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Mathematics for Physics

71 Maxwell’s Equations

A set of four field equations that is used to describe how electric and magnetic fields arise from charges and currents and how the fields generate each other, where a field equation is a differential equation that relates a field to its sources.

Gauss’s law. Electric charge produces electric field that spreads from the charge. Charge density is charge per unit volume. Divergence is a measure of how much a vector field spreads from a point. This principle is used to find the electric field of a given charge distribution.

Gauss’s law is

\[ \nabla\cdot\mathbf{E} = \dfrac{\rho}{\epsilon_{0}} \]

where

  • \(\nabla\cdot\) is the divergence.
  • \(\mathbf{E}\) is the electric field.
  • \(\rho\) is the volume charge density.
  • \(\epsilon_{0}\) is the permittivity of free space.

Gauss’s law for magnetism. Magnetic field lines form closed loops with no beginning and no end. A magnetic monopole is an isolated single magnetic pole. No magnetic monopole has been observed. This principle is used to constrain magnetic fields so they never diverge from a point source.

Gauss’s law for magnetism is

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

where

  • \(\nabla\cdot\) is the divergence.
  • \(\mathbf{B}\) is the magnetic field.

Faraday’s law of induction. A magnetic field that changes in time produces a swirling electric field around the changing magnetic field. Curl is a measure of the local swirl of a vector field. Electromagnetic induction is the generation of electric field by a changing magnetic field. This principle is used to design generators, transformers, and inductors.

Faraday’s law is

\[ \nabla\times\mathbf{E} = -\dfrac{\partial\mathbf{B}}{\partial t} \]

where

  • \(\nabla\times\) is the curl.
  • \(\mathbf{E}\) is the electric field.
  • \(\mathbf{B}\) is the magnetic field.
  • \(t\) is time.

The Ampère-Maxwell law. Magnetic fields are produced by electric current and by electric fields that change in time. Current density is charge flow per unit area. The displacement current is Maxwell’s addition: a changing electric field that sources magnetic field as a current does. This principle is used to compute fields of electromagnets and to show that electromagnetic waves travel in empty space.

The Ampère-Maxwell law is

\[ \nabla\times\mathbf{B} = \mu_{0}\mathbf{J} + \mu_{0}\epsilon_{0}\dfrac{\partial\mathbf{E}}{\partial t} \]

where

  • \(\nabla\times\) is the curl.
  • \(\mathbf{B}\) is the magnetic field.
  • \(\mu_{0}\) is the permeability of free space.
  • \(\mathbf{J}\) is the volume current density.
  • \(\epsilon_{0}\) is the permittivity of free space.
  • \(\mathbf{E}\) is the electric field.
  • \(t\) is time.

The continuity equation. Electric charge is never created and never destroyed: if charge inside a volume falls, the same charge must flow out through the surface. Local conservation is the requirement that a conserved quantity move continuously through space. This principle is used to tie charge density to current density as a consistency condition.

The continuity equation is

\[ \nabla\cdot\mathbf{J} = -\dfrac{\partial\rho}{\partial t} \]

where

  • \(\nabla\cdot\) is the divergence.
  • \(\mathbf{J}\) is the volume current density.
  • \(\rho\) is the volume charge density.
  • \(t\) is time.

Note: Gauss’s law for magnetism is also called the no-monopole law. The Ampère-Maxwell law is also called Ampère’s law with Maxwell’s correction. The continuity equation is also called local charge conservation.