic field is parallel to the current in the 13 cm side.
Hence, the magnetic field is along the hypotenuse of the right triangle.

We need the force on the 5 cm side.

In the $$5\text{-}12\text{-}13$$ right triangle, the angle between the 5 cm side and the 13 cm side is obtained using:

$$\sin\theta=\frac{12}{13}$$

Length of the 5 cm side:

$$L=5cm=\frac{5}{100}m$$

Now,

$$F=0.75\times2\times\frac{5}{100}\times\frac{12}{13}$$

$$F=\frac{90}{1300}$$

$$F=\frac{9}{130}N$$

Given,

$$F=\frac{x}{130}N$$

Therefore,

$$x=9$$

We need to find the magnetic field at the centre O of a semicircular arc carrying a current of 14 A with radius 2.2 cm.

We begin with the Biot-Savart law, which gives the magnetic field at the centre of a circular arc subtending an angle $$\theta$$ (in radians) at the centre: $$B = \frac{\mu_0 I \theta}{4\pi r}$$ where $$\mu_0 = 4\pi \times 10^{-7}$$ T m/A is the permeability of free space, $$I$$ is the current, and $$r$$ is the radius.

For a semicircular arc, $$\theta = \pi$$ radians. Substituting into the formula yields $$B = \frac{\mu_0 I \pi}{4\pi r} = \frac{\mu_0 I}{4r}$$.

Substituting the given values $$I = 14$$ A, $$r = 2.2$$ cm $$= 0.022$$ m, and $$\mu_0 = 4\pi \times 10^{-7}$$ T m/A into this expression gives $$B = \frac{4\pi \times 10^{-7} \times 14}{4 \times 0.022}$$.

To simplify, the numerator is $$4\pi \times 10^{-7} \times 14 = 56\pi \times 10^{-7}$$ and the denominator is $$4 \times 0.022 = 0.088$$. This gives $$B = \frac{56\pi \times 10^{-7}}{0.088} = \frac{56\pi}{0.088} \times 10^{-7}$$.

Next, $$\frac{56}{0.088} = \frac{56000}{88} = \frac{7000}{11} \approx 636.36$$ so $$B \approx 636.36 \times \pi \times 10^{-7} \approx 636.36 \times 3.14 \times 10^{-7}$$ and hence $$B \approx 1998.2 \times 10^{-7} = 2.0 \times 10^{-4} \text{ T}$$.

Converting to the required units confirms that $$B = 2.0 \times 10^{-4}$$ T = $$2 \times 10^{-4}$$ T. Therefore, the magnetic field at the centre of the semicircular arc is 2 $$\times 10^{-4}$$ T.

For a circular arc, the field at the center is $$B = \frac{\mu_0 I}{4\pi R} \theta$$, where $$\theta$$ is the angle in radians. A semicircle ($$\theta = \pi$$) produces $$B = \frac{\mu_0 I}{4R}$$

For the straight parts of the wire, the point $$O$$ lies exactly on the line of the conductor. This means the angle ($$\theta$$) between the current element $$d\vec{l}$$ and the position vector $$\vec{r}$$ is $$0^\circ$$ or $$180^\circ$$. Since $$\sin(0^\circ) = \sin(180^\circ) = 0$$, the magnetic field contribution from both straight sections is zero.

$$B = \frac{(4\pi \times 10^{-7}) \times 3}{4 \times \left(\frac{\pi}{10}\right)}$$

$$B = \frac{12\pi \times 10^{-7}}{\frac{4\pi}{10}} = \frac{12\pi \times 10^{-7} \times 10}{4\pi}$$

$$B = 3 \times 10 \times 10^{-7} = 3 \times 10^{-6}\text{ T}$$ 

$$\mathbf{B = 3\text{ }\mu\text{T}}$$

We are given that the magnetic intensity at the centre of a long current-carrying solenoid is $$H = 1.6 \times 10^3$$ A/m, and the number of turns is $$n = 8$$ turns/cm.

Convert turns per cm to turns per metre: $$ n = 8 \text{ turns/cm} = 8 \times 100 = 800 \text{ turns/m} $$

Use the formula for magnetic intensity in a solenoid. For a long solenoid, the magnetic intensity inside is:

$$ H = nI $$

where $$n$$ is turns per unit length and $$I$$ is the current.

Solve for current: $$ I = \frac{H}{n} = \frac{1.6 \times 10^3}{800} = \frac{1600}{800} = 2 \text{ A} $$

The current flowing through the solenoid is 2 A.

The magnetic field at the centre of a current-carrying coil of radius $$r$$ is:

$$B_{\text{centre}} = \frac{\mu_0 I}{2r}$$

The magnetic field at a distance $$r$$ from the centre on the axis of the coil is:

$$B_{\text{axis}} = \frac{\mu_0 I r^2}{2(r^2 + r^2)^{3/2}} = \frac{\mu_0 I r^2}{2(2r^2)^{3/2}} = \frac{\mu_0 I r^2}{2 \cdot 2\sqrt{2} \cdot r^3} = \frac{\mu_0 I}{4\sqrt{2} \cdot r}$$

Now taking the ratio:

$$\frac{B_{\text{centre}}}{B_{\text{axis}}} = \frac{\frac{\mu_0 I}{2r}}{\frac{\mu_0 I}{4\sqrt{2} r}} = \frac{4\sqrt{2}}{2} = 2\sqrt{2} = \sqrt{8}$$

We are given that the ratio is $$\sqrt{x} : 1$$, so:

$$\sqrt{x} = 2\sqrt{2} = \sqrt{8}$$

$$x = 8$$

Hence, the answer is $$8$$.

We have two identical circular loops of radius $$R = 20\;\text{cm} = 0.2\;\text{m}$$, carrying current $$I = \sqrt{2}\;\text{A}$$, placed in perpendicular planes.

The magnetic field at the centre of a circular loop is

Each loop produces a magnetic field at the common centre:

Since the two loops are in perpendicular planes, their magnetic fields are perpendicular to each other. So the net magnetic field is

Hence, the net magnetic field at the centre is $$628 \times 10^{-8}$$ T. So, the answer is $$628$$.

We need to find the pitch of the helical path taken by a proton with kinetic energy 2.0 eV in a magnetic field of magnitude $$\frac{\pi}{2} \times 10^{-3}$$ T, where the angle between the velocity and the magnetic field is 60°.

The kinetic energy relation $$KE = \frac{1}{2}mv^2$$ allows us to solve for the proton’s speed as $$v = \sqrt{\frac{2 \cdot KE}{m}}$$; substituting $$KE = 2.0 \times 1.6 \times 10^{-19}\,\text{J}$$ and $$m = 1.6 \times 10^{-27}\,\text{kg}$$ gives $$v = \sqrt{\frac{2 \times 2 \times 1.6 \times 10^{-19}}{1.6 \times 10^{-27}}} = \sqrt{4 \times 10^{8}} = 2 \times 10^4\,\text{m/s}$$.

The component of this velocity parallel to the magnetic field is $$v_{\parallel} = v\cos60^\circ = 2 \times 10^4 \times \tfrac12 = 10^4\,\text{m/s}$$.

The proton’s circular motion in the plane perpendicular to the field has period $$T = \frac{2\pi m}{qB}$$, and with $$m = 1.6 \times 10^{-27}\,\text{kg}$$, $$q = 1.6 \times 10^{-19}\,\text{C}$$, and $$B = \frac{\pi}{2} \times 10^{-3}\,\text{T}$$ one finds $$T = \frac{2\pi \times 1.6 \times 10^{-27}}{1.6 \times 10^{-19} \times \frac{\pi}{2} \times 10^{-3}} = \frac{2\pi \times 1.6 \times 10^{-27}}{0.8\pi \times 10^{-22}} = \frac{2 \times 1.6}{0.8} \times 10^{-5} = 4 \times 10^{-5}\,\text{s}$$.

The pitch of the helix follows from multiplying the parallel speed by the period as $$\text{Pitch} = v_{\parallel} \times T = 10^4 \times 4 \times 10^{-5} = 0.4\,\text{m} = 40\,\text{cm}$$, so that the result is $$\boxed{40\ \text{cm}}$$.

The rod of length $$l = 0.25\ \text{m}$$ slides vertically with speed $$v$$ in a uniform magnetic field $$B_0 = 4\ \text{T}$$ perpendicular to the plane of the rails. The rails are perfectly conducting, so the only resistance in the circuit is that of the rod, $$R = 10\ \Omega$$.

When a conductor of length $$l$$ moves with speed $$v$$ perpendicular to a field $$B$$, the motional emf is
$$\mathcal{E} = B l v$$.

The induced current is therefore
$$I = \frac{\mathcal{E}}{R} = \frac{B l v}{R}$$.

The magnetic force on the rod (direction upward, opposing its motion) is
$$F_B = B I l = B \left( \frac{B l v}{R} \right) l = \frac{B^2 l^2}{R}\,v$$.

Let $$m = 20\ \text{g} = 0.02\ \text{kg}$$. Applying Newton’s second law, downward positive:

$$m\,\frac{dv}{dt} = m g - \frac{B^2 l^2}{R}\,v \quad -(1)$$

Define the constant
$$k = \frac{B^2 l^2}{m R}.$$
With $$B^2 l^2 = 4^2 (0.25)^2 = 1.0,\; m R = 0.02 \times 10 = 0.2$$ we get
$$k = \frac{1.0}{0.2} = 5\ \text{s}^{-1}.$$

Equation $$-(1)$$ is a first-order linear ODE. With the initial condition $$v(0)=0$$ its solution is

$$v(t) = v_t\bigl(1 - e^{-k t}\bigr) \quad -(2)$$

where the terminal velocity $$v_t$$ is obtained from $$dv/dt=0$$ in $$-(1):$$
$$m g = \frac{B^2 l^2}{R}\,v_t \;\Longrightarrow\; v_t = \frac{m g R}{B^2 l^2}.$$

Substituting the numerical values:
$$v_t = \frac{0.02 \times 10 \times 10}{1.0} = 2.0\ \text{m s}^{-1}.$$
Thus List-II value 5 ($$2.00$$) matches item (S).

Now evaluate the quantities at $$t = 0.2\ \text{s}$$.

From $$-(2)$$:
$$v(0.2) = 2.0\bigl(1 - e^{-5 \times 0.2}\bigr) = 2.0\bigl(1 - e^{-1}\bigr).$$
Given $$e^{-1} = 0.4$$,
$$v(0.2) = 2.0(1 - 0.4) = 2.0 \times 0.6 = 1.2\ \text{m s}^{-1}.$$

Induced emf (List-I item P)
$$\mathcal{E}(0.2) = B l v = 4 \times 0.25 \times 1.2 = 1.20\ \text{V}.$$
This equals List-II value 3.

Magnetic force (List-I item Q)
Current $$I = \mathcal{E}/R = 1.20/10 = 0.12\ \text{A}.$$
$$F_B = B I l = 4 \times 0.12 \times 0.25 = 0.12\ \text{N}.$$
This equals List-II value 4.

Power dissipated as heat (List-I item R)
$$P_H = I^2 R = (0.12)^2 \times 10 = 0.0144 \times 10 = 0.144\ \text{W} \approx 0.14\ \text{W}.$$
This equals List-II value 2.

Collecting the matches:
P $$\to$$ 3, Q $$\to$$ 4, R $$\to$$ 2, S $$\to$$ 5.

Hence the correct option is
Option D: P $$\to$$ 3, Q $$\to$$ 4, R $$\to$$ 2, S $$\to$$ 5.

We need to find how the magnetic field $$B$$ varies with distance $$r$$ from the centre of a long straight wire of radius $$R$$ carrying a uniformly distributed current $$I$$, for $$r < R$$.

By applying Ampere’s circuital law to a circular Amperian loop of radius $$r$$ inside the wire, one writes $$\oint \vec{B}\cdot d\vec{l} = \mu_0 I_{enc}$$. Because the current is uniformly distributed over the cross section, the enclosed current is $$I_{enc} = I \frac{\pi r^2}{\pi R^2} = I \frac{r^2}{R^2}$$. By symmetry, $$B$$ is tangential and constant along the loop, so $$B\,(2\pi r) = \mu_0 I \frac{r^2}{R^2}$$, which yields $$B = \frac{\mu_0 I r}{2\pi R^2}$$.

Hence, inside the wire, the magnetic field varies as $$B \propto r$$. The correct answer is Option A: $$B \propto r$$.

Two charged particles with the same kinetic energy pass through a uniform magnetic field perpendicular to their motion. The ratio of radii is $$6:5$$, and the ratio of masses is $$9:4$$. We need to find the ratio of charges.

The radius of the circular path in a magnetic field is given by $$r = \frac{mv}{qB}$$, and since kinetic energy $$KE = \frac{1}{2}mv^2$$, we have $$v = \sqrt{\frac{2KE}{m}}$$.

Substituting this expression for velocity into the formula for the radius yields $$r = \frac{m}{qB}\sqrt{\frac{2KE}{m}} = \frac{\sqrt{2mKE}}{qB}$$.

Because both particles have the same kinetic energy and experience the same magnetic field, the ratio of their radii simplifies to $$\frac{r_1}{r_2} = \frac{\sqrt{m_1}/q_1}{\sqrt{m_2}/q_2} = \frac{q_2\sqrt{m_1}}{q_1\sqrt{m_2}}$$.

Using the given ratios $$\frac{r_1}{r_2} = \frac{6}{5}$$ and $$\frac{m_1}{m_2} = \frac{9}{4}$$, we substitute to obtain $$\frac{6}{5} = \frac{q_2}{q_1} \times \sqrt{\frac{9}{4}} = \frac{q_2}{q_1} \times \frac{3}{2}$$, which leads to $$\frac{q_2}{q_1} = \frac{6}{5} \times \frac{2}{3} = \frac{4}{5}$$.

