A
continuity equation in physics is a
differential equationA differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
that describes the transport of some kind of
conserved quantityIn mathematics, a conserved quantity of a dynamical system is a function H of the dependent variables that is a constant . A conserved quantity can be a useful tool for qualitative analysis...
. Since
massIn physics, mass commonly refers to any of three properties of matter, which have been shown experimentally to be equivalent: inertial mass, active gravitational mass and passive gravitational mass...
,
energyIn physics, energy is a scalar physical quantity that describes the amount of work that can be performed by a force, an attribute of objects and systems that is subject to a conservation law...
,
momentumIn classical mechanics, momentum is the product of the mass and velocity of an object . For more accurate measures of momentum, see the section "modern definitions of momentum" on this page...
,
electric chargeElectric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. Electrically charged matter is influenced by, and produces, electromagnetic fields...
and other natural quantities are conserved, a vast variety of physics may be described with continuity equations.
Continuity equations are the (stronger) local form of
conservation lawIn physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves....
s. All the examples of continuity equations below express the same idea, which is roughly that:
the total amount (of the conserved quantity) inside any region can only change by the amount that passes in or out of the region through the boundary.
A
continuity equation in physics is a
differential equationA differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
that describes the transport of some kind of
conserved quantityIn mathematics, a conserved quantity of a dynamical system is a function H of the dependent variables that is a constant . A conserved quantity can be a useful tool for qualitative analysis...
. Since
massIn physics, mass commonly refers to any of three properties of matter, which have been shown experimentally to be equivalent: inertial mass, active gravitational mass and passive gravitational mass...
,
energyIn physics, energy is a scalar physical quantity that describes the amount of work that can be performed by a force, an attribute of objects and systems that is subject to a conservation law...
,
momentumIn classical mechanics, momentum is the product of the mass and velocity of an object . For more accurate measures of momentum, see the section "modern definitions of momentum" on this page...
,
electric chargeElectric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. Electrically charged matter is influenced by, and produces, electromagnetic fields...
and other natural quantities are conserved, a vast variety of physics may be described with continuity equations.
Continuity equations are the (stronger) local form of
conservation lawIn physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves....
s. All the examples of continuity equations below express the same idea, which is roughly that:
the total amount (of the conserved quantity) inside any region can only change by the amount that passes in or out of the region through the boundary. A conserved quantity cannot increase or decrease, it can only move from place to place.
Any continuity equation has a "differential form" (in terms of the
divergenceIn vector calculus, the divergence is an operator that measures the magnitude of a vector field's source or sink at a given point; the divergence of a vector field is a scalar. For example, consider air as it is heated or cooled. The relevant vector field for this example is the velocity of the...
operator) and an "integral form" (in terms of a
fluxIn the various subfields of physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks.* In the study of transport phenomena , flux is defined as the amount that flows through a unit area per unit time...
integral). In this article, only the "differential form" versions will be given; see the article
divergence theoremIn vector calculus, the divergence theorem, also known as Gauss' theorem , Ostrogradsky's theorem , or Gauss–Ostrogradsky theorem is a result that relates the flow of a vector field through a surface to the behavior of the vector field inside the surface.More precisely, the divergence theorem...
for how to express any of these laws in "integral form".
General
The general form for a continuity equation is
where is some quantity, ƒ is a
functionIn mathematics, a function is a relation between a given set of elements and another set of elements , which associates each element in the domain with exactly one element in the codomain...
describing the
fluxIn the various subfields of physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks.* In the study of transport phenomena , flux is defined as the amount that flows through a unit area per unit time...
of , and
s describes the generation (or removal) rate of . This equation may be derived by considering the fluxes into an infinitesimal box. If is a conserved quantity, the generation or removal rate is zero:
This general equation may be used to derive any continuity equation, ranging from as simple as the volume continuity equation to as complicated as the Navier–Stokes equations. This equation also generalizes the advection equation.
Electromagnetic theory
In electromagnetic theory, the
continuity equation can either be regarded as an empirical law expressing (local)
charge conservationCharge conservation is the principle that electric charge can neither be created nor destroyed. The quantity of electric charge, the amount of positive charge minus the amount of negative charge in the universe, is always conserved....
, or can be
derived as a consequence of two of
Maxwell's equationsMaxwell's equations are a set of four partial differential equations that relate the electric and magnetic fields to their sources, charge density and current density. These equations can be combined to show that light is an electromagnetic wave...
. It states that the
divergenceIn vector calculus, the divergence is an operator that measures the magnitude of a vector field's source or sink at a given point; the divergence of a vector field is a scalar. For example, consider air as it is heated or cooled. The relevant vector field for this example is the velocity of the...
of the
current densityCurrent density is a measure of the density of flow of a conserved charge. Usually the charge is the electric charge, in which case the associated current density is the electric current per unit area of cross section, but the term current density can also be applied to other conserved...
is equal to the negative rate of change of the
charge densityThe linear, surface, or volume charge density is the amount of electric charge in a line, surface, or volume. It is measured in coulombs per metre , square metre , or cubic metre , respectively. Since there are positive as well as negative charges, the charge density can take on negative values....
