Detailed Notes on Electromagnetic Theory and Waves

MAXWELL'S EQUATIONS

• Gauss's Law:

  • $\nabla \cdot \mathbf{E} = \frac{Q<em>{encl}}{\epsilon</em>0}$

  • Relates electric field to charge density.

• Gauss's Law for Magnetism:

  • $\nabla \cdot \mathbf{B} = 0$

  • Indicates there are no magnetic monopoles.

• Ampere's Law:

  • $\nabla \times \mathbf{B} = \mu<em>0 \mathbf{j} + \epsilon</em>0 \frac{\partial \mathbf{E}}{\partial t}$

  • Relates magnetic field to current density and changing electric field.

• Faraday's Law:

  • $\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$

  • Indicates a changing magnetic field creates an electric field.

WAVE EQUATION

• By manipulating Maxwell's Equations, we derive the wave equations for electric and magnetic fields:-

  • $\frac{\partial^2 E(x,t)}{\partial x^2} = \frac{\mu<em>0 \epsilon</em>0}{c^2} \frac{\partial^2 E(x,t)}{\partial t^2}$

  • $\frac{\partial^2 B(x,t)}{\partial x^2} = \frac{\mu<em>0 \epsilon</em>0}{c^2} \frac{\partial^2 B(x,t)}{\partial t^2}$

  • Where $c = \frac{1}{\sqrt{\mu<em>0 \epsilon</em>0}}$ is the speed of light.

ELECTROMAGNETIC WAVES

• Intensity of light wave:

  • $I = \frac{1}{2} \epsilon<em>0 c E</em>0^2$

• The speed of electromagnetic waves:

  • $c = \frac{1}{\sqrt{\mu<em>0 \epsilon</em>0}}$

INTERFERENCE and SUPERPOSITION PRINCIPLE

• Constructive Interference: When waves are in-phase ($\Delta \phi = 0$)

• Destructive Interference: When waves are out-of-phase ($\Delta \phi = \pi$)

DOUBLE-SLIT INTERFERENCE

• Light waves passing through two closely spaced slits act as point sources, creating an interference pattern on the screen.

HUYGEN’S PRINCIPLE

• Each point on a wavefront can be considered a source of secondary wavelets, expanding outwards.