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All the examples of 'continuity equations' below express the same idea. Continuity equations are the (stronger) local form of conservation laws.

Contents

Electromagnetic theory

Derivation

Interpretation

Fluid dynamics

Quantum mechanics

Four-currents

See also

Electromagnetic theory

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,
:$$
abla cdot mathbf{J} = - {partial
ho over partial t}

Derivation

One of Maxwell's equations, Ampère's law, states that
:$$
abla imes mathbf{H} = mathbf{J} + {partial mathbf{D} over partial t}.
Taking the divergence of both sides results in
:$$
abla cdot
abla imes mathbf{H} =
abla cdot mathbf{J} + {partial
abla cdot mathbf{D} over partial t} ,
but the divergence of a curl is zero, so that
:$$
abla cdot mathbf{J} + {partial
abla cdot mathbf{D} over partial t} = 0. qquad qquad (1)
Another one of Maxwell's equations, Gauss's law, states that
:$$
abla cdot mathbf{D} =
ho.,
Substitute this into equation (1) to obtain
:$$
abla cdot mathbf{J} + {partial
ho over partial t} = 0,,
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

In fluid dynamics, a 'continuity equation' is an equation of conservation of mass. Its differential form is
:$\{partial$
ho over partial t} +
abla cdot (
ho mathbf{u}) = 0
where $$
ho is density, t is time, and 'u' is fluid velocity.

Quantum mechanics

In quantum mechanics, the conservation of probability also yields a 'continuity equation'. Let ''P''(''x'', ''t'') be a probability density and write
:$$
abla cdot mathbf{j} = -{ partial over partial t} P(x,t)
where 'J' is probability flux.

Four-currents

Conservation of a current is expressed compactly as the Lorentz invariant divergence of a four-current:
:$J^a\; =\; left(c$
ho, mathbf{j}
ight)
where
:''c'' is the speed of light
:ρ the density
:'j' the conventional current density.
:
ho}{partial t} +
abla cdot mathbf{j} = 0

See also

★ Conservation law

★ Euler equations

★ Incompressible fluid

★ Schrödinger equation

★ Probability density