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Mathematics for Physics

31 Conservation of Charge

A conservation law that is used to require that the total electric charge of an isolated system stay constant, where electric charge is the source of the electromagnetic field.

Global charge conservation. The total charge in an isolated region changes only if current flows through the boundary. This principle is used to write global charge conservation.

The integral conservation law is

\[ \dfrac{dQ_{\mathrm{enc}}}{dt} = -\oint\mathbf{J}\cdot d\mathbf{a} \]

where

  • \(Q_{\mathrm{enc}}\) is the charge enclosed by the surface.
  • \(\mathbf{J}\) is the current density.
  • \(d\mathbf{a}\) is the outward vector area element.
  • \(t\) is time.

The continuity equation. The local form of that statement is the continuity equation. This principle is used to enforce charge conservation at every point.

The continuity equation is

\[ \dfrac{\partial\rho}{\partial t} + \nabla\cdot\mathbf{J} = 0 \]

where

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

Consistency with Maxwell’s equations. Maxwell’s equations imply the continuity equation once the displacement current is included. This principle is used to show that charge conservation is not an extra postulate of electrodynamics.

The consistency identity is

\[ \nabla\cdot\bigl(\nabla\times\mathbf{B}\bigr) = 0 = \mu_{0}\Bigl(\nabla\cdot\mathbf{J} + \epsilon_{0}\dfrac{\partial}{\partial t}(\nabla\cdot\mathbf{E})\Bigr) \]

where

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