Magnetic Fields and EM Waves

Magnetic Fields

• Biot-Savart Law:

• Mathematical expression: $$d \vec{B}={\mu_0}{id\vec{l}\times\hat{r}}\over{4\pi r^2}$$

• Ampère’s Law:

• Relationship between magnetic field and current: $$\sum \vec{B}\cdot d\vec{l}=\mu_0 I_{enc}$$
• Applications in finding the magnetic field created by simple current-carrying conductors like straight wires, solenoids, and toroids.

Electromagnetic Induction

• Faraday’s Law:

• Induced EMF: $$\mathcal{E}=-\frac{d\Phi_B}{dt}$$

• Lenz’s Law:

• Determining the direction of induced EMF and current.

• Applications: Generators, transformers, and motors.

• Motional EMF: $$\mathcal{E}=vBL\sin\theta$$

EM Waves

• Maxwell’s Equations:

• Four fundamental equations governing electromagnetic phenomena.

• Wave Equation:

• Describes the propagation of electromagnetic waves: $$\left( \frac{\partial^2}{\partial t^2}-v^2\nabla^2\right) \vec{E}=0$$

• Transverse Nature:

• Oscillations are perpendicular to the direction of propagation.

• Plane EM Waves:

• Mathematical description: $$\vec{E}=\hat{E_0} \cos(kz-\omega t)$$

• Energy Density and Power:

• Poynting vector: $\vec{S}=\frac{1}{\mu_0}\vec{E}\times\vec{B}$

Additional Concepts:

• Displacement Current:

• Maxwell’s addition to Ampere’s law: $$I_d=\epsilon_0\frac{d\Phi_E}{dt}$$

• Boundary Conditions:

• Relations for EM fields at interfaces between different media.

• Reflection and Refraction:

• Behavior of EM waves at dielectric interfaces.

• Polarization:

• Linear, circular, and elliptical polarization states.

References

• NCERT Physics, Class 11: Chapter 4 (Motion in a Plane), Chapter 5 (Laws of Motion), Chapter 6 (Work, Energy, and Power).
• NCERT Physics, Class 12: Chapter 6 (Electrostatics), Chapter 7 (Current Electricity), Chapter 8 (Magnetic Effects of Current), Chapter 9 (Electromagnetic Induction).