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Magnetic Effects and Magnetism JEE Notes: Important Concepts
The chapter sits at the junction of electricity and magnetism and is a guaranteed question in both JEE Main and JEE Advanced. Revise these core ideas first:
- Biot–Savart Law – elemental field due to a current element.
- Ampere’s Circuital Law – integral form useful for symmetric conductors.
- Lorentz Force – force on moving charge and on current-carrying conductors.
- Magnetic Dipole & Torque – bar magnet analogy, potential energy in an external field.
- Classification of Materials – diamagnetic, paramagnetic, ferromagnetic, with susceptibility $$\chi_m$$ and relative permeability $$\mu_r$$.
- Earth’s Magnetism – magnetic elements (declination, dip, total intensity).
Mastering these pillars helps in solving multi-concept numerical problems that combine electricity, magnetism and mechanics.
Magnetic Field Due to Electric Currents
Biot–Savart Law (Microscopic Form)
For a current element $$I\,d\mathbf{l}$$ at position vector $$\mathbf{r}$$:
$$ d\mathbf{B}=\frac{\mu_0}{4\pi}\frac{I\,d\mathbf{l}\times \hat{\mathbf{r}}}{r^2} $$
- Direction: right-hand rule with thumb along $$I$$ and curl towards the point.
- Used for segments, arcs and loops with integration.
Standard Results Worth Memorising
| Conductor | Magnetic field magnitude $$B$$ | Direction |
|---|---|---|
| Infinite straight wire, distance $$r$$ | $$ B=\dfrac{\mu_0 I}{2\pi r} $$ | Circular around wire (right-hand rule) |
| Circular loop, centre, radius $$R$$ | $$ B=\dfrac{\mu_0 I}{2R} $$ | Along axis by right-hand rule |
| Long solenoid (n turns per unit length, current $$I$$) | $$ B=\mu_0 n I $$ (inside) | Parallel to axis; zero outside (ideal) |
| Toroid (mean radius $$R$$, N turns) | $$ B=\dfrac{\mu_0 N I}{2\pi R} $$ (inside core) | Tangential, confined within core |
Ampere’s Circuital Law (Macroscopic Form)
For closed loop $$\mathcal{C}$$:
$$ \oint_{\mathcal{C}}\mathbf{B}\cdot d\mathbf{l}= \mu_0 I_{\text{enc}} $$
This shortcut avoids integration when symmetry is high (infinite wire, solenoid, toroid, infinite sheet).
Velocity Selector and Cyclotron Motion
- When uniform $$\mathbf{E}$$ and $$\mathbf{B}$$ fields are perpendicular, only particles with velocity $$v=E/B$$ pass undeflected.
- Circular motion radius $$r=\dfrac{m v}{q B}$$, cyclotron frequency $$\omega=\dfrac{q B}{m}$$ independent of speed.
Solved Example
Example 1
A square loop of side 5 cm carries 4 A. Find the magnitude of the magnetic field at its centre.
Solution:
- Each side behaves like a finite straight segment of length $$a=5\;\text{cm}=0.05\;\text{m}$$, at centre distance $$r=\dfrac{a}{\sqrt{2}}$$ and angle subtended $$\theta=45^{\circ}$$ per end.
- Field due to one side: $$B_{\text{side}}=\dfrac{\mu_0 I}{4\pi r}(\sin\theta_1+\sin\theta_2)=\dfrac{\mu_0 I}{4\pi r}(2\sin45^{\circ})=\dfrac{\mu_0 I}{2\pi r}\dfrac{\sqrt2}{2}=\dfrac{\mu_0 I}{2\pi r}\dfrac{1}{\sqrt2}$$.
- Net field: $$B=4B_{\text{side}}=\dfrac{2\mu_0 I}{\pi r\sqrt2}=\dfrac{2\mu_0 (4)}{\pi (0.05/\sqrt2)\sqrt2}= \dfrac{8\mu_0}{\pi \times 0.05}= \dfrac{160\mu_0}{\pi}\;\text{T}$$.
- Putting $$\mu_0=4\pi\times10^{-7}$$ H/m gives $$B \approx 2.0\times10^{-4}\;\text{T}$$.
Answer: $$2.0\times10^{-4}\;\text{T}$$
For a larger problem set on current-carrying geometries, attempt the mixed difficulty questions inside JEE Questions after finishing this section.
Force on Charges and Current Elements
Lorentz Force on a Moving Charge
$$ \mathbf{F}=q\left(\mathbf{E}+\mathbf{v}\times\mathbf{B}\right) $$
- Magnitude in pure magnetic field: $$F=q v B\sin\theta$$.
- Direction: right-hand rule for $$\mathbf{v}\times\mathbf{B}$$, opposite for negative charge.
Force on Current-carrying Conductor
For length vector $$\mathbf{L}$$ in field $$\mathbf{B}$$:
$$ \mathbf{F}=I\,\mathbf{L}\times\mathbf{B} $$
Torque and Potential Energy of a Magnetic Dipole
- Magnetic dipole moment for current loop: $$\mathbf{m}=I\,\mathbf{A}$$ (direction by right-hand thumb).
- Torque: $$\boldsymbol{\tau}=\mathbf{m}\times\mathbf{B}$$.
- Potential energy: $$U=-\mathbf{m}\cdot\mathbf{B}$$.
Moving-Coil Galvanometer Philosophy
The pointer turns because coil experiences torque $$\tau=NIAB\sin\theta$$, opposed by spring torque $$k\phi$$ → deflection $$\phi \propto I$$ for small angles.
