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Encyclopedia > Continuity equation

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. In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves. ...

## Electromagnetic theory GA_googleFillSlot("encyclopedia_square");

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, Maxwells equations are the set of four equations, attributed to James Clerk Maxwell, that describe the behavior of both the electric and magnetic fields, as well as their interactions with matter. ... Maxwells equations (sometimes called the Maxwell equations) are the set of four equations, attributed to James Clerk Maxwell, that describe the behavior of both the electric and magnetic fields, as well as their interactions with matter. ... In electricity, current is the rate of flow of charges, usually through a metal wire or some other electrical conductor. ... Charge density is the amount of electric charge per unit volume. ... $nabla cdot mathbf{J} = - {partial rho over partial t}$

### Derivation

One of Maxwell's equations states that $nabla times mathbf{H} = mathbf{J} + {partial mathbf{D} over partial t}.$

Taking the divergence of both sides results in $nabla cdot nabla times mathbf{H} = nabla cdot mathbf{J} + {partial nabla cdot mathbf{D} over partial t}$,

but the divergence of a curl is zero, so that $nabla cdot mathbf{J} + {partial nabla cdot mathbf{D} over partial t} = 0. qquad qquad (1)$

Another one of Maxwell's equations states that $nabla cdot mathbf{D} = rho.,$

Substitute this into equation (1) to obtain $nabla cdot mathbf{J} + {partial rho over partial t} = 0,,$

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 Fluid dynamics is the subdiscipline of fluid mechanics that studies fluids in motion. ... law of conservation of mass/matter states that the mass of a system of substances is constant, regardless of the processes acting inside the system. ... ${partial rho over partial t} + nabla cdot (rho mathbf{u}) = 0$

where ρ 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(xt) be a probability density and write Fig. ... In quantum mechanics, a probability amplitude is a complex number-valued function which describes an uncertain or unknown quantity. ... $nabla cdot mathbf{j} = -{ partial over partial t} P(x,t)$

where J is probability flux. In quantum mechanics, the probability current (sometimes called probability flux) is a useful concept which describes the flow of probability density. ... Results from FactBites:

 Continuity equation - Wikipedia, the free encyclopedia (228 words) Continuity equations are the (stronger) local form of conservation laws. In electromagnetic theory, the continuity equation is derived from two of Maxwell's equations. In fluid dynamics, a continuity equation is an equation of conservation of mass.
 PlanetMath: continuity equation (1039 words) In the former case we apply equations governing the gross behavior of the flow through an integral formulation which is usually easier to treat analytically. Since the mentioned disciplines deal with the formulation of the basic laws in terms of finite systems, such formulations are the basis for deriving the control volume equations, concept that we shall develop to continuation. This is version 5 of continuity equation, born on 2006-05-31, modified 2006-09-04.
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