Magnetic Effect and Magnetism Formula For JEE 2026, PDF

The magnetic field $$d\vec{B}$$ at a point P due to a small current element $$Id\vec{l}$$ is:

$$$d\vec{B} = \frac{\mu_0}{4\pi} \cdot \frac{I \, d\vec{l} \times \hat{r}}{r^2}$$$

where:

• $$\mu_0 = 4\pi \times 10^{-7} \; \text{T}\cdot\text{m/A}$$ is the permeability of free space
• $$I$$ = current through the conductor
• $$d\vec{l}$$ = small length element in the direction of current
• $$\hat{r}$$ = unit vector from the element to point P
• $$r$$ = distance from the element to point P

The magnitude is: $$dB = \dfrac{\mu_0}{4\pi} \cdot \dfrac{I \, dl \, \sin\theta}{r^2}$$, where $$\theta$$ is the angle between $$d\vec{l}$$ and $$\hat{r}$$.

Magnetic Field Due to a Straight Wire

Finite wire (subtending angles $$\phi_1$$ and $$\phi_2$$ at perpendicular distance $$d$$):

$$$B = \frac{\mu_0 I}{4\pi d}(\sin\phi_1 + \sin\phi_2)$$$

Infinite wire ($$\phi_1 = \phi_2 = 90^\circ$$):

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

Semi-infinite wire (one end at the point, $$\phi_1 = 90^\circ$$, $$\phi_2 = 0^\circ$$):

$$$B = \frac{\mu_0 I}{4\pi d}$$$

Direction: Use the right-hand thumb rule.

At a point on the axis at distance $$x$$ from the centre of a circular loop of radius $$R$$ carrying current $$I$$:

$$$B = \frac{\mu_0 I R^2}{2(R^2 + x^2)^{3/2}}$$$

At the centre ($$x = 0$$):

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

For $$N$$ turns: multiply by $$N$$: $$B_{\text{centre}} = \dfrac{\mu_0 NI}{2R}$$

The line integral of the magnetic field around any closed loop (Amperian loop) equals $$\mu_0$$ times the net current enclosed:

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

• Choose an Amperian loop that exploits the symmetry of the problem
• $$I_{\text{enclosed}}$$ = net current passing through the area bounded by the loop
• The right-hand rule relates the direction of circulation to the positive current direction

Magnetic Field of a Solenoid

Inside a long solenoid (uniform field):

$$$B = \mu_0 n I$$$

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

At the end of a solenoid:

$$$B_{\text{end}} = \frac{\mu_0 n I}{2} = \frac{B_{\text{inside}}}{2}$$$

Magnetic Field of a Toroid

Inside the toroid (at mean radius $$r$$):

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

where $$N$$ = total turns and $$n = N/(2\pi r)$$ = turns per unit length.

$$B = 0$$ outside the toroid and inside the central hole.

The total electromagnetic force on a charge $$q$$ moving with velocity $$\vec{v}$$ in electric field $$\vec{E}$$ and magnetic field $$\vec{B}$$:

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

The magnetic part alone:

$$$\vec{F}_B = q\vec{v} \times \vec{B}$$$

• Magnitude: $$F_B = qvB\sin\theta$$, where $$\theta$$ is the angle between $$\vec{v}$$ and $$\vec{B}$$
• If $$\vec{v} \parallel \vec{B}$$ ($$\theta = 0^\circ$$): $$F = 0$$ (no force)
• If $$\vec{v} \perp \vec{B}$$ ($$\theta = 90^\circ$$): $$F = qvB$$ (maximum force)
• Direction: Use the right-hand rule (or Fleming's left-hand rule)
• The magnetic force does no work (it is always perpendicular to displacement)

Circular Motion in a Uniform Magnetic Field

When $$\vec{v} \perp \vec{B}$$, the particle moves in a circle:

• Radius: $$r = \dfrac{mv}{qB}$$
• Time period: $$T = \dfrac{2\pi m}{qB}$$ (independent of velocity!)
• Frequency: $$f = \dfrac{qB}{2\pi m}$$ (cyclotron frequency)
• Angular frequency: $$\omega = \dfrac{qB}{m}$$

When $$\vec{v}$$ has components both parallel and perpendicular to $$\vec{B}$$, the particle traces a helix (spiral path).