It follows that $$\frac{q_1}{q_2} = \frac{5}{4}$$.

Hence, the correct answer is Option B.

A charged particle moves in a uniform magnetic field $$\vec{B} = (2\hat{i} + 3\hat{j}) \text{ T}$$. Its acceleration is $$\vec{a} = (\alpha\hat{i} - 4\hat{j}) \text{ m/s}^2$$. We need to find $$\alpha$$.

The magnetic force on a charged particle is given by:

$$\vec{F} = q(\vec{v} \times \vec{B})$$

Since $$\vec{F} = m\vec{a}$$, the acceleration $$\vec{a}$$ is proportional to $$\vec{v} \times \vec{B}$$.

The magnetic force is always perpendicular to the magnetic field. Therefore, the acceleration must also be perpendicular to $$\vec{B}$$:

$$\vec{a} \cdot \vec{B} = 0$$

$$(\alpha\hat{i} - 4\hat{j}) \cdot (2\hat{i} + 3\hat{j}) = 0$$

$$2\alpha + (-4)(3) = 0$$

$$2\alpha - 12 = 0$$

$$\alpha = 6$$

The correct answer is Option B: $$6$$.

The magnetic field inside a long solenoid is given by:

$$B = \mu_0 n I$$

where $$n$$ is the number of turns per unit length and $$I$$ is the current.

Initially, the magnetic field is:

$$B = \mu_0 n I$$

When the current is doubled ($$I' = 2I$$) and the number of turns per cm is halved ($$n' = \frac{n}{2}$$):

$$B' = \mu_0 n' I' = \mu_0 \times \frac{n}{2} \times 2I = \mu_0 n I = B$$

The new magnetic field equals the original magnetic field.

Hence, the correct answer is Option D.

We need to find the magnetic field $$B$$ as a function of radial distance $$r$$ from the axis of an infinitely long hollow conducting cylinder of radius $$R$$ that carries a uniform current along its surface.

We apply Ampere's Law: $$\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}}$$.

For $$r < R$$: Choose a circular Amperian loop of radius $$r < R$$ centred on the axis. Since the current flows only on the surface at $$r = R$$, this loop encloses zero current. By Ampere's Law: $$B(2\pi r) = 0$$, so $$B = 0$$ for all $$r < R$$.

For $$r > R$$: Choose a circular Amperian loop of radius $$r > R$$. This loop encloses the entire surface current $$I$$. By Ampere's Law: $$B(2\pi r) = \mu_0 I$$, giving $$B = \dfrac{\mu_0 I}{2\pi r}$$, which decreases as $$\dfrac{1}{r}$$.

The correct graph shows $$B = 0$$ for $$r < R$$ and $$B \propto \dfrac{1}{r}$$ for $$r > R$$.

The answer is Option D.

We need to identify which factors increase the current sensitivity of a galvanometer.

Current sensitivity is the deflection per unit current:

$$\frac{\theta}{I} = \frac{nBA}{k}$$

where $$n$$ = number of turns, $$B$$ = magnetic field, $$A$$ = area of coil, and $$k$$ = torsional constant of the spring.

(A) Decreasing the number of turns $$n$$ → Sensitivity decreases (since $$\frac{\theta}{I} \propto n$$). Incorrect.

(B) Increasing the magnetic field $$B$$ → Sensitivity increases (since $$\frac{\theta}{I} \propto B$$). Correct.

(C) Decreasing the area of the coil $$A$$ → Sensitivity decreases (since $$\frac{\theta}{I} \propto A$$). Incorrect.

(D) Decreasing the torsional constant $$k$$ → Sensitivity increases (since $$\frac{\theta}{I} \propto \frac{1}{k}$$). Correct.

The correct combination is (B) and (D).

Hence, the correct answer is Option D: (B) and (D) only.

A cyclotron with magnetic field $$B = 1.0$$ T and radius $$R = 60$$ cm = 0.6 m accelerates protons. Find the kinetic energy in MeV.

Since for a charged particle in a cyclotron at the maximum radius the magnetic force equals the centripetal force, we have

$$qvB = \frac{mv^2}{R} \implies mv = qBR$$

From this relation one obtains

$$v = \frac{qBR}{m}$$

Substituting the expression for $$v$$ into the kinetic energy formula yields

$$KE = \frac{1}{2}mv^2 = \frac{1}{2}m\left(\frac{qBR}{m}\right)^2 = \frac{q^2B^2R^2}{2m}$$

Substituting $$q = e = 1.6 \times 10^{-19}$$ C, $$B = 1.0$$ T, $$R = 0.6$$ m, and $$m_p = 1.6 \times 10^{-27}$$ kg into this expression gives

$$KE = \frac{(1.6 \times 10^{-19})^2 \times (1.0)^2 \times (0.6)^2}{2 \times 1.6 \times 10^{-27}}$$

Numerator: $$(1.6)^2 \times 10^{-38} \times 0.36 = 2.56 \times 0.36 \times 10^{-38} = 0.9216 \times 10^{-38}$$

Denominator: $$3.2 \times 10^{-27}$$

Therefore,

$$KE = \frac{0.9216 \times 10^{-38}}{3.2 \times 10^{-27}} = 0.288 \times 10^{-11} = 2.88 \times 10^{-12} \text{ J}$$

Since $$1 \text{ eV} = 1.6 \times 10^{-19} \text{ J}$$ and $$1 \text{ MeV} = 1.6 \times 10^{-13} \text{ J}$$,

$$KE = \frac{2.88 \times 10^{-12}}{1.6 \times 10^{-13}} = 18 \text{ MeV}$$

Answer: Option B: 18

We need to find the ratio $$B_X : B_Y$$ for two coils carrying equal current. Coil $$X$$ has $$N_X = 200$$ turns and radius $$R_X = 20 \text{ cm}$$. Coil $$Y$$ has $$N_Y = 400$$ turns and radius $$R_Y = 20 \text{ cm}$$.

The magnetic field at the centre of a circular coil of $$N$$ turns, radius $$R$$, carrying current $$I$$ is:

$$B = \frac{\mu_0 N I}{2R}$$

For coil $$X$$:

$$B_X = \frac{\mu_0 N_X I}{2R_X} = \frac{\mu_0 \times 200 \times I}{2 \times 20}$$

For coil $$Y$$:

$$B_Y = \frac{\mu_0 N_Y I}{2R_Y} = \frac{\mu_0 \times 400 \times I}{2 \times 20}$$

Since both coils have the same radius ($$R_X = R_Y = 20 \text{ cm}$$) and carry the same current:

$$\frac{B_X}{B_Y} = \frac{N_X}{N_Y} = \frac{200}{400} = \frac{1}{2}$$

Therefore, $$B_X : B_Y = 1 : 2$$.

The correct answer is Option B: $$1 : 2$$.

The magnetic field at the center of a current-carrying circular loop of radius $$R$$ is $$B_1 = \frac{\mu_0 I}{2R}$$.

The magnetic field on the axis of the loop at a distance $$d$$ from the center is given by $$B = \frac{\mu_0 I R^2}{2(R^2 + d^2)^{3/2}}$$.

Here, $$d = \sqrt{3}R$$, so $$R^2 + d^2 = R^2 + 3R^2 = 4R^2$$, and $$(R^2 + d^2)^{3/2} = (4R^2)^{3/2} = 8R^3$$.

So $$B_2 = \frac{\mu_0 I R^2}{2 \times 8R^3} = \frac{\mu_0 I}{16R}$$.

Now, $$\frac{B_1}{B_2} = \frac{\mu_0 I / 2R}{\mu_0 I / 16R} = \frac{16R}{2R} = 8$$.

So $$\frac{B_1}{B_2} = 8:1$$.

Hence, the correct answer is Option C.

Two concentric circular loops of radii $$r_1 = 30 \text{ cm}$$ and $$r_2 = 50 \text{ cm}$$ in the X-Y plane carry current $$I = 7 \text{ A}$$. We need to find the net magnetic moment.

The magnetic moment of a circular loop is:

$$\vec{M} = I \cdot A \cdot \hat{n}$$

where $$A = \pi r^2$$ is the area and $$\hat{n}$$ is the unit normal determined by the direction of current (right-hand rule).

From the figure, the inner loop ($$r_1 = 0.3 \text{ m}$$) carries current in one direction and the outer loop ($$r_2 = 0.5 \text{ m}$$) carries current in the opposite direction.

For the inner loop (current gives $$+\hat{k}$$):

$$M_1 = 7 \times \pi \times (0.3)^2 = 7 \times 0.09\pi = 0.63\pi \text{ A m}^2$$

For the outer loop (current gives $$-\hat{k}$$):

$$M_2 = 7 \times \pi \times (0.5)^2 = 7 \times 0.25\pi = 1.75\pi \text{ A m}^2$$

$$\vec{M}_{net} = 0.63\pi \hat{k} - 1.75\pi \hat{k} = -1.12\pi \hat{k}$$

$$\vec{M}_{net} = -1.12 \times 3.14 \hat{k} \approx -3.52 \hat{k} \approx -\dfrac{7}{2} \hat{k} \text{ A m}^2$$

The correct answer is Option B: $$-\dfrac{7}{2} \hat{k} \text{ A m}^2$$.

Two parallel wires are separated by $$d = 0.20$$ m, each carrying current $$x$$ A. The force of attraction per meter is $$F/L = 2 \times 10^{-6}$$ N m$$^{-1}$$.

Write the formula for force per unit length between two parallel current-carrying wires.

$$\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}$$

Since both wires carry the same current $$x$$:

$$\frac{F}{L} = \frac{\mu_0 x^2}{2\pi d}$$

Substitute the known values.

$$2 \times 10^{-6} = \frac{4\pi \times 10^{-7} \times x^2}{2\pi \times 0.20}$$

$$2 \times 10^{-6} = \frac{2 \times 10^{-7} \times x^2}{0.20}$$

$$2 \times 10^{-6} = 10^{-6} \times x^2$$

Solve for $$x$$.

$$x^2 = 2$$

$$x = \sqrt{2} \approx 1.4 \text{ A}$$

The correct answer is Option C.

A velocity selector has $$\vec{E} = E\hat{k}$$ and $$\vec{B} = B\hat{j}$$ with $$B = 12 \text{ mT}$$. An electron has energy $$728 \text{ eV}$$ and moves along the positive x-axis. We need to find $$E$$.

The kinetic energy of the electron is $$KE = \dfrac{1}{2}mv^2 = 728 \text{ eV} = 728 \times 1.6 \times 10^{-19} \text{ J}$$, which simplifies to $$1164.8 \times 10^{-19} \text{ J} = 1.1648 \times 10^{-16} \text{ J}$$.

Substituting this value into the expression for velocity gives $$v^2 = \dfrac{2 \times KE}{m} = \dfrac{2 \times 1.1648 \times 10^{-16}}{9.1 \times 10^{-31}} = 2.56 \times 10^{14}$$, so that $$v = 1.6 \times 10^7 \text{ m s}^{-1}$$.

For the electron to pass undeflected through the selector, the electric force must balance the magnetic force, thus $$qE = qvB$$ and hence $$E = vB$$.

Substituting $$v = 1.6 \times 10^7 \text{ m s}^{-1}$$ and $$B = 12 \times 10^{-3}$$ yields $$E = 1.6 \times 10^7 \times 12 \times 10^{-3} = 19.2 \times 10^4 = 1.92 \times 10^5 \text{ V m}^{-1}$$, which is equal to $$192 \text{ kV m}^{-1}$$.

Therefore, the correct answer is Option A: $$192 \text{ kV m}^{-1}$$.

Assertion (A): In a uniform magnetic field, speed and energy remain the same for a moving charged particle. Reason (R): A moving charged particle experiences a magnetic force perpendicular to its direction of motion.

The magnetic force on a charged particle is $$\vec{F} = q(\vec{v} \times \vec{B})$$, which is always perpendicular to the velocity $$\vec{v}$$. Since the force is perpendicular to the velocity, the work done by the magnetic force is:

$$ W = \vec{F} \cdot \vec{v} \, dt = 0 $$

Since no work is done, kinetic energy $$\frac{1}{2}mv^2$$ remains constant and therefore the speed remains constant. Thus, the Assertion is true.

The cross product $$\vec{v} \times \vec{B}$$ is indeed perpendicular to $$\vec{v}$$, so the magnetic force is perpendicular to the direction of motion. Hence, the Reason is true.

It is precisely because the magnetic force is perpendicular to the velocity that no work is done on the particle, which directly leads to the speed and kinetic energy remaining constant. Therefore, the Reason is the correct explanation of the Assertion.

Therefore, the correct answer is Option A: Both Assertion and Reason are true and Reason is the correct explanation of Assertion.

A circular coil of 2 turns is unwound and rewound into 5 turns. We need to find $$\frac{B_2}{B_1}$$.

Relate the radii using the total wire length.

The total length of wire remains constant:

Write the magnetic field at the centre of each coil.

Calculate the ratio.

The correct answer is Option B: $$\dfrac{25}{4}$$.

The magnetic force between two parallel current-carrying wires is calculated using the formula derived from Ampere's Force Law: $$F = \frac{\mu_0 I_1 I_2 L}{2\pi d}$$

$$F = \frac{4\pi \times 10^{-7} \times 2 \times 3 \times 0.5}{2\pi \times 0.05}$$

$$F = \frac{6 \times 10^{-7}}{5 \times 10^{-2}} = 1.2 \times 10^{-5}\text{ N}$$

According to the magnetic force laws for parallel conductors, currents flowing in the same direction result in an attractive force. Consequently, wire $$Y$$ is pulled towards wire $$X$$.

We are given that the magnetic field at the centre of a circular coil of radius $$r$$ carrying current $$I$$ is $$B$$, and we need to find the magnetic field at a point on the axis at distance $$\frac{r}{2}$$ from the centre.

