Maxwell’s Equations

Classically, there are four fundamental forces: gravitational, electromagnetic, weak nuclear, and strong nuclear. Just like Newton’s inverse square law \(F = G \frac{m_1 m_2}{r^2}\) characterizes gravitational force, Maxwell’s equations (and the Lorentz force law) characterize electromagnetic forces. Recalling that \((\nabla\cdot) = \operatorname{div}\) and that \((\nabla\times) = \operatorname{curl},\) for an electric field \(\bm{E}\) and magnetic field \(\bm{B}\)

\(\displaystyle \nabla \cdot \bm{E} = \frac{\rho}{\varepsilon_0}\)
Gauss’s Law
\(\displaystyle \nabla \cdot \bm{B} = 0\)
Gauss’s Law
for Magnetism
\(\displaystyle \nabla \times \bm{E} = -\frac{\partial \bm{B}}{\partial t} \)
Faraday’s Law
\(\displaystyle \nabla \times \bm{B} = \mu_0\biggl(\bm{J} + \varepsilon_0\frac{\partial \bm{E}}{\partial t}\biggr) \)
The Ampère-Maxwell Law

where \(\rho\) is the electric charge density, \(\bm{J}\) is the current density, \(\varepsilon_0\) is the vacuum permittivity, and \(\mu_0\) is the vacuum permeability. Each of these laws are usually expressed as differential equations, however they can also be expressed in an integral form per Stokes’s Theorem and the Divergence Theorem.

\(\displaystyle \oiint_{\partial E} \bm{E}\cdot\mathrm{d}\bm{S} = \frac{1}{\varepsilon_0}\iiint_{E} \rho \,\mathrm{d}V \)
Gauss’s Law
\(\displaystyle \oiint_{\partial E} \bm{B}\cdot\mathrm{d}\bm{S} = 0 \)
Gauss’s Law
for Magnetism
\(\displaystyle \oint_{\partial S} \bm{E}\cdot\mathrm{d}\bm{\ell} = -\frac{\mathrm{d}}{\mathrm{d}t}\iint_S \bm{B}\cdot\mathrm{d}\bm{S} \)
Faraday’s Law
\(\displaystyle \oint_{\partial S} \bm{B}\cdot\mathrm{d}\bm{\ell} = \mu_0 \biggl(\iint_S \bm{J}\cdot\mathrm{d}\bm{S} +\varepsilon_0 \frac{\mathrm{d}}{\mathrm{d}t}\iint_S \bm{E}\cdot\mathrm{d}\bm{S} \biggr) \)
The Ampère-Maxwell Law