Solved Example
Example 2
An electron enters a uniform magnetic field of 0.2 T with speed $$3\times10^6$$ m/s perpendicular to the field. Calculate the radius of its path.
Solution: $$ r=\dfrac{m v}{q B}=\dfrac{9.11\times10^{-31}\times3\times10^{6}}{1.6\times10^{-19}\times0.2}=8.54\times10^{-4}\;\text{m}=0.854\;\text{mm}. $$
Answer: 0.85 mm
A similar concept often appears in JEE Advanced Previous Papers where the magnetic field varies with time or space.
Magnetism in Materials and Earth’s Magnetism
Key Definitions
| Quantity | Symbol | Relation |
|---|---|---|
| Magnetisation | $$\mathbf{M}$$ | Magnetic moment per unit volume |
| Magnetic susceptibility | $$\chi_m$$ | $$\mathbf{M}=\chi_m \mathbf{H}$$ |
| Relative permeability | $$\mu_r$$ | $$\mu_r=1+\chi_m$$ |
| Intensity of magnetisation (or magnetising field) | $$\mathbf{H}$$ | $$\mathbf{B}=\mu_0(\mathbf{H}+\mathbf{M})$$ |
Types of Magnetic Materials
- Diamagnetic: $$\chi_m < 0$$, weak repulsion, examples: Bi, Cu.
- Paramagnetic: $$\chi_m>0$$ but small, weak attraction, examples: Al, Pt.
- Ferromagnetic: large $$\chi_m$$ (>1000), spontaneous magnetisation, hysteresis loop, examples: Fe, Co, Ni.
Hysteresis Curve
Area enclosed equals energy loss per cycle per unit volume: $$W=\mu_0 \oint H\,dB$$. Soft iron has narrow loop (low energy loss), steel has wide loop (permanent magnets).
Earth’s Magnetism
- Magnetic declination (D) – angle between geographic meridian and magnetic meridian.
- Magnetic dip (I) – angle made by total intensity $$\mathbf{B}_E$$ with horizontal.
- Components: $$B_H=B_E\cos I$$ and $$B_V=B_E\sin I$$.
Solved Example
Example 3
At a location $$B_H = 25\;\mu\text{T}$$ and dip $$I = 60^{\circ}$$. Calculate the earth’s total magnetic field magnitude.
Solution: $$ B_E=\dfrac{B_H}{\cos I}=\dfrac{25\times10^{-6}}{0.5}=50\times10^{-6}\;\text{T}=50\;\mu\text{T}. $$
Answer: 50 µT
Important Formulas and Results at a Glance
Download the full physics sheet from JEE Formula Sheets after you tick every box in the table below.
| Topic | Formula | Remarks |
|---|---|---|
| Biot–Savart elemental field | $$ dB=\frac{\mu_0}{4\pi}\frac{I\,dl\sin\theta}{r^2} $$ | Integration gives standard results. |
| Magnetic field on axis of loop | $$ B=\frac{\mu_0 I R^2}{2(R^2+x^2)^{3/2}} $$ | $$x\to0$$ gives centre value $$\mu_0 I/2R$$. |
| Force between two parallel wires | $$ \frac{F}{L}=\frac{\mu_0 I_1 I_2}{2\pi d} $$ | Attractive for same direction currents. |
| Magnetic moment of current loop | $$ m=I A $$ | Vector normal by right-hand rule. |
| Torque on dipole | $$ \tau = m B \sin\theta $$ | Equilibrium when $$\theta=0,\,\pi$$. |
| Lorentz force | $$ F=q v B\sin\theta $$ | Centripetal in uniform $$B$$. |
| Cyclotron frequency | $$ f=\dfrac{q B}{2\pi m} $$ | Independent of radius. |
| Magnetic energy density | $$ u=\dfrac{B^2}{2\mu_0} $$ | Used in energy problems for solenoid. |
| Hysteresis loss | $$ W=\mu_0\oint H\,dB $$ | Area of BH loop. |
| Relation of $$B,\,H,\,M$$ | $$ B=\mu_0(H+M) $$ | In linear materials $$M=\chi_m H$$. |
$$ B=\mu_0 n I $$ (Ideal long solenoid)
$$ \oint\mathbf{B}\cdot d\mathbf{l} = \mu_0 I_{\text{enc}} $$ (Ampere’s circuital law)
JEE Important Points, Common Mistakes and Quick Revision
High-Yield Points
- Remember the sign of susceptibility. Many students wrongly write $$\chi_m$$ positive for diamagnets.
- For finite conductors, always use the angle form $$B=\dfrac{\mu_0 I}{4\pi r}(\sin\theta_1+\sin\theta_2)$$, not the infinite-wire shortcut.
- Dipole approximation holds only when distance $$\gt$$ 10 × size of magnet. JEE sometimes checks this limit.
- In questions mixing $$\mathbf{E}$$ and $$\mathbf{B}$$, check whether electric part does work; pure magnetic force never changes speed.
Common Calculation Traps
- Missing $$\mu_0$$ in force between wires; write units to catch it.
- Using $$B_H$$ instead of $$B_E$$ while applying dip and declination formulas.
- Mistaking number of turns per unit length for total turns in solenoid questions.
Last-Minute Revision Routine
- Re-derive Biot–Savart for a straight wire without looking – takes 3 min, fixes sign errors.
- Write every standard $$B$$ value on flash cards; flip daily.
- Practise 15 mixed problems from the Jan and April papers in JEE Mains Previous Papers.
- End with two hysteresis numerical questions to keep units (J/m3) fresh
