Maxwell’s equations

Maxwell's equations are a set of four fundamental equations that describe the behavior of electric and magnetic fields and their interactions. These equations played a crucial role in unifying the theories of electricity and magnetism.

Integral Forms

Gauss's Law for Electricity:

\[ \oint \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enc}}}{\varepsilon_0}\]

This equation relates the electric field \(\vec{E}\) to the electric charge \(Q_{\text{enc}}\) within a closed surface.

Gauss's Law for Magnetism:

\[ \oint \vec{B} \cdot d\vec{A} = 0\]

This equation states that the magnetic field \(\vec{B}\) has no sources or sinks (magnetic monopoles).

Faraday's Law of Electromagnetic Induction:

\[\oint \vec{E} \cdot d\vec{l} = -\frac{d\Phi_B}{dt}\]

This equation describes how a changing magnetic field induces an electromotive force (EMF) and, consequently, an electric field along a closed loop.

Ampère's Law with Maxwell's Addition:

\[\oint \vec{B} \cdot d\vec{l} = \mu_0 \left( I_{\text{enc}} + \varepsilon_0 \frac{d\Phi_E}{dt} \right)\]

This equation relates the circulation of the magnetic field around a closed loop to the total current \(I_{\text{enc}}\) passing through the loop and the rate of change of the electric flux \(\Phi_E\) through the loop.

Differential forms

In addition to pervious integral forms, Maxwell's equations also have equivalent differential forms, which express the equations in terms of partial derivatives and are particularly useful in describing the local behavior of fields.

Gauss's Law for Electricity (Divergence of Electric Field):

\[ \nabla \cdot \vec{E} = \frac{\rho}{\varepsilon_0}\]

This equation states that the divergence of the electric field \(\vec{E}\) at a point in space is proportional to the electric charge density \(\rho\) at that point.

Gauss's Law for Magnetism (Divergence of Magnetic Field):

\[\nabla \cdot \vec{B} = 0 \]

This equation states that the divergence of the magnetic field \(\vec{B}\) at a point in space is zero, indicating the absence of magnetic monopoles.

Faraday's Law of Electromagnetic Induction (Curl of Electric Field):

\[\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t} \]

This equation describes how a changing magnetic field induces an electric field, and it relates the curl of the electric field \(\vec{E}\) to the rate of change of the magnetic field \(\vec{B}\) with respect to time.

Ampère's Law with Maxwell's Addition (Curl of Magnetic Field):

\[\nabla \times \vec{B} = \mu_0 \left( \vec{J} + \varepsilon_0 \frac{\partial \vec{E}}{\partial t} \right)\]

This equation relates the curl of the magnetic field \(\vec{B}\) to the total current density \(\vec{J}\) and the rate of change of the electric field \(\vec{E}\) with respect to time. The additional term involving \(\varepsilon_0\) accounts for the displacement current.

Here:

\(\nabla \cdot\) represents the divergence operator. \(\nabla \times\) represents the curl operator.

\(\rho\) is charge density.

\(\vec{J}\) is current density.

\(\varepsilon_0\) is the permittivity of free space.

\(\mu_0\) is the permeability of free space.