The magnetic field at the centre of a circular coil is $$ B_{centre} = \frac{\mu_0 I}{2r} = B $$ and on the axis at a distance $$x$$ from the centre it is $$ B_{axis} = \frac{\mu_0 I r^2}{2(r^2 + x^2)^{3/2}} $$. Substituting $$x = \frac{r}{2}$$ gives $$ B_{axis} = \frac{\mu_0 I r^2}{2\left(r^2 + \frac{r^2}{4}\right)^{3/2}} = \frac{\mu_0 I r^2}{2\left(\frac{5r^2}{4}\right)^{3/2}} $$. Since $$ \left(\frac{5r^2}{4}\right)^{3/2} = \frac{(5)^{3/2}}{(4)^{3/2}} \cdot r^3 = \frac{5\sqrt{5}}{8} \cdot r^3 $$, it follows that $$ B_{axis} = \frac{\mu_0 I r^2}{2 \cdot \frac{5\sqrt{5}}{8} \cdot r^3} = \frac{\mu_0 I}{2r} \cdot \frac{8}{5\sqrt{5}} = B \cdot \frac{8}{5\sqrt{5}} $$. Noting $$ \frac{8}{5\sqrt{5}} = \frac{2^3}{(\sqrt{5})^2 \cdot \sqrt{5}} = \frac{2^3}{(\sqrt{5})^3} = \left(\frac{2}{\sqrt{5}}\right)^3 $$, we conclude

$$ B_{axis} = \left(\frac{2}{\sqrt{5}}\right)^3 B $$

The correct answer is Option C.

Two long parallel conductors are 8 cm apart. The magnetic field at the midpoint is 300 $$\mu$$T. Find the equal current and direction.

The midpoint is at 4 cm = 0.04 m from each conductor.

Magnetic field due to each conductor at the midpoint:

$$B = \frac{\mu_0 I}{2\pi d} = \frac{4\pi \times 10^{-7} \times I}{2\pi \times 0.04} = \frac{2 \times 10^{-7} \times I}{0.04} = 5 \times 10^{-6} I \text{ T}$$

If currents are in the same direction: At the midpoint, the fields due to the two wires point in opposite directions (by right-hand rule). The net field = 0. This contradicts 300 $$\mu$$T.

If currents are in opposite directions: At the midpoint, the fields add up:

$$B_{\text{net}} = 2B = 2 \times 5 \times 10^{-6} I = 10^{-5} I$$

$$300 \times 10^{-6} = 10^{-5} I$$

$$I = \frac{300 \times 10^{-6}}{10^{-5}} = 30 \text{ A}$$

Hence, the correct answer is Option C: 30 A in the opposite direction.

Step 1: Magnetic field at point P

$$B_{\text{net}}=B_1-B_2$$

$$B_1=\frac{\mu_0\cdot4}{2\pi(0.04)},\quad$$

$$B_2=\frac{\mu_0\cdot2}{2\pi(0.06)}$$

$$B_{\text{net}}=\frac{\mu_0}{2\pi}\left(\frac{4}{0.04}-\frac{2}{0.06}\right)$$

= $$\frac{\mu_0}{2\pi}\left(100-\frac{100}{3}\right)$$

=$$\frac{\mu_0}{2\pi}\cdot\frac{200}{3}$$

Direction: $$-\hat{k}$$

$$\vec{B} = \frac{\mu_0}{2\pi}\cdot \frac{200}{3}(-\hat{k})$$

Step 2: Magnetic force on charge

$$\vec{F} = q(\vec{v} \times \vec{B})$$

= $$3\pi \left[(2\hat{i} + 3\hat{j}) \times \left(\frac{\mu_0}{2\pi}\cdot \frac{200}{3}(-\hat{k})\right)\right]=3π[(2i^+3j^​)×(2πμ0​​⋅\ \frac{\ 200}{3}​(−k^))]$$

= $$3\pi \cdot \frac{\mu_0}{2\pi} \cdot \frac{200}{3} \left[2\hat{j} - 3\hat{i}\right]$$

Step 3: Simplify
Using $$\mu_0 = 4\pi \times 10^{-7}$$4:

$$\vec{F} = (4\pi \times 10^{-7})(100)\left(-3\hat{i} + 2\hat{j}\right)$$

= $$4\pi \times 10^{-5} \left(-3\hat{i} + 2\hat{j}\right)$$

Compare with given:

$$\vec{F} = 4\pi \times 10^{-5}(-x\hat{i} + 2\hat{j})$$

⇒ x = 3

Final Answer:

3

Given: A proton, a deuteron, and an $$\alpha$$-particle enter a uniform magnetic field with the same kinetic energy, perpendicular to the field.

Recall the formula for the radius of circular motion in a magnetic field: When a charged particle moves in a magnetic field, the radius of its circular path is:

$$r = \frac{mv}{qB}$$

Express radius in terms of kinetic energy: Since $$KE = \frac{1}{2}mv^2$$, we get $$v = \sqrt{\frac{2 \cdot KE}{m}}$$, so $$mv = \sqrt{2m \cdot KE}$$.

Therefore:

$$r = \frac{\sqrt{2m \cdot KE}}{qB}$$

Since $$KE$$ and $$B$$ are the same for all three particles:

$$r \propto \frac{\sqrt{m}}{q}$$

Identify mass and charge of each particle: Let $$m_p$$ be the proton mass and $$e$$ be the elementary charge.

Proton: $$m = m_p$$, $$q = e$$

Deuteron: $$m = 2m_p$$, $$q = e$$

$$\alpha$$-particle: $$m = 4m_p$$, $$q = 2e$$

Calculate the ratio of radii: $$r_p : r_d : r_\alpha = \frac{\sqrt{m_p}}{e} : \frac{\sqrt{2m_p}}{e} : \frac{\sqrt{4m_p}}{2e}$$

$$= \frac{\sqrt{m_p}}{e} : \frac{\sqrt{2m_p}}{e} : \frac{2\sqrt{m_p}}{2e}$$

$$= \frac{\sqrt{m_p}}{e} : \frac{\sqrt{2m_p}}{e} : \frac{\sqrt{m_p}}{e}$$

Cancelling the common factor $$\frac{\sqrt{m_p}}{e}$$:

$$= 1 : \sqrt{2} : 1$$

The ratio of the radii of their circular paths is $$1 : \sqrt{2} : 1$$.

The correct answer is Option B.

We need to evaluate two statements about electric and magnetic forces.

Statement I: "The electric force changes the speed of the charged particle and hence changes its kinetic energy; whereas the magnetic force does not change the kinetic energy of the charged particle."

This is CORRECT. The electric force $$\vec{F} = q\vec{E}$$ can do work on a charged particle, thereby changing its speed and kinetic energy. The magnetic force $$\vec{F} = q\vec{v} \times \vec{B}$$ is always perpendicular to the velocity, so it does zero work. Hence, the magnetic force cannot change the speed or kinetic energy of a charged particle.

Statement II: "The electric force accelerates the positively charged particle perpendicular to the direction of electric field. The magnetic force accelerates the moving charged particle along the direction of magnetic field."

This is INCORRECT. Both parts are wrong:

- The electric force acts along the direction of the electric field (for positive charges), not perpendicular to it: $$\vec{F} = q\vec{E}$$.

- The magnetic force $$\vec{F} = q\vec{v} \times \vec{B}$$ is perpendicular to both $$\vec{v}$$ and $$\vec{B}$$, not along the direction of $$\vec{B}$$.

Therefore, Statement I is correct and Statement II is incorrect.

Hence, the correct answer is Option C: Statement I is correct but Statement II is incorrect.

When the magnetic flux linked with a closed conducting loop changes, an emf is induced according to Faraday’s law
$$\varepsilon = -\dfrac{d\Phi_B}{dt} = -A\dfrac{dB}{dt}$$

The area of the circuit is $$A = 1\;{\rm m^2}$$ and the magnetic field varies as $$B = B_0 + \beta t$$ with $$\beta = 0.04\;{\rm T\,s^{-1}}$$, so the rate of change of field is constant:
$$\dfrac{dB}{dt} = \beta = 0.04\;{\rm T\,s^{-1}}$$

Hence the magnitude of the induced emf is
$$\varepsilon = A\,\beta = 1 \times 0.04 = 0.04\;{\rm V}$$

Let $$q(t)$$ be the charge on the capacitor plates (positive on one plate) and $$i(t)=\dfrac{dq}{dt}$$ the current through the inductor. For an ideal LC loop the Kirchhoff equation is
$$L\dfrac{di}{dt} + \dfrac{q}{C} = \varepsilon$$

Substituting $$i=\dfrac{dq}{dt}$$ gives the second-order equation
$$L\dfrac{d^{2}q}{dt^{2}} + \dfrac{q}{C} = \varepsilon \qquad -(1)$$

The homogeneous part of $$(1)$$ is the standard LC oscillator with natural angular frequency
$$\omega = \dfrac{1}{\sqrt{LC}}$$

Inductance $$L = 0.1\;{\rm H}$$, capacitance $$C = 10^{-3}\;{\rm F}$$, so
$$LC = 0.1 \times 10^{-3} = 10^{-4}$$ $$\sqrt{LC} = 10^{-2}$$ $$\therefore \;\omega = \dfrac{1}{10^{-2}} = 100\;{\rm rad\,s^{-1}}$$

Equation $$(1)$$ has a particular (steady-state) solution $$q_p = C\varepsilon$$ because substituting $$q = C\varepsilon$$ makes the left side equal to $$\varepsilon$$. Hence the full solution is
$$q(t) = C\varepsilon + A\cos\omega t + B\sin\omega t \qquad -(2)$$

Before $$t=0$$ the field was constant, so there was no induced emf; therefore the capacitor was uncharged and no current flowed:
$$q(0)=0,\; i(0)=0$$

Imposing $$q(0)=0$$ in $$(2)$$:
$$0 = C\varepsilon + A \Longrightarrow A = -C\varepsilon$$

Current is the time derivative of charge:
$$i(t) = \dfrac{dq}{dt} = -A\omega\sin\omega t + B\omega\cos\omega t$$

With $$i(0)=0$$ we have $$B\omega = 0 \Longrightarrow B = 0$$.

Thus
$$q(t) = C\varepsilon\bigl(1-\cos\omega t\bigr)$$ $$i(t) = C\varepsilon\,\omega\sin\omega t$$

The sine function varies between $$-1$$ and $$+1$$, so the maximum magnitude of the current is
$$I_{\max} = C\varepsilon\,\omega$$

Substituting the numerical values:
$$I_{\max} = (10^{-3}\;{\rm F})(0.04\;{\rm V})(100\;{\rm rad\,s^{-1}})$$ $$I_{\max} = 4 \times 10^{-3}\;{\rm A} = 0.004\;{\rm A} = 4\;{\rm mA}$$

Therefore, the maximum magnitude of the current in the circuit is 4 mA.

The magnetic field at the center of a closely wound circular coil is given by the formula:

Here, $$ B $$ is the magnetic field, $$ \mu_0 $$ is the permeability of free space ($$ 4\pi \times 10^{-7} $$ T m/A), $$ N $$ is the number of turns, $$ I $$ is the current, and $$ R $$ is the radius of the coil.

We are given:

• Radius $$ R = 5 $$ cm = 0.05 m (converted to meters for SI units),
• Magnetic field $$ B = 37.68 \times 10^{-4} $$ T,
• Number of turns $$ N = 100 $$,
• $$ \pi = 3.14 $$.

First, express $$ \mu_0 $$ using the given $$ \pi $$:

Now, rearrange the formula to solve for $$ I $$:

Substitute the known values:

Compute the numerator:

Compute the denominator:

Now, divide the numerator by the denominator:

Perform the division:

Therefore, the current $$ I $$ is 3 A.

Hence, the correct answer is 3.

We need to find the separation between two parallel current-carrying wires.

Since the length of each wire is $$L = 10$$ cm = $$0.1$$ m, each carries a current of $$I_1 = I_2 = 5$$ A, and the force experienced by each wire is $$F = 10^{-5}$$ N, we apply the expression for the force between two parallel currents. The formula is $$F = \frac{\mu_0 I_1 I_2 L}{2\pi d}$$ where $$d$$ is the separation between the wires and $$\mu_0 = 4\pi \times 10^{-7}$$ T m/A.

Rearranging this expression for the distance gives

$$d = \frac{\mu_0 I_1 I_2 L}{2\pi F}$$

Substituting the known values yields

$$d = \frac{4\pi \times 10^{-7} \times 5 \times 5 \times 0.1}{2\pi \times 10^{-5}}$$

$$= \frac{4\pi \times 10^{-7} \times 2.5}{2\pi \times 10^{-5}}$$

$$= \frac{4 \times 2.5 \times 10^{-7}}{2 \times 10^{-5}}$$

$$= \frac{10 \times 10^{-7}}{2 \times 10^{-5}}$$

$$= \frac{10^{-6}}{2 \times 10^{-5}}$$

$$= \frac{1}{20} = 0.05$$ m = $$5$$ cm

Therefore, the separation between the wires is 5 cm.

We need to find the radius of the circular path of a singly ionized magnesium ion in a magnetic field. The mass number, $$A = 24$$, so mass $$m = 24 \times 1.67 \times 10^{-27}$$ kg $$= 24 \times 1.67 \times 10^{-27}$$ kg; the kinetic energy, $$KE = 5$$ keV $$= 5 \times 10^3 \times 1.6 \times 10^{-19}$$ J $$= 8 \times 10^{-16}$$ J; the magnetic field, $$B = 0.5$$ T; and the charge, $$q = 1.6 \times 10^{-19}$$ C (singly ionized).

Since $$KE = \frac{1}{2}mv^2$$, it follows that $$v = \sqrt{\frac{2 \times KE}{m}}$$.

