Maxwell equations

Maxwell’s equations relate electric and magnetic fields to charge and current:

Gauss's law for magnetism is $\nabla \cdot \mathbf{B}=0$. Its integral form says that the net magnetic flux through every closed surface vanishes, so the normal magnetic field is continuous across an ordinary interface.

For the electromagnetic two-form $F$, the source-free Maxwell equations are $dF=0$ and $d(\ast F)=0$. The first is the Bianchi identity and the second is the dynamical equation.

In macroscopic matter with free charge density $\rho$ and free current density $J$,The electric displacement $D$ and magnetic field strength $H$ encode the material response.

Constitutive relations specify how matter responds to fields. A homogeneous isotropic linear electromagnetic medium obeys $\mathbf{D}=\epsilon \mathbf{E}$ and $\mathbf{B}=\mu \mathbf{H}$.

Electromagnetic permeability relates magnetic flux density to magnetic field strength through $\mathbf{B}=\mu \mathbf{H}$ in an isotropic linear medium.

A linear anisotropic medium has constitutive relations $D_i={\epsilon }_{ij}{E}_j$ and $B_i={\mu }_{ij}{H}_j$. Symmetric, time-independent tensors make the field-energy density a quadratic form.

The Ampère-Maxwell equationrelates magnetic circulation to conduction and displacement current.

An Amperian loop is a closed curve used in the integral form of Ampère's circuital law or the Ampère-Maxwell equation.

In vacuum magnetostatics, the Ampère-Maxwell equation reduces to $\nabla \times \mathbf{B}={\mu }_0\mathbf{J}$. Applying Stokes theorem gives ${\oint }_C\mathbf{B}\cdot d\mathbf{\ell }={\mu }_0{I}_{\mathrm{enc}}$, where $I_{\mathrm{enc}}$ is the electric current crossing an oriented spanning surface. Time-dependent electric fields require the additional displacement current term in the Ampère-Maxwell equation.

In vacuum, the displacement-current density is ${ϵ}_0\partial E/\partial t$. Its contribution to magnetic circulation makes the Ampère-Maxwell equation compatible with changing electric charge.

Faraday's law is

The induced current flows so that its magnetic effect opposes the change of flux that produced it.

If a loop of resistance $R$ has magnetic flux ${\Phi }_B=-kz$, then its induced current is $I=k\dot{z}/R$. When its magnetic force is $-kI$ in the vertical direction, its equation under gravity isso its downward terminal speed is $Rmg/k^2$.

Gauss's law states

A Gaussian surface is a closed surface used with the integral form of Gauss's law to relate electric flux to enclosed charge.

A Gaussian pillbox is a thin cylindrical Gaussian surface crossing an interface. Shrinking its height converts a divergence law into a jump condition for the field's normal component.

Taking the divergence of the Ampère-Maxwell equation and using Gauss's law givesHence the charge in a fixed volume changes only through current crossing its boundary.

The local form of conservation of electric charge relates charge density to current density. Taking the divergence of the Ampère-Maxwell equation and substituting Gauss's law proves the equation, using that the divergence of a curl is zero.

With metric signature $(-,+,+,+)$, $A_0=-\Phi /c$, and $j^{\mu }=(c\rho ,J)$, the field tensor satisfiesIts components are $F_{0i}=-E_i/c$ and $F_{ij}={ϵ}_{ijk}{B}_k$, so these tensor equations reproduce all four Maxwell equations.

For $D=ϵE$ and $B=\mu H$ with no free charge or current,

A plane wave in a homogeneous linear medium satisfies

The source-free divergence equations imply $\mathbf{k}\cdot {\mathbf{E}}_0=\mathbf{k}\cdot {\mathbf{B}}_0=0$, so both fields are transverse to the wavevector.