Maxwell's equations

Maxwell’s equations are the fundamental equations describing electromagnetism. Here $\mathbf E$ is the electric vector field in space, $\mathbf B$ is the magnetic vector field, and $\hat{\mathbf n}$ is the outward-pointing unit normal vector to a closed surface.

The constant $\epsilon_0$ is the vacuum permittivity and $\mu_0$ is the vacuum permeabiliy. These are the two fundamental constants related to electricity and magnetism, respectively. In material, $\epsilon$ and $\mu$ may be different, substitute them instead of $\epsilon_0$ and $\mu_0$.

Gauss’s law

Gauss’s law tells us that electric charges create electric fields. Here $\rho$ is the electric charge density (units $C\cdot m^{-3}$). The right hand side of the equation is $Q_\mathrm{enclosed}$, the charge enclosed in $V$, and the left is the electric flux through the surface $S$ that encloses it.

In differential form

Magnetic Gauss’s law

Magnetic Gauss’s law tells us that there are no magnetic monopoles (isolated + or -). All magnetic fields are created by dipoles (which have both a + and a -), so the magnetic flux through a closed surface is always net zero.

In differential form

Faraday’s law

Faraday’s law tells us that there is another way to create an electric field, which is using a time-varying magnetic field. The electric field is called the induced electric field, and the principle which Faraday’s law describes is called electromagnetic induction.

Faraday’s law is also written as $\oint_C \mathbf E \cdot \mathbf{ds} = -\frac{d\Phi_B}{dt}$. In this form it is sometimes called Lenz’s law.

In differential form

Maxwell-Ampere’s law

Maxwell-Ampere’s law tells us that there are two ways of creating magnetic fields. The first term (Ampere’s discovery) tells us that currents create a magnetic field, and the second term (Maxwell’s discovery) tells us that a time-varying electric field creates a magnetic field.

In differential form