The radius of circular motion in a magnetic field is given by $$r = \frac{mv}{qB}$$. Substituting $$v = \sqrt{\frac{2 \times KE}{m}}$$ gives

$$r = \frac{m}{qB} \sqrt{\frac{2 \times KE}{m}} = \frac{\sqrt{2m \times KE}}{qB}$$

Substituting the values:

$$m = 24 \times 1.67 \times 10^{-27} = 4.008 \times 10^{-26} \text{ kg}$$

$$2m \times KE = 2 \times 4.008 \times 10^{-26} \times 8 \times 10^{-16}$$

$$= 64.128 \times 10^{-42} = 6.4128 \times 10^{-41}$$

$$\sqrt{2m \times KE} = \sqrt{6.4128 \times 10^{-41}} = 8.008 \times 10^{-21}$$

Now calculate $$r$$:

$$r = \frac{8.008 \times 10^{-21}}{1.6 \times 10^{-19} \times 0.5}$$

$$r = \frac{8.008 \times 10^{-21}}{8 \times 10^{-20}}$$

$$r = \frac{8.008}{80} = 0.1001 \text{ m}$$

$$r \approx 0.1 \text{ m} = 10 \text{ cm}$$

Hence, the radius of the path is 10 cm.

We have a wire of length $$L = 314$$ cm $$= 3.14$$ m carrying a current of $$I = 14$$ A, bent to form a circle. We need to find the magnetic moment of this circular coil.

When the wire is bent into a circle, the circumference equals the length of the wire: $$2\pi r = L$$, so $$r = \dfrac{L}{2\pi} = \dfrac{3.14}{2 \times 3.14} = \dfrac{1}{2} = 0.5$$ m.

The area of the circular loop is $$A = \pi r^2 = 3.14 \times (0.5)^2 = 3.14 \times 0.25 = 0.785$$ m$$^2$$.

The magnetic moment of a current loop is $$M = nIA$$, where $$n$$ is the number of turns. Since the wire forms a single loop, $$n = 1$$. Therefore, $$M = 1 \times 14 \times 0.785 = 10.99 \approx 11$$ A-m$$^2$$.

Hence, the correct answer is 11.

Recall the formula for radius of circular motion in a magnetic field.

A charged particle moving with kinetic energy $$K$$ in a magnetic field $$B$$ follows a circular path of radius:

$$r = \frac{mv}{qB} = \frac{\sqrt{2mK}}{qB}$$

(since $$K = \frac{1}{2}mv^2 \Rightarrow mv = \sqrt{2mK}$$)

Write the radii for deuteron and proton.

For the proton: mass $$m_p$$, charge $$q_p = e$$

$$r_p = \frac{\sqrt{2m_pK}}{eB}$$

For the deuteron: mass $$m_d = 2m_p$$, charge $$q_d = e$$

$$r_d = \frac{\sqrt{2 \times 2m_p \times K}}{eB} = \frac{\sqrt{2} \cdot \sqrt{2m_pK}}{eB}$$

Find the ratio.

$$\frac{r_d}{r_p} = \frac{\sqrt{2} \cdot \sqrt{2m_pK}}{eB} \times \frac{eB}{\sqrt{2m_pK}} = \sqrt{2}$$

So $$\frac{r_d}{r_p} = \sqrt{2} : 1$$.

Comparing with $$\sqrt{x} : 1$$, we get $$x = 2$$.

The answer is $$\boxed{2}$$.

The magnetic field at any point is obtained by adding the contributions from each current-carrying segment separately.
For every straight segment lying in the $$xy$$-plane the field at the point O is perpendicular to the plane, i.e. along $$\pm \hat{k}$$. We therefore have to calculate only the magnitudes and check the sign with the right-hand rule.

Magnetic field of a straight wire.
For an infinitely long straight wire at a perpendicular distance $$d$$ from the observation point, $$B=\dfrac{\mu_0 I}{2\pi d}$$.

Magnetic field of a finite straight wire.
For a straight piece of length $$AB$$ the field at a point P lying at a perpendicular distance $$r$$ from the wire is $$B=\dfrac{\mu_0 I}{4\pi r}\left(\sin\theta_1+\sin\theta_2\right)$$ $$-(1)$$ where $$\theta_1$$ and $$\theta_2$$ are the angles made by the lines PA and PB with the perpendicular from P to the wire.

The geometry given in the figure consists of two parts: (i) an infinitely long wire whose closest distance from O is $$d=\dfrac{L}{2\pi}$$, and (ii) a straight segment of length $$L$$ lying parallel to the $$x$$-axis at a height $$y=L$$ above O, its right end being at $$(L,L)$$. Both carry the same current $$I$$.

(i) Field due to the infinitely long wire
Using the expression written above, $$B_1=\dfrac{\mu_0 I}{2\pi d}=\dfrac{\mu_0 I}{2\pi\left(L/2\pi\right)} =\dfrac{\mu_0 I}{L}.$$ With the current direction shown in the figure, the right-hand rule gives the direction as $$-\hat{k}$$.

(ii) Field due to the finite segment of length $$L$$
For this segment the perpendicular distance from O is $$r=L$$. The nearer end of the segment lies directly above O, so $$\theta_1=0^{\circ}$$ and hence $$\sin\theta_1=0$$. For the farther end at $$(L,L)$$, the line joining it to O makes an angle of $$45^{\circ}$$ with the perpendicular; therefore $$\sin\theta_2=\sin45^{\circ}=\dfrac{1}{\sqrt2}$$. Substituting these values in $$(1)$$, $$B_2=\dfrac{\mu_0 I}{4\pi L}\left(0+\dfrac{1}{\sqrt2}\right) =\dfrac{\mu_0 I}{4\sqrt2\pi L}.$$ The same right-hand rule shows that this field is also along $$-\hat{k}$$.

Net magnetic field at O
Both contributions point along $$-\hat{k}$$, so the magnitudes simply add: $$\vec{B}= -\hat{k}\left(\dfrac{\mu_0 I}{L} +\dfrac{\mu_0 I}{4\sqrt2\pi L}\right) =\dfrac{-\mu_0 I}{L}\left(1+\dfrac{1}{4\sqrt2\pi}\right)\hat{k}.$$

Comparing with the given options, this is exactly Option C.

Option C which is: $$\vec{B} = \dfrac{-\mu_0 I}{L}\left(1 + \dfrac{1}{4\sqrt{2}\pi}\right)\hat{k}.$$

The magnetic force on a moving charge is given by $$\vec{F} = Q(\vec{v} \times \vec{B})$$. By the properties of the cross product, this force is always perpendicular to the velocity $$\vec{v}$$ of the charge.

The work done by a force is $$dW = \vec{F} \cdot d\vec{l}$$, where $$d\vec{l}$$ is along the direction of displacement (and hence along $$\vec{v}$$). Since $$\vec{F} \perp \vec{v}$$, the dot product $$\vec{F} \cdot d\vec{l} = 0$$ at every instant.

Therefore, the work done by the magnetic field $$\vec{B}$$ on the charge is zero.

All three particles — proton, deuteron, and $$\alpha$$ particle — move with the same momentum $$p$$ in a uniform magnetic field $$B$$.

The magnetic force on a charged particle is $$F = qvB$$. Since $$p = mv$$, we have $$v = \frac{p}{m}$$, so $$F = \frac{qpB}{m}$$. The force ratio depends on $$\frac{q}{m}$$ for each particle.

For a proton: $$q_p = e$$, $$m_p = m$$, so $$\frac{q_p}{m_p} = \frac{e}{m}$$.

For a deuteron: $$q_d = e$$, $$m_d = 2m$$, so $$\frac{q_d}{m_d} = \frac{e}{2m}$$.

For an $$\alpha$$ particle: $$q_\alpha = 2e$$, $$m_\alpha = 4m$$, so $$\frac{q_\alpha}{m_\alpha} = \frac{2e}{4m} = \frac{e}{2m}$$.

The ratio of magnetic forces is $$F_p : F_d : F_\alpha = \frac{e}{m} : \frac{e}{2m} : \frac{e}{2m} = 2 : 1 : 1$$.

The speed of each particle is $$v = \frac{p}{m}$$, so the speed ratio is $$v_p : v_d : v_\alpha = \frac{1}{m} : \frac{1}{2m} : \frac{1}{4m} = 4 : 2 : 1$$.

Therefore the ratio of forces is $$2 : 1 : 1$$ and the speeds are in the ratio $$4 : 2 : 1$$, which corresponds to Option (1).

We want the magnetic field at the centre of a flat spiral coil that starts from an inner radius $$a$$ and ends at an outer radius $$b$$. The coil carries a steady current $$I$$ and has a total of $$N$$ closely packed turns.

For a single circular loop of radius $$r$$ carrying current $$I$$, the magnetic field at its centre is given by the well-known formula

$$B_{\text{single}}=\dfrac{\mu_0 I}{2r}.$$

In the spiral, the turns are so tightly wound that as we move radially outwards by a small distance $$dr$$ we encounter a small fractional turn $$dN$$. Because the winding is uniform, the number of turns per unit radial length is the same everywhere. The total radial width of the spiral is $$b-a$$, so

$$\text{Turns per unit length}= \dfrac{N}{\,b-a\,},\qquad\text{hence}\qquad dN=\dfrac{N}{\,b-a\,}\,dr.$$

Every elementary circular ring between radii $$r$$ and $$r+dr$$ behaves like $$dN$$ separate loops, each contributing the field $$\dfrac{\mu_0 I}{2r}$$ at the centre. Therefore the differential contribution to the field is

$$dB = \left(\dfrac{\mu_0 I}{2r}\right)\,dN =\left(\dfrac{\mu_0 I}{2r}\right)\left(\dfrac{N}{\,b-a\,}\,dr\right) =\dfrac{\mu_0 I N}{2(b-a)}\;\dfrac{dr}{r}.$$

All these elemental fields point in the same direction (perpendicular to the plane of the coil), so we add them directly. Integrating $$dB$$ from the inner radius $$a$$ to the outer radius $$b$$ gives the total magnetic field at the centre:

$$ \begin{aligned} B &= \int_{a}^{b} dB = \dfrac{\mu_0 I N}{2(b-a)}\int_{a}^{b}\dfrac{dr}{r} \\ &= \dfrac{\mu_0 I N}{2(b-a)}\Bigl[\ln r\Bigr]_{a}^{b} \\ &= \dfrac{\mu_0 I N}{2(b-a)}\;\ln\!\left(\dfrac{b}{a}\right). \end{aligned} $$

This expression matches option 4 exactly.

Hence, the correct answer is Option 4.

The magnetic field at a point on the axis of a circular coil of radius $$R$$ at a distance $$x$$ from the centre is given by $$B = \frac{\mu_0 N I R^2}{2(R^2 + x^2)^{3/2}}$$.

Given that the fields at distances $$x_1 = 0.05$$ m and $$x_2 = 0.2$$ m from the centre are in the ratio $$8 : 1$$, we have:

$$\frac{B_1}{B_2} = \frac{(R^2 + x_2^2)^{3/2}}{(R^2 + x_1^2)^{3/2}} = 8$$

Taking the cube root of both sides (since $$8^{1/3} = 2$$) and then squaring:

$$\frac{R^2 + x_2^2}{R^2 + x_1^2} = 8^{2/3} = 4$$

Substituting $$x_1 = 0.05$$ m and $$x_2 = 0.2$$ m:

$$R^2 + (0.2)^2 = 4(R^2 + (0.05)^2)$$

$$R^2 + 0.04 = 4R^2 + 0.01$$

$$0.03 = 3R^2$$

$$R^2 = 0.01$$

$$R = 0.1$$ m

The correct answer is Option (3): 0.1 m.

We need the magnetic field produced by a circular coil carrying a steady current. The standard expression for the magnitude of the magnetic field on the axis of a single loop of radius $$a$$, at a point whose distance from the centre measured along the axis is $$r$$, is

$$B(r)=\frac{\mu_{0}\,I\,a^{2}}{2\left(a^{2}+r^{2}\right)^{3/2}}.$$

(This formula can be derived from the Biot-Savart law, but we take it here as a known result.) At the centre of the same coil we simply place $$r=0$$, giving

$$B_{\text{centre}}=B(0)=\frac{\mu_{0}\,I\,a^{2}}{2\left(a^{2}+0\right)^{3/2}} =\frac{\mu_{0}\,I\,a^{2}}{2a^{3}} =\frac{\mu_{0}\,I}{2a}.$$

We now form the ratio of the field at the off-axis point to the field at the centre:

$$\frac{B(r)}{B_{\text{centre}}} =\frac{\dfrac{\mu_{0}I a^{2}}{2\left(a^{2}+r^{2}\right)^{3/2}}} {\dfrac{\mu_{0}I}{2a}} =\frac{a^{3}}{\left(a^{2}+r^{2}\right)^{3/2}} =\frac{1}{\left(1+\dfrac{r^{2}}{a^{2}}\right)^{3/2}}.$$

To find the fractional change we write

$$\text{Fractional change}=\frac{B(r)-B_{\text{centre}}}{B_{\text{centre}}} =\frac{B(r)}{B_{\text{centre}}}-1 =\left(1+\frac{r^{2}}{a^{2}}\right)^{-3/2}-1.$$

The question specifies $$r<a$$, so the quantity $$\dfrac{r^{2}}{a^{2}}$$ is small. We therefore expand the power using the binomial approximation. For a small $$x$$, the expansion

$$(1+x)^{n}\;\approx\;1+nx$$

is adequate to first order. Here $$n=-\dfrac{3}{2}$$ and $$x=\dfrac{r^{2}}{a^{2}}$$, so

$$\left(1+\frac{r^{2}}{a^{2}}\right)^{-3/2} \approx 1-\frac{3}{2}\frac{r^{2}}{a^{2}}.$$

Substituting this approximation into the expression for the fractional change gives

$$\text{Fractional change} \approx\Bigl[1-\frac{3}{2}\frac{r^{2}}{a^{2}}\Bigr]-1 =-\frac{3}{2}\frac{r^{2}}{a^{2}}.$$

The negative sign merely tells us that the magnetic field at the point on the axis is smaller than the field at the centre. The magnitude of the fractional change is therefore

$$\left|\frac{B(r)-B_{\text{centre}}}{B_{\text{centre}}}\right| =\frac{3}{2}\frac{r^{2}}{a^{2}}.$$

Among the given options, this matches

$$\frac{3}{2}\frac{r^{2}}{a^{2}}.$$

Hence, the correct answer is Option C.

