Maxwell’s Equations in Differential Form:
- Gauss’s Law for Electricity: ∇⋅E = ρ/ε₀
- – Gauss’s Law for Magnetism: ∇⋅B = 0
- – Faraday’s Law: ∇×E = -∂B/∂t
- – Ampere-Maxwell Law: ∇×B = μ₀J + μ₀ε₀∂E/∂t
These four equations, together with the Lorentz force law F = q(E + v×B), completely describe classical electromagnetism.
Wave Equation: Maxwell’s equations lead to the electromagnetic wave equation: ∇²E = (1/c²)∂²E/∂t² and ∇²B = (1/c²)∂²B/∂t², where c = 1/√(μ₀ε₀) is the speed of light.
Properties of EM Waves: Electromagnetic waves are transverse waves with E and B fields perpendicular to each other and to the direction of propagation. They carry energy and momentum, with energy density u = ½(ε₀E² + B²/μ₀) and Poynting vector S = (1/μ₀)E×B describing energy flow.