JEE Physics: Magnetism Complete Guide

Magnetism is a fundamental branch of physics essential for understanding natural phenomena and modern technology. In the JEE exam, magnetism is a vital topic involving magnetic fields, forces, dipoles, Earth's magnetism, and properties of materials. This comprehensive guide covers all crucial concepts, formulas, and sample problems to help you master magnetism for JEE preparation.

1. Introduction to Magnetism

Magnetism is a force experienced by moving charges and magnetic materials. It is closely related to electricity and is described by magnetic fields produced by electric currents and magnetic dipoles. Magnetic phenomena are caused by the motion of charges at atomic or macroscopic scales.

2. Magnetic Field and Magnetic Flux

The magnetic field \(\vec{B}\) is a vector field representing the magnetic influence on moving charges and magnets. It is measured in Tesla (T) or Weber per square meter.

Magnetic flux \(\Phi_B\) through an area \(A\) with magnetic field \(\vec{B}\) making angle \(\theta\):

$$\Phi_B = \int \vec{B} \cdot d\vec{A} = B A \cos \theta$$

3. Magnetic Force on Moving Charges and Currents

A charged particle \(q\) moving with velocity \(\vec{v}\) in magnetic field \(\vec{B}\) experiences Lorentz force:

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

This force is always perpendicular to both velocity and magnetic field, causing circular or helical motion.

3.1 Radius of Circular Path

For velocity perpendicular to \(\vec{B}\), radius \(r\):

$$r = \frac{m v}{|q| B}$$

3.2 Cyclotron Frequency

$$f = \frac{|q| B}{2 \pi m}$$

Frequency of revolution of charged particle in uniform magnetic field.

4. Magnetic Field Due to a Current

Electric currents produce magnetic fields. The magnitude and direction depend on current shape and position.

4.1 Biot-Savart Law

Magnetic field \(d\vec{B}\) due to current element \(I d\vec{l}\) at distance \(r\):

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

4.2 Magnetic Field Around Long Straight Wire

Using Ampere’s Law, magnetic field at distance \(r\):

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

4.3 Magnetic Field on Axis of Circular Loop

For loop radius \(R\), field at center:

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

4.4 Magnetic Field Inside Solenoid

For solenoid with turns per unit length \(n\):

$$B = \mu_0 n I$$

5. Force Between Current-Carrying Conductors

Two parallel wires carrying currents \(I_1\), \(I_2\) separated by distance \(d\) exert force per unit length:

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

Force is attractive if currents flow in same direction; repulsive if opposite.

6. Magnetic Dipole Moment

Current loops behave like magnetic dipoles with magnetic moment:

$$\vec{\mu} = I \vec{A}$$

Where \(A\) is loop area and direction is given by right-hand thumb rule.

7. Torque on a Current Loop in Magnetic Field

Torque experienced by loop carrying current \(I\) in magnetic field \(B\):

$$\tau = I A B \sin \theta$$

\(\theta\) is angle between normal to loop and magnetic field.

8. Magnetic Properties of Materials

Materials respond differently to magnetic fields:

Magnetic susceptibility \(\chi_m\) and permeability \(\mu\) define material response:

$$\vec{B} = \mu \vec{H} = \mu_0 (1 + \chi_m) \vec{H}$$

9. Earth's Magnetism

Earth behaves like a giant magnet with magnetic field approximated by a dipole.

10. Magnetic Circuits

Magnetic circuits work like electric circuits but for magnetic flux:

$$\Phi = \frac{NI}{\mathcal{R}}$$

Where \(N\) is number of turns, \(I\) current, and \(\mathcal{R}\) reluctance of magnetic path.

11. Electromagnetic Induction (Brief Overview)

Changing magnetic flux induces electromotive force (emf):

$$\mathcal{E} = -\frac{d\Phi_B}{dt}$$

Detailed study of this falls under Faraday’s laws and Lenz’s law.

12. Important Formulas Summary

Concept Formula Notes
Magnetic Flux $$\Phi_B = B A \cos \theta$$ Area \(A\), angle \(\theta\)
Magnetic Force on Charge $$\vec{F} = q \vec{v} \times \vec{B}$$ Velocity \(\vec{v}\), charge \(q\)
Radius of Circular Motion $$r = \frac{m v}{|q| B}$$ Mass \(m\), velocity \(v\)
Magnetic Field Long Wire $$B = \frac{\mu_0 I}{2\pi r}$$ Distance \(r\)
Force Between Wires $$\frac{F}{L} = \frac{\mu_0}{2\pi} \frac{I_1 I_2}{d}$$ Distance \(d\)
Magnetic Moment $$\vec{\mu} = I \vec{A}$$ Loop area \(A\)
Torque on Loop $$\tau = I A B \sin \theta$$ Angle \(\theta\)

13. Practice Problems

Problem 1: Magnetic Field at Center of Circular Loop

A circular loop of radius 0.1 m carries 5 A current. Find magnetic field at the center.

Solution:

$$B = \frac{\mu_0 I}{2 R} = \frac{4\pi \times 10^{-7} \times 5}{2 \times 0.1} = 3.14 \times 10^{-5} \, \text{T}$$

Problem 2: Force Between Two Parallel Wires

Two wires 0.5 m apart carry 3 A and 4 A currents in same direction. Find force per meter.

Solution:

$$\frac{F}{L} = \frac{\mu_0}{2\pi} \frac{I_1 I_2}{d} = \frac{4\pi \times 10^{-7}}{2\pi} \times \frac{3 \times 4}{0.5} = 4.8 \times 10^{-6} \, \text{N/m}$$

Problem 3: Torque on a Current Loop

A rectangular loop with area 0.02 \(m^2\) carries 2 A current in a 0.3 T magnetic field. Find torque if plane makes 60° angle with field.

Solution:

$$\tau = I A B \sin \theta = 2 \times 0.02 \times 0.3 \times \sin 60^\circ = 0.0208 \, \text{Nm}$$

Problem 4: Radius of Circular Path for Electron

Electron moves at speed \(10^6\) m/s perpendicular to 0.01 T magnetic field. Find radius of circular path. (Mass of electron \(9.1 \times 10^{-31} kg\), charge \(1.6 \times 10^{-19} C\))

Solution:

$$r = \frac{m v}{q B} = \frac{9.1 \times 10^{-31} \times 10^6}{1.6 \times 10^{-19} \times 0.01} = 5.7 \times 10^{-3} \, m$$

Conclusion

Understanding magnetism requires mastering the behavior of magnetic fields, forces on charges and currents, magnetic dipoles, and material properties. Regular practice of problems and conceptual clarity are key for excelling in JEE Physics. Use this comprehensive guide as a reference and practice platform to strengthen your fundamentals in magnetism.