For a charged particle moving in a magnetic field, the radius of the circular path is $$r = \frac{mv}{qB} = \frac{\sqrt{2mK}}{qB}$$, where $$K$$ is the kinetic energy.

For the deuteron: mass $$m_d = 2u$$, charge $$q_d = e$$. For the alpha particle: mass $$m_\alpha = 4u$$, charge $$q_\alpha = 2e$$.

Both particles have equal kinetic energy $$K$$ and enter the same magnetic field $$B$$. Taking the ratio:

$$\frac{r_d}{r_\alpha} = \frac{\sqrt{2m_d K}/(q_d B)}{\sqrt{2m_\alpha K}/(q_\alpha B)} = \frac{q_\alpha}{q_d} \cdot \sqrt{\frac{m_d}{m_\alpha}} = \frac{2e}{e} \cdot \sqrt{\frac{2u}{4u}} = 2 \times \frac{1}{\sqrt{2}} = \sqrt{2}$$

The radius of the circular trajectory of a charged particle in a magnetic field is $$r = \frac{mv}{qB} = \frac{\sqrt{2mK}}{qB}$$, where $$K$$ is the kinetic energy.

For a proton: $$r_p = \frac{\sqrt{2m_p K_p}}{eB}$$. For an $$\alpha$$-particle (mass $$4m_p$$, charge $$2e$$): $$r_\alpha = \frac{\sqrt{2 \cdot 4m_p \cdot K_\alpha}}{2eB} = \frac{\sqrt{8m_p K_\alpha}}{2eB}$$.

The ratio is: $$\frac{r_p}{r_\alpha} = \frac{\sqrt{2m_p K_p}}{eB} \cdot \frac{2eB}{\sqrt{8m_p K_\alpha}} = \frac{2\sqrt{K_p}}{2\sqrt{K_\alpha}} = \sqrt{\frac{K_p}{K_\alpha}}$$.

Given $$\frac{r_p}{r_\alpha} = 2$$, we get $$\frac{K_p}{K_\alpha} = 4$$, so $$K_p : K_\alpha = 4 : 1$$.

The magnetic flux density inside a solenoid with a core of relative permeability $$\mu_r$$ is given by $$B = \mu_r \mu_0 n I$$, where $$n$$ is the number of turns per metre and $$I$$ is the current.

Substituting the given values: $$B = 500 \times 4\pi \times 10^{-7} \times 1000 \times 5$$. Computing step by step: $$4\pi \times 10^{-7} \times 1000 = 4\pi \times 10^{-4}$$, then $$500 \times 4\pi \times 10^{-4} = 2000\pi \times 10^{-4} = 0.2\pi$$, and finally $$0.2\pi \times 5 = \pi$$ T.

The correct answer is option 1: $$\pi$$ T.

We have two ions that both enter a region where the magnetic field $$\vec B$$ is uniform and perpendicular to their velocities. Whenever a charged particle moves perpendicular to a magnetic field, it describes a circular path. The required formula for the radius of that circular path is first recalled:

For a particle of mass $$m$$, charge $$q$$ and speed $$v$$ in a magnetic field of magnitude $$B$$ (with velocity perpendicular to the field), the magnetic force provides the necessary centripetal force. Hence

$$q\,v\,B = \dfrac{m v^{2}}{r}.$$

Solving this equation for the radius $$r$$ gives

$$r = \dfrac{m\,v}{q\,B} \;.$$

In the present question, both ions possess the same kinetic energy. Let us denote that common kinetic energy by $$K$$. The kinetic energy of any particle is related to its speed by the well-known relation

$$K = \dfrac{1}{2} m v^{2}.$$

From this we can express the speed $$v$$ of each ion in terms of its mass and the common kinetic energy:

$$v = \sqrt{\dfrac{2K}{m}}.$$

Now we substitute this expression for $$v$$ into the radius formula. Performing the substitution, we get

$$r = \dfrac{m}{q\,B}\;\sqrt{\dfrac{2K}{m}}.$$

By taking the square root of $$m$$ in the numerator, the algebraic simplification proceeds as follows:

$$r = \dfrac{\sqrt{m}\;\sqrt{m}}{q\,B}\;\sqrt{\dfrac{2K}{m}} = \dfrac{\sqrt{m}\;}{q\,B}\;\sqrt{2K} = \dfrac{\sqrt{2K}}{B}\;\dfrac{\sqrt{m}}{q}.$$

The magnetic field $$B$$ and the factor $$\sqrt{2K}$$ are identical for both ions, so the only quantity that decides the relative radii is the ratio $$\dfrac{\sqrt{m}}{q}$$.

Let us calculate this ratio for each ion separately.

Ion 1 (lighter ion)
Mass $$m_{1}=4\ \text{amu},\qquad q_{1}=+2e.$$

$$\dfrac{\sqrt{m_{1}}}{q_{1}} = \dfrac{\sqrt{4}}{2e} = \dfrac{2}{2e} = \dfrac{1}{e}.$$

Ion 2 (heavier ion)
Mass $$m_{2}=16\ \text{amu},\qquad q_{2}=+3e.$$

$$\dfrac{\sqrt{m_{2}}}{q_{2}} = \dfrac{\sqrt{16}}{3e} = \dfrac{4}{3e}.$$

We now compare these two numerical factors:

$$\dfrac{\sqrt{m_{2}}}{q_{2}} = \dfrac{4}{3e}\;>\;\dfrac{1}{e} = \dfrac{\sqrt{m_{1}}}{q_{1}}.$$

The radius of curvature is directly proportional to this factor, so

$$r_{2} > r_{1}.$$

Because the heavier ion (Ion 2) travels in a circle of larger radius, it is bent less sharply by the magnetic field. Conversely, the lighter ion (Ion 1) follows a circle of smaller radius, meaning its trajectory bends more.

In ordinary language, a smaller radius corresponds to greater deflection; a larger radius corresponds to less deflection. Therefore the lighter ion is deflected more, and the heavier ion is deflected less.

Hence, the correct answer is Option B.

We have a long straight coaxial system. Its inner solid conductor has radius $$a$$ and carries a total current $$I_0$$ uniformly over its cross-section. Its outer cylindrical shell occupies the space from $$r=b$$ to $$r=c$$ and carries an equal current $$I_0$$ but in the opposite direction, also distributed uniformly through its own metal. Because the currents are steady and the geometry is perfectly coaxial, we take every magnetic field line to be a circle centred on the common axis, so the magnitude of the field depends only on the radial distance $$r$$ from that axis.

For such circular symmetry we apply Ampère’s circuital law, first stating it explicitly:

$$\oint \vec B \cdot d\vec l = \mu_0 I_{\text{enclosed}}.$$

Choosing a circular Amperian path of radius $$r$$ (so $$d\vec l$$ is everywhere tangential and of magnitude $$r\,d\theta$$), the left side becomes $$B(2\pi r)$$. Hence

$$B(2\pi r)=\mu_0 I_{\text{enclosed}}\quad\Longrightarrow\quad B=\dfrac{\mu_0 I_{\text{enclosed}}}{2\pi r}.$$

We now find the enclosed current in the two separate radial zones asked for and then form their ratio.

Case (i)  $$r=x\lt a$$

The current density $$J$$ in the inner wire is uniform, so

$$J=\dfrac{I_0}{\text{cross-sectional area}}=\dfrac{I_0}{\pi a^2}.$$

The area enclosed by our path of radius $$x$$ is $$\pi x^2$$, hence the enclosed current is

$$I_{\text{enc}}^{(i)} = J(\pi x^2)=\dfrac{I_0}{\pi a^2}\,(\pi x^2)=I_0\,\dfrac{x^2}{a^2}.$$

Substituting this into Ampère’s result gives

$$B_{(i)}=\dfrac{\mu_0}{2\pi x}\,I_0\,\dfrac{x^2}{a^2}=\dfrac{\mu_0 I_0\,x}{2\pi a^2}.$$

Case (ii)  $$a\lt r=x\lt b$$

Now the entire inner conductor lies inside the Amperian loop, while none of the outer shell (which starts only at $$r=b$$) is enclosed. Therefore

$$I_{\text{enc}}^{(ii)}=I_0.$$

Inserting this into Ampère’s expression, we find

$$B_{(ii)}=\dfrac{\mu_0 I_0}{2\pi x}.$$

Forming the required ratio

$$\dfrac{B_{(i)}}{B_{(ii)}}=\dfrac{\dfrac{\mu_0 I_0\,x}{2\pi a^2}}{\dfrac{\mu_0 I_0}{2\pi x}} =\dfrac{x^2}{a^2}.$$

This matches Option D in the given list.

Hence, the correct answer is Option D.

First recall the magnetic-field formula for a straight, finite conductor. For a wire segment of length $$L$$ that carries current $$I$$, the field at a point that is at a perpendicular distance $$r$$ from the wire and that “sees’’ the two ends under angles $$\theta_1$$ and $$\theta_2$$ (measured from the perpendicular) is

$$ B=\dfrac{\mu_0 I}{4\pi r}\,(\sin\theta_1+\sin\theta_2), $$

and the direction is obtained from the right-hand rule for the element $$I\,\vec{\mathrm d\ell}\times\vec r$$.

The triangle is equilateral with side

$$ a = 9\ \text{cm}=0.09\ \text{m}, $$

and the current is $$I = 1.5\ \text{A}$$, flowing clockwise. The point at which we need the field is the centroid $$G$$. In an equilateral triangle the perpendicular distance from the centroid to any side is one third of the altitude. The altitude is

$$ h = a\dfrac{\sqrt{3}}{2}, $$

so the required distance is

$$ r = \dfrac{h}{3}= \dfrac{a}{3}\,\dfrac{\sqrt{3}}{2} = a\,\dfrac{\sqrt{3}}{6} = 0.09\ \text{m}\times\dfrac{\sqrt{3}}{6}. $$

Numerically, using $$\sqrt{3}=1.732$$,

$$ r = 0.09\times\dfrac{1.732}{6} = 0.09\times0.2887 \approx 0.02598\ \text{m}. $$

Because the centroid lies on the perpendicular bisector of every side, the two angles for each side are equal: $$\theta_1=\theta_2=\theta$$. The half-length of a side is $$a/2$$, so

$$ \tan\theta = \dfrac{a/2}{r} = \dfrac{0.09/2}{0.02598} = \dfrac{0.045}{0.02598} \approx 1.732 =\sqrt{3}. $$

Hence $$\theta=60^\circ$$ and

$$ \sin\theta = \sin60^\circ = \dfrac{\sqrt{3}}{2}. $$

Substituting $$\theta_1=\theta_2=60^\circ$$ in the straight-wire formula gives the field from one side:

$$ B_1 = \dfrac{\mu_0 I}{4\pi r}\, (\sin\theta+\sin\theta) = \dfrac{\mu_0 I}{4\pi r}\,(2\sin60^\circ) = \dfrac{\mu_0 I}{4\pi r}\, \left(2\times\dfrac{\sqrt{3}}{2}\right) = \dfrac{\mu_0 I\sqrt{3}}{4\pi r}. $$

Now insert the numerical values. We have $$\mu_0/4\pi = 10^{-7}\,{\rm T\,m\,A^{-1}}$$, so

$$ B_1 = 10^{-7} \times\dfrac{1.5\sqrt{3}}{0.02598} = 10^{-7} \times\dfrac{1.5\times1.732}{0.02598}. $$

The numerator is $$1.5\times1.732 = 2.598$$. Dividing,

$$ \dfrac{2.598}{0.02598}\approx 100.0. $$

Therefore

$$ B_1 \approx 100.0\times10^{-7}\ \text{T} = 1.0\times10^{-5}\ \text{T}. $$

The triangle has three identical sides, and by symmetry each contributes the same magnitude and the same axial direction, so the net field is

$$ B = 3B_1 = 3\times1.0\times10^{-5}\ \text{T} = 3.0\times10^{-5}\ \text{T}. $$

Finally, we determine the sense (into or out of the plane). Curl the fingers of your right hand in the direction of the current (clockwise as viewed from the front); the thumb then points into the page, that is, inside the plane of the triangle. All three side contributions add with this same orientation, so the resultant field is into the plane.

Hence, the correct answer is Option D.

For a charged particle that enters a uniform magnetic field $$\vec B$$ normally (that is, with its velocity perpendicular to the field), the magnetic force supplies the centripetal force required for uniform circular motion. First we write the force equation:

Magnetic force: $$F_{\text{mag}} = q\,v\,B$$

Centripetal force: $$F_{\text{cent}} = \dfrac{m v^{2}}{r}$$

Equating the two forces because the magnetic force itself acts as the centripetal force, we have

$$q\,v\,B = \dfrac{m v^{2}}{r}$$

Now we solve this equation for the radius $$r$$ of the circular path:

$$q\,v\,B = \dfrac{m v^{2}}{r} \;\;\Longrightarrow\;\; q\,v\,B\,r = m v^{2} \;\;\Longrightarrow\;\; r = \dfrac{m v}{q B}$$

So the radius is given by the formula $$r = \dfrac{m v}{q B}.$$

The problem involves two ions. We denote their physical quantities with subscripts 1 and 2:

Same mass: $$m_{1} = m_{2} = m$$

Charge ratio: $$q_{1} : q_{2} = 1 : 2 \;\;\Longrightarrow\;\; q_{1} = q,\; q_{2} = 2q$$

Speed ratio: $$v_{1} : v_{2} = 2 : 3 \;\;\Longrightarrow\;\; v_{1} = 2v,\; v_{2} = 3v$$

Both ions move in the same magnetic field $$B$$, so $$B$$ is common.

