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Mathematics for Physics

42 Electric Field

A vector field produced by electric charges or changing magnetic fields that is used to give the electric force on a charge placed in it, where a vector field is an assignment of a vector to each point in space.

The definition of the electric field. The electric field is the electric force per unit charge on a test charge. A test charge is a small charge used to probe the field. This principle is used to define \(\mathbf{E}\) independently of which charge feels the force.

The electric field from a force is

\[ \mathbf{E} = \dfrac{\mathbf{F}}{q} \]

where

  • \(\mathbf{E}\) is the electric field.
  • \(\mathbf{F}\) is the electric force.
  • \(q\) is the test charge.

The Coulomb field of a charge distribution. A stationary charge distribution produces an electric field given by Coulomb’s law integrated over the sources. This principle is used to map the electrostatic field of a known charge collection.

The electric field of a charge distribution is

\[ \mathbf{E}(\mathbf{r}) = \dfrac{1}{4\pi\epsilon_{0}}\displaystyle\int\dfrac{\rho(\mathbf{r}')}{r^{2}}\hat{\mathbf{r}}\,d\tau' \]

where

  • \(\mathbf{E}(\mathbf{r})\) is the electric field at the field point.
  • \(\rho\) is the volume charge density.
  • \(r\) is the distance from the source element to the field point.
  • \(\hat{\mathbf{r}}\) is the unit vector from the source element to the field point.
  • \(d\tau'\) is the volume element.
  • \(\epsilon_{0}\) is the permittivity of free space.

The induced electric field of Faraday’s law. A changing magnetic field produces a circulating electric field. This principle is used to compute induced electric fields.

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.