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Mathematics for Physics

32 Continuity Equation

A local conservation law relating the time derivative of charge density to the divergence of current that is used to express conservation of charge.

The continuity equation. Charge is conserved locally: any decrease of charge in a volume equals the net current flowing out. This principle is used to write charge conservation as a differential equation.

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.
  • \(\nabla\cdot\) is the divergence.

Static charge density. If there is no current, the charge density is static. This principle is used to recover electrostatics as the case \(\mathbf{J}=\mathbf{0}\).

The static-density condition is

\[ \dfrac{\partial\rho}{\partial t} = 0 \]

where

  • \(\rho\) is the charge density.
  • \(t\) is time.

Divergenceless steady current. In a steady state the current is divergenceless. This principle is used to treat magnetostatics, where charge density does not change with time.

The steady-current condition is

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

where

  • \(\mathbf{J}\) is the current density.

Note: Also written \(\dfrac{\partial\rho}{\partial t}+\nabla\cdot\mathbf{J}=0\).

32.1 References

  1. Griffiths, D. J. Introduction to Electrodynamics. Cambridge University Press, 2024. — \(\dfrac{\partial\rho}{\partial t}+\nabla\cdot\mathbf{J}=0\).
  2. Knight, R. D. Physics for Scientists and Engineers. Pearson, 2023. — charge conservation and continuity.
  3. Susskind, L., & Cabannes, A. General Relativity: The Theoretical Minimum. Penguin Books, 2023. — continuity equation for conserved densities.