Using the formula for each radius, we write

$$r_{1} = \dfrac{m v_{1}}{q_{1} B} = \dfrac{m (2v)}{q B} = \dfrac{2 m v}{q B}$$

$$r_{2} = \dfrac{m v_{2}}{q_{2} B} = \dfrac{m (3v)}{2q B} = \dfrac{3 m v}{2 q B}$$

To find the required ratio of the radii, we divide $$r_{1}$$ by $$r_{2}$$:

$$\dfrac{r_{1}}{r_{2}} = \dfrac{\dfrac{2 m v}{q B}}{\dfrac{3 m v}{2 q B}}$$

Because $$m$$, $$v$$, $$q$$, and $$B$$ are common factors, we can cancel them step by step:

$$\dfrac{r_{1}}{r_{2}} = \dfrac{2}{1}\;\cdot\;\dfrac{1}{1}\;\cdot\;\dfrac{1}{1}\;\cdot\;\dfrac{2}{3} = \dfrac{4}{3}$$

Hence, $$r_{1} : r_{2} = 4 : 3.$$

Checking the options, $$4 : 3$$ corresponds to Option B.

Hence, the correct answer is Option B.

We have a cyclotron in which the proton is accelerated every time it passes through the gap between the two dees. The radio-frequency oscillator provides a maximum potential difference of 12 kV across the gap. Whenever the signs of the dees are opposite, the proton experiences an accelerating potential of this magnitude, gains kinetic energy, enters the circular dee, describes a semicircle, and then emerges at the gap again, where the potential has reversed so that it is once more accelerated. Hence, in one complete revolution the proton crosses the gap twice and is accelerated twice.

The energy gained in one single crossing is given by the elementary relation

$$\Delta E_{\text{cross}} = eV,$$

where $$e = 1.6 \times 10^{-19}\ \text{C}$$ is the charge on the proton and $$V = 12\ \text{kV} = 12 \times 10^{3}\ \text{V}.$$ Substituting the numbers,

$$\Delta E_{\text{cross}} = \bigl(1.6 \times 10^{-19}\bigr)\bigl(12 \times 10^{3}\bigr) = 1.6 \times 12 \times 10^{-19+3} = 19.2 \times 10^{-16}\ \text{J} = 1.92 \times 10^{-15}\ \text{J}.$$

Because the proton crosses the gap twice in one full revolution, the energy gained per revolution is

$$\Delta E_{\text{rev}} = 2 \,\Delta E_{\text{cross}} = 2 \times 1.92 \times 10^{-15}\ \text{J} = 3.84 \times 10^{-15}\ \text{J}.$$

The target speed of the proton is one sixth of the speed of light, i.e.

$$v = \frac{1}{6}\,c = \frac{1}{6}\,\bigl(3 \times 10^{8}\bigr)\ \text{m s}^{-1} = 5.0 \times 10^{7}\ \text{m s}^{-1}.$$

For this (non-relativistic) speed, the kinetic energy acquired is obtained from the classical formula

$$K = \frac{1}{2}mv^{2},$$

where $$m = 1.67 \times 10^{-27}\ \text{kg}$$ is the proton’s mass. Substituting,

$$K = \frac{1}{2}\,\bigl(1.67 \times 10^{-27}\bigr)\,\bigl(5.0 \times 10^{7}\bigr)^{2}.$$

First evaluate the square of the speed:

$$\bigl(5.0 \times 10^{7}\bigr)^{2} = 25 \times 10^{14} = 2.5 \times 10^{15}.$$

Now multiply:

$$K = \frac{1}{2} \times 1.67 \times 10^{-27} \times 2.5 \times 10^{15} = 0.835 \times 10^{-27} \times 2.5 \times 10^{15} = 2.0875 \times 10^{-12}\ \text{J}.$$

This is the total kinetic energy the proton must gain. Each revolution supplies $$\Delta E_{\text{rev}} = 3.84 \times 10^{-15}\ \text{J}.$$ Therefore the number of revolutions required is

$$N = \frac{K}{\Delta E_{\text{rev}}} = \frac{2.0875 \times 10^{-12}}{3.84 \times 10^{-15}} = \frac{2.0875}{3.84} \times 10^{3} \approx 0.5435 \times 10^{3} \approx 5.435 \times 10^{2}.$$

Keeping only whole revolutions, we need approximately 543 revolutions.

So, the answer is $$543$$.

We have an equilateral-triangle coil carrying a current in a uniform magnetic field. For a current-carrying loop, the magnitude of the magnetic torque is given by the formula

$$\tau = N\,I\,A\,B\,\sin\phi,$$

where $$N$$ is the number of turns, $$I$$ is the current, $$A$$ is the area of the coil, $$B$$ is the magnetic-field magnitude, and $$\phi$$ is the angle between the field $$\vec B$$ and the normal to the plane of the coil.

Only one turn is mentioned, so we set $$N = 1.$$

The coil is an equilateral triangle of side $$a = 10\text{ cm} = 0.1\text{ m}.$$ The area of an equilateral triangle is

$$A = \frac{\sqrt{3}}{4}\,a^{2}.$$

Substituting $$a = 0.1\text{ m}$$, we get

$$A = \frac{\sqrt{3}}{4}\,(0.1)^{2} = \frac{\sqrt{3}}{4}\,(0.01) = 0.0025\,\sqrt{3}\;\text{m}^{2}.$$

The magnetic field is horizontal with magnitude $$B = 20\text{ mT} = 20\times10^{-3}\text{ T} = 2\times10^{-2}\text{ T}.$$

The current in the coil is $$I = 0.2\text{ A}.$$

The question states that the plane of the coil becomes parallel to the magnetic field. When the plane of the coil is parallel to $$\vec B$$, the normal to the plane is perpendicular to $$\vec B$$, giving $$\phi = 90^{\circ}$$ and hence $$\sin\phi = 1.$$

Now we substitute every quantity into the torque formula:

$$\tau = (1)\,(0.2)\,(0.0025\sqrt{3})\,(2\times10^{-2})\,(1).$$

First multiply $$0.2$$ and $$0.0025$$:

$$0.2 \times 0.0025 = 0.0005.$$

Next multiply this by $$2\times10^{-2}$$:

$$0.0005 \times 2\times10^{-2} = (0.0005 \times 2)\times10^{-2} = 0.001\times10^{-2} = 1\times10^{-5}.$$

Finally, attach the factor $$\sqrt{3}$$:

$$\tau = \sqrt{3} \times 10^{-5}\;\text{N\,m}.$$

The problem states the torque in the form $$\sqrt{x}\times10^{-5}\text{ N\,m}$$. Comparing coefficients, we have

$$x = 3.$$

So, the answer is $$3.$$

We recall the definition of magnetic moment of a current-carrying planar coil. For any coil, the magnetic moment $$m$$ is given by the formula

$$m = N\,I\,A,$$

where $$N$$ is the number of turns, $$I$$ is the current flowing through the wire and $$A$$ is the area enclosed by a single turn of the coil. This relation will be applied successively to the triangular and the square coils made from the same wire.

The total length of the conducting wire is specified to be $$24a$$. Its resistance is given as $$R$$, so when it is connected to the same voltage source of emf $$V_0$$ in either shape, the current through the wire will be identical in the two cases. Using Ohm’s law, we have

$$I = \dfrac{V_0}{R}.$$

Because $$I$$ is the same for both coils, the ratio of their magnetic moments will depend only on the respective products $$N\,A$$. Let us therefore determine $$N$$ and $$A$$ for each geometry separately.

Triangular coil  The shape is an equilateral triangle with side length $$a$$. The perimeter of one turn is $$3a$$, so the number of turns that can be formed from a total length $$24a$$ is

$$N_{\triangle}= \dfrac{24a}{3a}=8.$$

The area of a single equilateral triangle of side $$a$$ is obtained from the standard formula

$$A_{\triangle}= \dfrac{\sqrt{3}}{4}\,a^{2}.$$

Square coil  The square has side length $$a$$. Its perimeter is $$4a$$, therefore the number of turns obtained from the same wire is

$$N_{\square}= \dfrac{24a}{4a}=6.$$

The area of a square of side $$a$$ is simply

$$A_{\square}= a^{2}.$$

Ratio of magnetic moments  Using $$m=NIA$$ for each coil and canceling the common factor $$I$$, we write

$$\dfrac{m_{\triangle}}{m_{\square}} \;=\;\dfrac{N_{\triangle} A_{\triangle}} {N_{\square} A_{\square}}.$$ Substituting the numerical values just obtained, we have

$$\dfrac{m_{\triangle}}{m_{\square}} = \dfrac{\,8 \times \dfrac{\sqrt{3}}{4}a^{2}\,} {\,6 \times a^{2}\,}.$$

Simplifying step by step,

$$\dfrac{m_{\triangle}}{m_{\square}} = \dfrac{8}{4}\;\times\;\dfrac{\sqrt{3}}{6} = 2 \times \dfrac{\sqrt{3}}{6} = \dfrac{\sqrt{3}}{3}.$$

The statement of the problem expresses this same ratio in the form

$$\dfrac{m_{\triangle}}{m_{\square}} = \dfrac{1}{\sqrt{y}}.$$

Accordingly, we equate

$$\dfrac{1}{\sqrt{y}} = \dfrac{\sqrt{3}}{3}.$$

Cross-multiplying,

$$\sqrt{y} = \dfrac{3}{\sqrt{3}} = \sqrt{3}.$$

Finally, squaring both sides gives

$$y = 3.$$

So, the answer is $$3$$.

We have a particle of mass $$m$$ and charge $$q$$. Its initial velocity is purely along the $$y$$-axis, so we may write it as $$\vec v(0)=v_0\,\hat j$$. An electric field $$\vec E$$ and a magnetic field $$\vec B$$ are both directed along the $$x$$-axis; in symbols we write $$\vec E=E_0\,\hat i$$ and $$\vec B=B_0\,\hat i$$.

The force acting on a charged particle in simultaneous electric and magnetic fields is given by the Lorentz formula

$$\vec F = q\,(\vec E + \vec v \times \vec B).$$

Dividing by the mass, we obtain the acceleration

$$\frac{d\vec v}{dt}= \frac{q}{m}\,\bigl(\vec E + \vec v \times \vec B\bigr).$$

Let us resolve every quantity into Cartesian components. We denote the velocity as $$\vec v=(v_x,\,v_y,\,v_z).$$ The electric field has only an $$x$$-component $$E_0$$, while the magnetic field has only an $$x$$-component $$B_0$$. Therefore

$$\vec E=(E_0,\,0,\,0), \qquad \vec B=(B_0,\,0,\,0).$$

The magnetic part of the force requires the cross-product $$\vec v \times \vec B$$. Writing out the determinant

$$ \vec v \times \vec B= \begin{vmatrix} \hat i & \hat j & \hat k\\[2pt] v_x & v_y & v_z\\[2pt] B_0 & 0 & 0 \end{vmatrix} = \hat i\,(v_y\cdot 0-v_z\cdot 0)\;-\; \hat j\,(v_x\cdot 0-v_z\cdot B_0)\;+\; \hat k\,(v_x\cdot 0-v_y\cdot B_0). $$

Simplifying term by term we get

$$\vec v \times \vec B =(0,\;v_z B_0,\;-v_y B_0).$$

Substituting the electric and magnetic contributions into the acceleration equation gives the three component equations

$$m\,\frac{dv_x}{dt}=qE_0,$$

$$m\,\frac{dv_y}{dt}=q\,v_z B_0,$$

$$m\,\frac{dv_z}{dt}=-q\,v_y B_0.$$

We now analyse each direction.

1. Along the $$x$$-axis the acceleration is constant because the right-hand side is the constant $$qE_0$$. Hence

$$\frac{dv_x}{dt}= \frac{qE_0}{m}\equiv a,$$

where we have introduced $$a=\dfrac{qE_0}{m}$$ for convenience. Integrating with the initial condition $$v_x(0)=0$$ (because the initial velocity had no $$x$$-component) we get

$$v_x(t)=a\,t=\frac{qE_0}{m}\,t.$$

2. The $$y$$- and $$z$$-components are coupled. It is helpful to introduce the cyclotron frequency

$$\omega=\frac{qB_0}{m}.$$

The two differential equations now read

$$\frac{dv_y}{dt}= \omega\,v_z,$$

$$\frac{dv_z}{dt}= -\omega\,v_y.$$

Differentiate the first of these once more with respect to time:

$$\frac{d^2 v_y}{dt^2}= \omega\,\frac{dv_z}{dt}=\omega\,(-\omega v_y)=-\omega^2\,v_y.$$

This is the standard simple-harmonic-motion equation $$\dfrac{d^2y}{dt^2}=-\omega^2 y$$ whose solution is a sinusoid. Using the initial conditions $$v_y(0)=v_0$$ and $$v_z(0)=0$$ we obtain

$$v_y(t)=v_0\cos(\omega t),$$

and, from the relation $$\dfrac{dv_y}{dt}=\omega v_z,$$

$$v_z(t)= -\,v_0\sin(\omega t).$$

We now compute the square of the speed at any instant. The speed is the magnitude of $$\vec v$$, so

$$ \begin{aligned} v^2(t)&=v_x^2(t)+v_y^2(t)+v_z^2(t)\\[4pt] &=\bigl(a t\bigr)^2+\bigl(v_0\cos\omega t\bigr)^2+\bigl(-v_0\sin\omega t\bigr)^2\\[4pt] &=a^2 t^2+v_0^2\bigl(\cos^2\omega t+\sin^2\omega t\bigr). \end{aligned} $$

Because $$\cos^2\theta+\sin^2\theta=1$$, the trigonometric terms combine neatly to unity, leaving

$$v^2(t)=v_0^2 + a^2 t^2.$$

We are asked to find the time at which the speed doubles, i.e. when $$v(t)=2v_0$$. Squaring this requirement gives

$$\bigl(2v_0\bigr)^2 = v_0^2 + a^2 t^2.$$

Simplifying step by step,

$$4v_0^2 = v_0^2 + a^2 t^2,$$

$$4v_0^2 - v_0^2 = a^2 t^2,$$

$$3v_0^2 = a^2 t^2,$$

$$t^2 = \frac{3v_0^2}{a^2}.$$

Taking the positive square root (time is positive) we have

$$t = \frac{\sqrt{3}\,v_0}{a}.$$

Finally substitute back $$a=\dfrac{qE_0}{m}$$:

$$t = \frac{\sqrt{3}\,v_0}{\dfrac{qE_0}{m}} =\frac{\sqrt{3}\,m\,v_0}{qE_0}.$$

This time corresponds exactly to Option C in the given list.

