Maxwell's Equations

Maxwell's equations unify electricity, magnetism, and optics into a single framework. They predict electromagnetic waves traveling at the speed of light and form the foundation of classical electrodynamics.

Key Concepts

Gauss's law: ∇·E = ρ/ε₀
Gauss's law for magnetism: ∇·B = 0 (no magnetic monopoles)
Faraday's law: ∇×E = -∂B/∂t
Ampere-Maxwell law: ∇×B = μ₀J + μ₀ε₀∂E/∂t
Displacement current: J_D = ε₀∂E/∂t

Key Equations

Gauss's law

\nabla\cdot\mathbf{E} = \frac{\rho}{\varepsilon_0}

Faraday's law

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

Ampere-Maxwell

\nabla\times\mathbf{B} = \mu_0\mathbf{J} + \mu_0\varepsilon_0\frac{\partial\mathbf{E}}{\partial t}

Wave equation

\nabla^2\mathbf{E} = \mu_0\varepsilon_0\frac{\partial^2\mathbf{E}}{\partial t^2}

Worked Example

Example Problem

Problem

A capacitor is being charged with current I=2.0 A. The plate area is A=0.01 m². Find the displacement current density J_D between the plates.

Solution

J_D = I/A = 2.0/0.01 = 200 A/m². This equals the conduction current density, consistent with continuity.

Practice

Exercises

7 problems

1 of 7

A region has uniform charge density ρ=10 nC/m³. Find ∇·E in V/m².

Your answer

V/m²

2 of 7

A parallel-plate capacitor (A=0.05 m²) is charged by I=0.5 A. Find the rate of change of electric field dE/dt in V/(m·s).

Your answer

V/(m·s)

3 of 7

An EM wave in vacuum has E₀=1000 V/m. Find B₀ in μT.

Unlock Exercise 3