All the examples of continuity equations below express the same idea; they are all really examples of the same concept. Continuity equations are the (stronger) local form of conservation laws.

Electromagnetic theory

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In electromagnetic theory, the continuity equation is derived from two of Maxwell's equations. It states that the divergence of the current density is equal to the negative rate of change of the charge density,

$\backslash nabla\; \backslash cdot\; \backslash mathbf\{J\}\; =\; -\; \{\backslash partial\; \backslash rho\; \backslash over\; \backslash partial\; t\}$

Derivation

One of Maxwell's equations states that

$\backslash nabla\; \backslash times\; \backslash mathbf\{H\}\; =\; \backslash mathbf\{J\}\; +\; \{\backslash partial\; \backslash mathbf\{D\}\; \backslash over\; \backslash partial\; t\}.$

Taking the divergence of both sides results in

$\backslash nabla\; \backslash cdot\; \backslash nabla\; \backslash times\; \backslash mathbf\{H\}\; =\; \backslash nabla\; \backslash cdot\; \backslash mathbf\{J\}\; +\; \{\backslash partial\; \backslash nabla\; \backslash cdot\; \backslash mathbf\{D\}\; \backslash over\; \backslash partial\; t\}$,

but the divergence of a curl is zero, so that

$\backslash nabla\; \backslash cdot\; \backslash mathbf\{J\}\; +\; \{\backslash partial\; \backslash nabla\; \backslash cdot\; \backslash mathbf\{D\}\; \backslash over\; \backslash partial\; t\}\; =\; 0.\; \backslash qquad\; \backslash qquad\; (1)$

Another one of Maxwell's equations states that

$\backslash nabla\; \backslash cdot\; \backslash mathbf\{D\}\; =\; \backslash rho.\backslash ,$

Substitute this into equation (1) to obtain

$\backslash nabla\; \backslash cdot\; \backslash mathbf\{J\}\; +\; \{\backslash partial\; \backslash rho\; \backslash over\; \backslash partial\; t\}\; =\; 0,\backslash ,$

which is the continuity equation.

Interpretation

Current density is the movement of charge density. The continuity equation says that if charge is moving out of a differential volume (i.e. divergence of current density is positive) then the amount of charge within that volume is going to decrease, so the rate of change of charge density is negative. Therefore the continuity equation amounts to a conservation of charge.

Fluid dynamics

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In fluid dynamics, a continuity equation is an equation of conservation of mass. Its differential form is

$\{\backslash partial\; \backslash rho\; \backslash over\; \backslash partial\; t\}\; +\; \backslash nabla\; \backslash cdot\; (\backslash rho\; \backslash mathbf\{u\})\; =\; 0$

where $\backslash rho$ is density, t is time, and u is fluid velocity.

Quantum mechanics

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In quantum mechanics, the conservation of probability also yields a continuity equation. Let P(x, t) be a probability density and write

$\backslash nabla\; \backslash cdot\; \backslash mathbf\{j\}\; =\; -\{\; \backslash partial\; \backslash over\; \backslash partial\; t\}\; P(x,t)$

where J is probability flux.

See also

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• Conservation law
• Euler equations
• Incompressible fluid
• Schrödinger equation
• Probability density

Equations | Fluid dynamics | Conservation laws

Rovnice kontinuity | Kontinuitätsgleichung | Ecuación de continuidad | Equazione di continuità | Kontinuitási egyenlet | Persamaan keselanjaran | 連続の方程式 | Kontinuitetna enačba