Hence, the correct answer is Option C.

We first write the Lorentz-force law for a charged particle moving in a magnetic field:

$$\vec F = q\,\vec v \times \vec B.$$

The magnetic field is along the positive $$z$$-axis,

$$\vec B = B_0 \hat k,$$

while the particle is shot towards the screen with velocity

$$\vec v = -\,v\hat i \qquad (v \neq 0).$$

Taking the cross-product we get

$$\vec v \times \vec B \;=\; (-\,v\hat i)\times(B_0\hat k) = -\,vB_0\;(\hat i \times \hat k) = -\,vB_0\,(-\hat j) = +\,vB_0\,\hat j,$$

so

$$\vec F = q\,vB_0\,\hat j.$$

The force (and therefore the acceleration) is along $$+y$$, perpendicular to the initial velocity. A constant force perpendicular to the velocity produces uniform circular motion in the plane perpendicular to $$\vec B$$ (the $$x\!-\!y$$ plane).

For such motion the magnetic force supplies the necessary centripetal force. Stating the formula,

$$\text{Centripetal force} = \frac{mv^{2}}{r}, \qquad \text{Magnetic force} = qvB_0,$$

and equating them gives

$$qvB_0 = \frac{mv^{2}}{r}.$$

Solving for the radius $$r$$ of the circular path,

$$r = \frac{mv}{qB_0}.$$

Let the screen be the plane $$x = 0$$. The particle starts from the point $$P(d,0)$$ and its initial velocity is towards $$-x$$. Because the acceleration is along $$+y$$, the centre of the circle lies a distance $$r$$ above the starting point, i.e. at $$C(d,\,r)$$. Hence the circle has centre abscissa $$x = d$$ and radius $$r$$. The left-most point of this circle (the point of smallest $$x$$) is therefore at

$$x_{\min} = d - r.$$

If the particle is to miss the screen, this left-most point must stay to the right of the screen, i.e.

$$x_{\min} > 0 \;\;\Longrightarrow\;\; d - r > 0 \;\;\Longrightarrow\;\; r < d.$$

The borderline (minimum) speed that just manages to avoid the screen occurs when the trajectory merely grazes it, i.e. when

$$r = d.$$

Substituting $$r = d$$ in the expression $$r = \dfrac{mv}{qB_0}$$, we obtain

$$d = \frac{mv_{\min}}{qB_0} \;\;\Longrightarrow\;\; v_{\min} = \frac{qB_0\,d}{m}.$$

Thus the least speed required so that the particle never reaches the screen is

$$v_{\min} = \frac{q\,d\,B_0}{m}.$$

Comparing with the given options, this corresponds to Option C.

Hence, the correct answer is Option C.

We have a stream of protons that enter a uniform magnetic field with speed $$v = 4 \times 10^{5}\,\text{m s}^{-1}$$ making an angle $$\theta = 60^{\circ}$$ with the field. Because the velocity has a component parallel to the field and another component perpendicular to it, each proton follows a helical path. The physical quantity we are asked to find is the pitch of that helix, i.e. the distance travelled parallel to the magnetic field in one complete revolution about the field lines.

First we resolve the velocity into two perpendicular components. Using $$v_{\parallel}=v\cos\theta$$ for the component parallel to the field and $$v_{\perp}=v\sin\theta$$ for the component perpendicular to the field, we write

$$v_{\parallel}=v\cos 60^{\circ}=4\times10^{5}\times\frac12=2\times10^{5}\;\text{m s}^{-1}.$$

The perpendicular component is not directly required for the pitch, but it is this component that causes the circular motion. The circular motion has an angular frequency $$\omega$$ determined by the magnetic force. For a charge $$q$$ moving in a uniform magnetic field $$B$$, the formula for the angular frequency is stated as

$$\omega=\frac{qB}{m},$$

where $$m$$ is the mass of the particle. Substituting the given values $$q=1.69\times10^{-19}\;\text{C},\; B=0.3\;\text{T},\; m=1.67\times10^{-27}\;\text{kg},$$ we obtain

$$\omega=\frac{1.69\times10^{-19}\times0.3}{1.67\times10^{-27}}.$$

Multiplying in the numerator gives

$$1.69\times0.3=0.507,$$

so

$$\omega=\frac{0.507\times10^{-19}}{1.67\times10^{-27}} =\frac{5.07\times10^{-20}}{1.67\times10^{-27}} =3.03\times10^{7}\;\text{s}^{-1}\;(\text{approximately}).$$

The time period $$T$$ of one revolution is the reciprocal of the frequency, and for angular frequency we use

$$T=\frac{2\pi}{\omega}.$$

Substituting $$\omega=3.03\times10^{7}\;\text{s}^{-1}$$ gives

$$T=\frac{2\pi}{3.03\times10^{7}} =\frac{6.283}{3.03}\times10^{-7} =2.07\times10^{-7}\;\text{s}.$$

Now the pitch $$p$$ is the distance travelled along the magnetic field in one time period:

$$p=v_{\parallel}\,T.$$

Using $$v_{\parallel}=2\times10^{5}\;\text{m s}^{-1}$$ and $$T=2.07\times10^{-7}\;\text{s},$$ we have

$$p=2\times10^{5}\times2.07\times10^{-7} =4.14\times10^{-2}\;\text{m}.$$

To express this in centimetres, we multiply by 100:

$$p=4.14\times10^{-2}\times100=4.14\;\text{cm}.$$

The closest value among the options is 4 cm.

Hence, the correct answer is Option D.

We need the magnetic force on a moving charge. The magnetic (Lorentz) force formula is stated first:

$$\vec{F}=q\,\left(\vec{v}\times\vec{B}\right).$$

Here the charge is $$q = 1\,\mu\text{C}=1\times10^{-6}\,\text{C},$$ the velocity is $$\vec{v}=2\hat i+3\hat j+4\hat k\;\text{m s}^{-1},$$ and the magnetic field is $$\vec{B}=\left(5\hat i+3\hat j-6\hat k\right)\times10^{-3}\,\text{T}.$$

We first evaluate the cross product $$\vec{v}\times\vec{B}.$$ Writing out the determinant:

$$$ \vec{v}\times\vec{B}= \begin{vmatrix} \hat i & \hat j & \hat k\\[4pt] 2 & 3 & 4\\[4pt] 5\times10^{-3} & 3\times10^{-3} & -6\times10^{-3} \end{vmatrix}. $$$

Because every component of $$\vec{B}$$ already contains the common factor $$10^{-3},$$ we keep that factor outside for clarity and perform the determinant with the numerical parts $$5,\,3,\, -6$$ only. Thus,

$$$ \vec{v}\times\vec{B}=10^{-3}\! \begin{vmatrix} \hat i & \hat j & \hat k\\[4pt] 2 & 3 & 4\\[4pt] 5 & 3 & -6 \end{vmatrix}. $$$

Expanding the 3×3 determinant step by step:

$$$ \begin{aligned} \vec{v}\times\vec{B} &=10^{-3}\Bigl[\hat i\,(3\!\cdot\!(-6)-4\!\cdot\!3)\;-\;\hat j\,(2\!\cdot\!(-6)-4\!\cdot\!5)\;+\;\hat k\,(2\!\cdot\!3-3\!\cdot\!5)\Bigr] \\[4pt] &=10^{-3}\Bigl[\hat i\,(-18-12)\;-\;\hat j\,(-12-20)\;+\;\hat k\,(6-15)\Bigr] \\[4pt] &=10^{-3}\Bigl[\hat i\,(-30)\;+\;\hat j\,(32)\;+\;\hat k\,(-9)\Bigr] \\[4pt] &=\left(-30\hat i+32\hat j-9\hat k\right)\times10^{-3}. \end{aligned} $$$

Now we multiply this result by the charge $$q=1\times10^{-6}\,\text{C}$$ to obtain the force:

$$$ \vec{F}=q\left(\vec{v}\times\vec{B}\right) =1\times10^{-6}\,\text{C}\;\times\;\left(-30\hat i+32\hat j-9\hat k\right)\times10^{-3}. $$$

Combining the powers of ten, $$10^{-6}\times10^{-3}=10^{-9},$$ so

$$$ \vec{F}=\left(-30\hat i+32\hat j-9\hat k\right)\times10^{-9}\,\text{N}. $$$

The problem statement itself writes the force in the form $$\vec{F}\times10^{-9}\,\text{N},$$ so the required vector $$\vec{F}$$ is simply

$$\vec{F}=-30\hat i+32\hat j-9\hat k.$$ Hence, the correct answer is Option B.

We have an electron whose velocity is strictly along the $$+x$$-axis, so we write

$$\vec v = 6 \times 10^{6}\,\text{m s}^{-1}\,\hat i.$$

The region also possesses a uniform electric field pointing along $$+y$$, that is

$$\vec E = 300\,\text{V cm}^{-1}\,\hat j.$$

First, we convert this electric field into SI units. Since $$1\,\text{cm}=10^{-2}\,\text{m},$$ we obtain

$$300\,\text{V cm}^{-1}=300 \times 10^{2}\,\text{V m}^{-1}=3.0 \times 10^{4}\,\text{V m}^{-1}.$$

For a charge $$q$$ moving in simultaneous electric and magnetic fields, the Lorentz force formula is stated as

$$\vec F = q\bigl(\vec E + \vec v \times \vec B\bigr).$$

We want the electron to continue moving only along the $$x$$-direction, which means the net transverse force must vanish. Hence we require

$$\vec F = \vec 0 \Longrightarrow \vec E + \vec v \times \vec B = \vec 0.$$

Re-arranging gives

$$\vec v \times \vec B = -\vec E.$$

Because the electron carries charge $$q=-e,$$ its electric force is $$\vec F_E = -e\vec E.$$ Since $$\vec E$$ is along $$+y,$$ $$\vec F_E$$ is along $$-y.$$ Therefore the magnetic force $$\vec F_B = q(\vec v \times \vec B)$$ must point along $$+y$$ in order to exactly cancel $$\vec F_E.$$ To secure that direction we inspect the cross-product:

$$\vec v = v\,\hat i, \qquad \vec B = B\,\hat k \;(\text{assume}).$$

Using the right-hand rule,

$$\hat i \times \hat k = -\hat j,$$

so

$$\vec v \times \vec B = vB(-\hat j).$$

Multiplying by the electron’s charge $$q=-e$$ gives

$$q(\vec v \times \vec B) = (-e)\bigl(vB(-\hat j)\bigr)=+e vB\,\hat j,$$

which indeed points along $$+y$$ as required. Hence our initial assumption that $$\vec B$$ is along $$+z$$ ($$\hat k$$) is correct. The magnetic field must therefore be directed along $$+z$$.

To find its magnitude, we equate the magnitudes of the electric and magnetic forces:

$$eE = e v B \quad\Longrightarrow\quad B = \frac{E}{v}.$$

Substituting $$E = 3.0 \times 10^{4}\,\text{V m}^{-1}$$ and $$v = 6 \times 10^{6}\,\text{m s}^{-1},$$ we obtain

$$B = \frac{3.0 \times 10^{4}}{6 \times 10^{6}}\,\text{T} = 0.5 \times 10^{-2}\,\text{T} = 5 \times 10^{-3}\,\text{T}.$$

Thus, a magnetic field of magnitude $$5 \times 10^{-3}\,\text{T}$$ directed along $$+z$$ will ensure the electron continues straight along the $$x$$-axis.

Hence, the correct answer is Option C.

We consider a very long straight wire of radius $$a$$ that carries a steady current $$I$$. The current is said to be uniformly distributed over the circular cross-section, which means the current density $$J$$ (current per unit area) is the same at every point inside the wire.

Because the situation is cylindrically symmetric, we use Ampère’s circuital law in its integral form, which states

$$\oint \vec B \cdot d\vec l \;=\;\mu_0 I_{\text{encl}}$$

where $$I_{\text{encl}}$$ is the current enclosed by the chosen circular Amperian path and $$\mu_0$$ is the permeability of free space. For a circle of radius $$r$$ centred on the axis of the wire, $$\vec B$$ is tangential and has the same magnitude everywhere on the circle, so the left side becomes $$B\,(2\pi r)$$. Thus

$$B\,(2\pi r)=\mu_0 I_{\text{encl}} \quad\Longrightarrow\quad B=\frac{\mu_0 I_{\text{encl}}}{2\pi r}$$

Now we need the magnetic field at two different distances from the axis: (i) at $$r_1=\dfrac{a}{3}$$ (inside the wire) and (ii) at $$r_2=2a$$ (outside the wire).

