Maxwell 方程式

The four equations (SI units)

Electric field $\mathbf E$, magnetic field $\mathbf B$, charge density $\rho$, current density $\mathbf J$, constants $\varepsilon_0, \mu_0$:

$$\begin{aligned} \nabla \cdot \mathbf{E} &= \frac{\rho}{\varepsilon_0}, & \text{(Gauss's law)} \\ \nabla \cdot \mathbf{B} &= 0, & \text{(no magnetic monopoles)} \\ \nabla \times \mathbf{E} &= -\frac{\partial \mathbf{B}}{\partial t}, & \text{(Faraday)} \\ \nabla \times \mathbf{B} &= \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}. & \text{(Ampère–Maxwell)} \end{aligned}$$

That last term, $\mu_0\varepsilon_0\,\partial_t\mathbf E$ (the displacement current), was Maxwell’s insertion — and it predicted electromagnetic waves.

Electromagnetic waves

In vacuum ($\rho=0,\ \mathbf J=0$), taking the curl of Faraday’s law and substituting Ampère–Maxwell:

$$\nabla^2 \mathbf{E} - \mu_0\varepsilon_0 \frac{\partial^2 \mathbf{E}}{\partial t^2} = 0.$$

A wave equation. The propagation speed is

$$c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}} \approx 2.998 \times 10^{8}\,\mathrm{m/s}.$$

Light is an electromagnetic wave.

k₁ k₂ ω₁ ω₂

4-form (covariant)

Introduce the four-potential $A^\mu = (\phi/c,\, \mathbf A)$ and the field strength tensor

$$F^{\mu\nu} = \partial^\mu A^\nu - \partial^\nu A^\mu.$$

The four equations collapse into two:

$$\partial_\mu F^{\mu\nu} = \mu_0 J^\nu, \qquad \partial_{[\alpha} F_{\beta\gamma]} = 0.$$

Manifestly invariant under Lorentz transformations.

Lagrangian density

$$\mathcal L = -\tfrac{1}{4\mu_0} F_{\mu\nu} F^{\mu\nu} - J_\mu A^\mu.$$

Varying with respect to $A^\mu$ (Euler–Lagrange) reproduces the inhomogeneous Maxwell equations — the field-theoretic generalization of Lagrangian mechanics.

Boundary conditions, summarized

Quantity	Across an interface
$\mathbf E_{\parallel}$	continuous
$\mathbf B_{\perp}$	continuous
$\mathbf D_{\perp}$	jumps by $\sigma_f$
$\mathbf H_{\parallel}$	jumps by $\mathbf K_f$