Notes from Toppers

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).


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