Magnetic Fields and EM Waves

Magnetic Fields

• Biot-Savart Law:

• Mathematical expression: $$\frac{d\overrightarrow{B}={\mu }_{0}id\overrightarrow{l}\times \hat{r}}{4\pi r^2}$$

• Ampère’s Law:

• Relationship between magnetic field and current: $$\sum \overrightarrow{B}\cdot d\overrightarrow{l}={\mu }_0{I}_{enc}$$
• 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: $$\mathcal{E}=-\frac{d{\mathrm{\Phi }}_B}{dt}$$

• 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: $$(\frac{{\partial }^2}{\partial t^2}-v^2{\mathrm{\nabla }}^2)\overrightarrow{E}=0$$

• Transverse Nature:

• Oscillations are perpendicular to the direction of propagation.

• Plane EM Waves:

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

• Energy Density and Power:

• Poynting vector: $\overrightarrow{S}=\frac{1}{{\mu }_0}\overrightarrow{E}\times \overrightarrow{B}$

Additional Concepts:

• Displacement Current:

• Maxwell’s addition to Ampere’s law: $$I_d={ϵ}_0\frac{d{\mathrm{\Phi }}_E}{dt}$$

• 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).