,
Derivation from Maxwell's equations
One of
Maxwell's equationsMaxwell's equations are a set of four partial differential equations that relate the electric and magnetic fields to their sources, charge density and current density. These equations can be combined to show that light is an electromagnetic wave...
,
Ampère's lawIn classical electromagnetism, Ampère's circuital law, discovered by André-Marie Ampère in 1826, relates the integrated magnetic field around a closed loop to the electric current passing through the loop...
, states that
Taking the divergence of both sides results in
but the divergence of a curl is zero, so that
Another one of Maxwell's equations,
Gauss's lawIn physics, Gauss's law, also known as Gauss's flux theorem, is a law relating the distribution of electric charge to the resulting electric field. Gauss's law states that:...
, states that
Substitute this into equation (1) to obtain
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 dynamicsIn physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...
, the
continuity equation is a mathematical statement that, in any
steady stateA system in a steady state has numerous properties that are unchanging in time. The concept of steady state has relevance in many fields, in particular thermodynamics. Steady state is a more general situation than dynamic equilibrium. If a system is in steady state, then the recently observed...
process, the rate at which mass enters a system is equal to the rate at which mass leaves the system.
In fluid dynamics, the continuity equation is analogous to Kirchhoff's Current Law in electric circuits.
The differential form of the
continuity equation is:
where is fluid density, t is time, and
u is fluid velocity. If density is a constant, as in the case of
incompressible flowIn fluid mechanics or more generally continuum mechanics, an incompressible flow is solid or fluid flow in which the divergence of velocity is zero. This is more precisely termed isochoric flow. It is an idealization used to simplify analysis. In reality, all materials are compressible to some...
, the mass continuity equation simplifies to a volume continuity equation:
which means that the divergence of velocity field is zero everywhere. Physically, this is equivalent to saying that the local volume dilation rate is zero.
Further, the
Navier-Stokes equationsThe Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of fluid substances, that is substances which can flow...
form a vector continuity equation describing the conservation of linear momentum.
Quantum mechanics
In
quantum mechanicsQuantum mechanics is a set of principles describing the physical reality at the atomic level of matter and the subatomic . These descriptions include the simultaneous wave-like and particle-like behavior of both matter and radiation...
, the conservation of probability also yields a
continuity equation. If
P(
x,
t) be a
probability density functionIn probability theory, a probability density function —often referred to as a probability distribution function—or density, of a random variable is a function that describes the density of probability at each point in the sample space...
,
where
j is probability flux.
Four-currents
Conservation of a current (not necessarily an electromagnetic current) is expressed compactly as the Lorentz invariant
divergenceIn vector calculus, the divergence is an operator that measures the magnitude of a vector field's source or sink at a given point; the divergence of a vector field is a scalar. For example, consider air as it is heated or cooled. The relevant vector field for this example is the velocity of the...
of a
four-currentIn special and general relativity, the four-current is the Lorentz covariant four-vector that replaces the electromagnetic current density, or indeed any conventional charge current density. Its four components are given by:where...
:
where
- c is the speed of light
In physics, the speed of light is a physical constant, the speed at which electromagnetic radiation, such as light, travels in free space . Its value is 299,792,458 metres per second...
- ρ the charge density
The linear, surface, or volume charge density is the amount of electric charge in a line, surface, or volume. It is measured in coulombs per metre , square metre , or cubic metre , respectively. Since there are positive as well as negative charges, the charge density can take on negative values....
- j the conventional current density
Current density is a measure of the density of flow of a conserved charge. Usually the charge is the electric charge, in which case the associated current density is the electric current per unit area of cross section, but the term current density can also be applied to other conserved...
.
- μ labels the space-time
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space being three-dimensional and time playing the role of a fourth dimension that is of a different sort than the spatial dimensions...
dimensionIn mathematics and physics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify each point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
so that since
then
implies that the current is conserved:
See also
- Conservation law
In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves....
- Noether's Theorem
Noether's theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. This seminal theorem was proved by Emmy Noether in 1915 and published in 1918...
- Groundwater energy balance
The groundwater energy balance is the energy balance of a groundwater body in terms of incoming hydraulic energy associated with groundwater inflow into the body, energy associated with the outflow, energy conversion into heat due to friction of flow, and the resulting change of energy status and...
- Euler equations
In fluid dynamics, the Euler equations govern inviscid flow. They correspond to the Navier–Stokes equations with zero viscosity and heat conduction terms. They are usually written in the conservation form shown below to emphasize that they directly represent conservation of mass, momentum, and energy...
- Incompressible fluid
- Schrödinger equation
In physics, specifically quantum mechanics, the Schrödinger equation is an equation that describes how the quantum state of a physical system changes in time...
- Probability density function
In probability theory, a probability density function —often referred to as a probability distribution function—or density, of a random variable is a function that describes the density of probability at each point in the sample space...