The Maxwell Equations

The four Maxwell equations can be written in differential or integral form:

$$\begin{array}{rlrl}\vec{\nabla }\cdot \vec{E}& =\frac{\rho }{{\epsilon }_0}& & \text{(Gauss’s Law)}\\ \vec{\nabla }\cdot \vec{B}& =0& & \text{(Gauss’s Law for Magnetism)}\\ \vec{\nabla }\times \vec{E}& =-\frac{\partial \vec{B}}{\partial t}& & \text{(Faraday’s Law)}\\ \vec{\nabla }\times \vec{B}& ={\mu }_0\vec{J}+{\mu }_0{\epsilon }_0\frac{\partial \vec{E}}{\partial t}& & \text{(Ampeˋre–Maxwell Law)}\end{array}$$ $$\begin{array}{rlrl}{\oint }_{\partial V}\mathbf{E}\cdot d\mathbf{A}& =\frac{1}{{\epsilon }_0}{\int }_V\rho dV& & \text{(Gauss’s Law)}\\ {\oint }_{\partial V}\mathbf{B}\cdot d\mathbf{A}& =0& & \text{(Gauss’s Law for Magnetism)}\\ {\oint }_{\partial S}\mathbf{E}\cdot d\mathbf{\ell }& =-\frac{d}{dt}{\iint }_S\mathbf{B}\cdot d\mathbf{A}& & \text{(Faraday’s Law)}\\ {\oint }_{\partial S}\mathbf{B}\cdot d\mathbf{\ell }& ={\mu }_0{\iint }_S\mathbf{J}\cdot d\mathbf{A}+{\mu }_0{\epsilon }_0\frac{d}{dt}{\iint }_S\mathbf{E}\cdot d\mathbf{A}& & \text{(Ampeˋre–Maxwell Law)}\end{array}$$

Derivation

Gauss’s Law

Gauss’s Law for Magnetism

Faraday’s Law

The Flux Rule

$${\Phi }_B={\iint }_S\vec{B}\cdot d\vec{A}$$

Lenz’s Law

Electromagnetic induction is the process where a changing magnetic field induces an electric current in a conductor. Michael Faraday recorded experimentally that a voltage $V$ was induced within a conductive coil with $n$ turns when the magnetic field passing through that coil was altered, and that the induced voltage was proportional to the change in the magnetic field. We can write this simple relationship as,

$$V=n\frac{d\phi }{dt}$$

where $\phi$ is the magnetic flux passing through the coil, and $t$ is of course time.

Magnetic flux is a measurement of the magnitude of the magnetic field which passes through a surface in space

Ampère-Maxwell Law

$$\begin{array}{rl}\nabla \cdot \mathbf{D}& ={\rho }_{\mathrm{free}}\\ \nabla \cdot \mathbf{B}& =0\\ \nabla \times \mathbf{E}& =-\frac{\partial \mathbf{B}}{\partial t}\\ \nabla \times \mathbf{H}& ={\mathbf{J}}_{\mathrm{free}}+\frac{\partial \mathbf{D}}{\partial t}\end{array}$$ $$\nabla \cdot \mathbf{J}+\frac{\partial \rho }{\partial t}=0$$

See The Integral, The Differential.

References

1. The Maxwell Equations - Part of the Feynman Lectures
2. The Maxwell Equations - U.S. Particle Accelerator School