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Mathematics for Physics

43 Electric Fields

A physical entity in space that is used to distinguish a conservative electrostatic field from a nonconservative induced field, where a conservative field is a field whose work around every closed path vanishes.

The conservative electrostatic field. A static charge distribution produces a conservative electric field whose curl vanishes. This principle is used to write that field as minus the gradient of a scalar potential.

The electrostatic curl condition is

\[ \nabla\times\mathbf{E} = 0 \]

where

  • \(\nabla\times\) is the curl.
  • \(\mathbf{E}\) is the electrostatic field.

Gauss’s law. The divergence of the electric field is proportional to the local charge density. This principle is used to relate the field to its sources.

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.

The nonconservative induced field. A changing magnetic field produces a nonconservative electric field whose curl does not vanish. A nonconservative field is a field whose work around a closed path need not vanish. This principle is used to describe 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.