Continuity Equation Article Index for
Continuity 		Website Links For
Continuity 		 

Information About

Continuity Equation
  CATEGORIES ABOUT CONTINUITY EQUATION
   
equations of fluid dynamics
   
conservation laws
   
APPAREL
BABY
BEAUTY
BOOKS
CAR TOYS
CELL PHONES
DVD'S
ELECTRONICS
GOURMET FOOD
GROCERIES
HEALTH & PERSONAL
HOME & GARDEN
JEWELRY
MUSIC
MUSIC INSTRUMENTS
OFFICE PRODUCTS
SOFTWARE
SPORTING GOODS
TOOLS & HARDWARE
TOYS
VIDEO GAMES
SHOPPING HOME
MORE SHOPPING...




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

:ρ the Density
:j the conventional Current Density .

:\partial_a J^a = rac{\partial ho}{\partial t} +
abla \cdot \mathbf{j} = 0


SEE ALSO