Pitch of helix: $$p = v_{\parallel} T = \dfrac{2\pi m v\cos\theta}{qB}$$

$$$\vec{F} = I\vec{l} \times \vec{B}$$$

• Magnitude: $$F = BIl\sin\theta$$
• $$I$$ = current, $$l$$ = length of conductor in the field, $$\theta$$ = angle between $$\vec{l}$$ and $$\vec{B}$$
• Maximum force when $$\theta = 90^\circ$$: $$F = BIl$$
• Zero force when $$\theta = 0^\circ$$ (wire parallel to field)
• Direction: Fleming's left-hand rule (Fore finger = Field, Middle finger = Current, Thumb = Force)

Force Between Parallel Wires

The force per unit length between two long parallel wires separated by distance $$d$$:

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

• Same direction currents: attract each other
• Opposite direction currents: repel each other

• Torque on coil: $$\tau = NBIA$$ (with radial field, $$\sin\theta = 1$$ always)
• At equilibrium: $$NBIA = k\phi$$, where $$k$$ = torsional spring constant, $$\phi$$ = angular deflection
• Current sensitivity: $$\dfrac{\phi}{I} = \dfrac{NBA}{k}$$ (deflection per unit current)
• Voltage sensitivity: $$\dfrac{\phi}{V} = \dfrac{NBA}{kR}$$ (deflection per unit voltage)

Conversion of Galvanometer

To Ammeter (measures large currents): Connect a small resistance $$S$$ (shunt) in parallel with the galvanometer:

$$$S = \frac{I_g G}{I - I_g}$$$

where $$G$$ = galvanometer resistance, $$I_g$$ = full-scale deflection current, $$I$$ = desired range.

To Voltmeter (measures voltage): Connect a high resistance $$R$$ in series:

$$$R = \frac{V}{I_g} - G$$$

where $$V$$ = desired voltage range.

Torque on a Magnetic Dipole

In a uniform magnetic field $$\vec{B}$$:

$$$\vec{\tau} = \vec{m} \times \vec{B}$$$

Magnitude: $$\tau = mB\sin\theta$$

• Maximum torque when $$\theta = 90^\circ$$ (dipole perpendicular to field)
• Zero torque when $$\theta = 0^\circ$$ or $$180^\circ$$ (dipole aligned with field)
• Potential energy: $$U = -\vec{m} \cdot \vec{B} = -mB\cos\theta$$
• Stable equilibrium: $$\theta = 0^\circ$$ ($$U = -mB$$, minimum)
• Unstable equilibrium: $$\theta = 180^\circ$$ ($$U = +mB$$, maximum)

A bar magnet with pole strength $$m_p$$ and length $$2l$$:

• Magnetic moment: $$M = m_p \times 2l$$
• Axial field (at distance $$r \gg l$$): $$B = \dfrac{\mu_0}{4\pi} \cdot \dfrac{2M}{r^3}$$
• Equatorial field (at distance $$r \gg l$$): $$B = \dfrac{\mu_0}{4\pi} \cdot \dfrac{M}{r^3}$$
• $$B_{\text{axial}} = 2 \times B_{\text{equatorial}}$$

Components of Earth's Magnetic Field

If $$B$$ is the total Earth's field and $$\phi$$ is the angle of dip:

• Horizontal component: $$B_H = B\cos\phi$$
• Vertical component: $$B_V = B\sin\phi$$
• $$\tan\phi = \dfrac{B_V}{B_H}$$
• $$B = \sqrt{B_H^2 + B_V^2}$$

Classification of Magnetic Materials

Key Concepts of Ferromagnetism

• Curie Temperature ($$T_C$$): Above this temperature, a ferromagnetic material becomes paramagnetic. For iron, $$T_C \approx 770^\circ$$C.
• Hysteresis: The lag of magnetization ($$B$$) behind the magnetizing field ($$H$$) when the material is taken through a cycle of magnetization.
• Retentivity: The magnetization remaining when $$H = 0$$ (after the material was fully magnetized).
• Coercivity: The reverse magnetizing field needed to bring the magnetization to zero.
• Area of hysteresis loop = energy lost per cycle as heat.