1. Field at $$r_1=\dfrac{a}{3}$$ (inside the wire)

Because the point lies inside the conductor, only the current contained in the smaller circle of radius $$r_1$$ contributes. The current density is

$$J=\frac{I}{\pi a^{2}}\;,$$

so the enclosed current is

$$I_{\text{encl}}=J\;(\text{area within }r_1)=\frac{I}{\pi a^{2}}\;(\pi r_1^{2})=\frac{I r_1^{2}}{a^{2}}.$$

Substituting $$r_1=\dfrac{a}{3}$$ gives

$$I_{\text{encl}}=\frac{I}{a^{2}}\left(\frac{a}{3}\right)^{2}=\frac{I a^{2}}{9a^{2}}=\frac{I}{9}.$$

Putting this value into the expression for $$B$$, we get the magnetic field at $$r_1$$:

$$B_1=\frac{\mu_0 I_{\text{encl}}}{2\pi r_1} =\frac{\mu_0 \left(\dfrac{I}{9}\right)}{2\pi \left(\dfrac{a}{3}\right)} =\frac{\mu_0 I}{9}\;\frac{3}{2\pi a} =\frac{\mu_0 I}{6\pi a}.$$

2. Field at $$r_2=2a$$ (outside the wire)

For a point outside, the entire current $$I$$ is enclosed, i.e., $$I_{\text{encl}}=I$$. Using the same formula, we have

$$B_2=\frac{\mu_0 I}{2\pi r_2} =\frac{\mu_0 I}{2\pi (2a)} =\frac{\mu_0 I}{4\pi a}.$$

3. Required ratio $$\dfrac{B_1}{B_2}$$

We now divide $$B_1$$ by $$B_2$$:

$$\frac{B_1}{B_2} =\frac{\dfrac{\mu_0 I}{6\pi a}}{\dfrac{\mu_0 I}{4\pi a}} =\left(\frac{\mu_0 I}{6\pi a}\right)\left(\frac{4\pi a}{\mu_0 I}\right) =\frac{4}{6} =\frac{2}{3}.$$

Hence, the ratio of the magnetic fields at the two specified distances is $$\dfrac{2}{3}$$.

Hence, the correct answer is Option A.

Let the centre of the circular portion be O and its radius be $$R$$. We have to add the magnetic fields produced at O by

(i) the complete circular turn $$DMND$$ and
(ii) the two very long straight parts $$AB$$ and $$BC$$ which touch the circle tangentially at the points $$N$$ and $$D$$ respectively.

Field due to the circular turn
For a circular loop carrying current $$I$$, the magnetic field at its centre is given by the well-known result

$$B_{\text{circle}}=\frac{\mu_0 I}{2R}\;.$$

It will be directed perpendicular to the plane of the loop. Writing the same expression with a denominator $$2\pi R$$ (so that all later terms have a common factor) we have

$$B_{\text{circle}}=\frac{\mu_0 I}{2\pi R}\,\pi\;.$$

Field due to the straight part AB
For a finite straight conductor the Biot-Savart result is

$$B=\frac{\mu_0 I}{4\pi r}\,(\sin\theta_1+\sin\theta_2),$$

where $$r$$ is the perpendicular distance of the point from the wire and $$\theta_1,\;\theta_2$$ are the angles which the lines joining the point to the two ends of the conductor make with the conductor itself.

The perpendicular distance of O from AB is clearly $$R$$ (AB is a tangent). One end of AB (the point A) is at infinity, so the line AO is almost along AB itself; hence $$\theta_A=0^\circ$$ and $$\sin\theta_A=0$$. The other end is B. Because B has the coordinates $$(R,\;R)$$ while the wire AB is horizontal, the line OB makes an angle of $$45^\circ$$ with AB, giving $$\theta_B=45^\circ$$ and $$\sin\theta_B=\dfrac1{\sqrt2}$$. Substituting,

$$B_{AB}=\frac{\mu_0 I}{4\pi R}\left(0+\frac1{\sqrt2}\right)=\frac{\mu_0 I}{4\pi R\sqrt2}\;.$$

Field due to the straight part BC
Exactly the same geometry holds for BC, because BC is a vertical tangent lying a distance $$R$$ from O. Again one end (C) is at infinity giving $$\sin\theta_C=0$$, while the nearer end B subtends $$45^\circ$$ with BC giving $$\sin\theta_B=\dfrac1{\sqrt2}$$. Therefore

$$B_{BC}=\frac{\mu_0 I}{4\pi R}\left(\frac1{\sqrt2}+0\right)=\frac{\mu_0 I}{4\pi R\sqrt2}\;.$$

Directions of the straight-wire fields
If the current reaches the junction B from A (i.e.\ it flows to the left along AB) and then turns to flow upward along BC, the right-hand rule shows that both $$B_{AB}$$ and $$B_{BC}$$ point out of the plane of the paper. Hence the two contributions add:

$$B_{\text{straight}}=B_{AB}+B_{BC}=\frac{\mu_0 I}{4\pi R\sqrt2}+\frac{\mu_0 I}{4\pi R\sqrt2} =\frac{\mu_0 I}{2\pi R\sqrt2}\;.$$

Total field at the centre
Adding the circular and straight-wire fields (both are perpendicular to the plane and hence add algebraically)

$$\begin{aligned} B_{\text{net}}&=B_{\text{circle}}+B_{\text{straight}}\\[4pt] &=\frac{\mu_0 I}{2\pi R}\,\pi+\frac{\mu_0 I}{2\pi R}\,\frac1{\sqrt2}\\[4pt] &=\frac{\mu_0 I}{2\pi R}\left(\pi+\frac1{\sqrt2}\right). \end{aligned}$$

Hence, the correct answer is Option A.

We are given that a proton possesses a kinetic energy of $$1\ \text{MeV}$$ and is travelling from south to north. First, we convert this energy into joules because the usual mechanical formulas employ SI units.

$$1\ \text{MeV}=1\times10^{6}\ \text{eV}=1\times10^{6}\times1.6\times10^{-19}\ \text{J}=1.6\times10^{-13}\ \text{J}$$

The classical relation between kinetic energy and speed is stated as

$$K=\frac12\,m\,v^{2}$$

where $$K$$ is the kinetic energy, $$m$$ the mass and $$v$$ the speed. Substituting $$K=1.6\times10^{-13}\ \text{J}$$ and $$m=1.6\times10^{-27}\ \text{kg}$$ (rest mass of the proton), we get

$$1.6\times10^{-13}=\frac12\,(1.6\times10^{-27})\,v^{2}$$

Multiplying both sides by 2 to remove the fraction,

$$2\times1.6\times10^{-13}=(1.6\times10^{-27})\,v^{2}$$

Notice that the factor $$1.6$$ appears on both sides, so it cancels:

$$2\times10^{-13}=10^{-27}\,v^{2}$$

Dividing by $$10^{-27}$$, we have

$$v^{2}=2\times10^{14}$$

Taking the square root,

$$v=\sqrt{2}\times10^{7}\ \text{m s}^{-1}\approx1.414\times10^{7}\ \text{m s}^{-1}$$

Now, the proton experiences an acceleration of $$10^{12}\ \text{m s}^{-2}$$ due to a magnetic field directed from west to east, while its velocity is south to north. The magnetic force formula is first stated:

$$\mathbf{F}=q\,\mathbf{v}\times\mathbf{B}$$

The magnitude of this force when $$\mathbf{v}$$ and $$\mathbf{B}$$ are perpendicular is

$$F=q\,v\,B$$

According to Newton’s second law, this force produces an acceleration $$a$$, so

$$F=m\,a$$

Equating the two expressions for the force gives

$$m\,a=q\,v\,B$$

Solving for the magnetic field magnitude $$B$$,

$$B=\frac{m\,a}{q\,v}$$

We substitute $$m=1.6\times10^{-27}\ \text{kg}$$, $$a=10^{12}\ \text{m s}^{-2}$$, $$q=1.6\times10^{-19}\ \text{C}$$ and $$v=1.414\times10^{7}\ \text{m s}^{-1}$$:

$$B=\frac{(1.6\times10^{-27})(10^{12})}{(1.6\times10^{-19})(1.414\times10^{7})}$$

Simplifying the numerator,

$$1.6\times10^{-27}\times10^{12}=1.6\times10^{-15}$$

Simplifying the denominator,

$$(1.6\times10^{-19})(1.414\times10^{7})=1.6\times1.414\times10^{-12}=2.2624\times10^{-12}$$

Hence,

$$B=\frac{1.6\times10^{-15}}{2.2624\times10^{-12}}=\frac{1.6}{2.2624}\times10^{-3}\ \text{T}$$

Calculating the numerical ratio,

$$\frac{1.6}{2.2624}\approx0.707$$

Therefore,

$$B\approx0.707\times10^{-3}\ \text{T}=0.707\ \text{mT}$$

This value matches closely with $$0.71\ \text{mT}$$.

Hence, the correct answer is Option A.

We start with the definition of magnetic moment for a planar current loop. A charge $$q$$ completing one revolution in time period $$T$$ constitutes a current

$$I \;=\;\frac{q}{T}.$$

The particle moves in a circle of radius $$r$$ with speed $$v$$, so its period is obtained from the well-known relation $$T=\frac{\text{circumference}}{\text{speed}}$$ :

$$T \;=\;\frac{2\pi r}{v}.$$

Substituting this value of $$T$$ in the formula for current, we obtain

$$I \;=\;\frac{q}{\dfrac{2\pi r}{v}} \;=\;\frac{qv}{2\pi r}.$$

The area of the circular path is

$$A \;=\;\pi r^{2}.$$

By definition the magnitude of the magnetic moment is

$$\mu \;=\; I\,A.$$

Inserting the expressions for $$I$$ and $$A$$ gives

$$\mu \;=\;\left(\frac{qv}{2\pi r}\right)\!(\pi r^{2}) \;=\;\frac{qvr}{2}.$$

Its direction is perpendicular to the plane of the circle, along the unit vector $$\hat n$$ obtained from the right-hand rule for a positive charge. Hence in vector form

$$\vec\mu \;=\;\frac{qvr}{2}\,\hat n.$$

Next, we express $$r$$ in terms of the given magnetic field $$\vec B$$. Because the velocity is perpendicular to $$\vec B$$, the magnetic Lorentz force provides the necessary centripetal force. Writing this equality explicitly,

$$qvB \;=\;\frac{mv^{2}}{r}.$$

Solving for the radius,

$$r \;=\;\frac{mv}{qB}.$$

Substituting this value of $$r$$ back into $$\vec\mu$$, we obtain

$$\vec\mu \;=\;\frac{qv}{2}\left(\frac{mv}{qB}\right)\hat n \;=\;\frac{mv^{2}}{2B}\,\hat n.$$

The angular momentum $$\vec L$$ of the particle is $$\vec L = mvr\,\hat n$$, and for a positive charge $$\vec\mu$$ is parallel to $$\vec L$$, whereas for a negative charge it is antiparallel. The options given contain a negative sign, so the particle is to be regarded as negatively charged. Thus we write

$$\vec\mu \;=\;-\frac{mv^{2}}{2B}\,\hat B,$$

because $$\hat n$$ is opposite to $$\hat B$$ for a negative charge in this standard configuration. Since $$\hat B = \dfrac{\vec B}{B}$$, we finally have

$$\vec\mu \;=\;-\frac{mv^{2}}{2B^{2}}\;\vec B.$$

This matches Option D.

Hence, the correct answer is Option D.

We have a flat circular coil whose plane is kept vertical at the beginning, so the magnetic moment vector $$\vec \mu$$ is horizontal while the external uniform magnetic field $$\vec B$$ is vertical. Hence the initial angle between the two vectors is $$\theta_0 = 90^{\circ}$$.

The coil can rotate freely about one of its horizontal diameters. When the field is suddenly switched on, the magnetic torque tries to turn the coil so that $$\vec \mu$$ may align with $$\vec B$$. We are told that the coil actually rotates through an angle of $$60^{\circ}$$ about the horizontal diameter. Therefore the new angle between $$\vec \mu$$ and $$\vec B$$ becomes

$$\theta = 90^{\circ} - 60^{\circ} = 60^{\circ}.$$

The magnetic potential energy of a magnetic dipole in a uniform magnetic field is given first - we state the formula explicitly:

$$U = -\,\vec \mu \cdot \vec B = -\,\mu B \cos\theta.$$

Initially (at $$\theta_0 = 90^{\circ}$$),

$$U_0 = -\,\mu B \cos 90^{\circ} = -\,\mu B \,(0) = 0.$$

After the coil has turned (at $$\theta = 60^{\circ}$$),

$$U = -\,\mu B \cos 60^{\circ} = -\,\mu B \left(\dfrac12\right) = -\dfrac{\mu B}{2}.$$

Thus the change in potential energy experienced by the coil is

$$\Delta U = U - U_0 = \left(-\dfrac{\mu B}{2}\right) - 0 = -\dfrac{\mu B}{2}.$$

The negative sign shows that the energy of the system has decreased; that lost potential energy appears as rotational kinetic energy. By conservation of mechanical energy:

$$\dfrac12 I \omega^{2} = -\Delta U = \dfrac{\mu B}{2}.$$

We now insert the numerical values given in the problem:

Moment of inertia: $$I = 0.8 \text{ kg m}^2,$$
Magnetic moment: $$\mu = 20 \text{ A m}^2,$$
Magnetic field: $$B = 4 \text{ T}.$$

Substituting these values into the energy-conservation equation, we get

$$\dfrac12 \,(0.8)\,\omega^{2} = \dfrac{(20)(4)}{2}.$$

Simplifying the right-hand side first,

$$\dfrac{(20)(4)}{2} = \dfrac{80}{2} = 40,$$

so the equation becomes

$$0.4\,\omega^{2} = 40.$$

Solving step by step:

$$\omega^{2} = \dfrac{40}{0.4} = 100,$$

and therefore

$$\omega = \sqrt{100} = 10 \text{ rad s}^{-1}.$$

Hence, the correct answer is Option A.