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Charge density
 
 
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The linear, surface, or volume charge density is the amount of electric charge
Electric charge

Electric 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....
 in a line
Line (mathematics)

In geometry, a line is a Curvature curve. When geometry is used to model the real world, lines are used to represent straight objects with negligible width and height....
, surface
Surface

In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space E3....
, or volume
Volume

The volume of any solid, liquid, plasma, vacuum or theoretical object is how much three-dimensional space it occupies, often quantified numerically....
. It is measured in coulomb
Coulomb

The coulomb is the SI unit of electric charge. It is named after Charles-Augustin de Coulomb....
s per metre (C/m), square metre (C/m²), or cubic metre (C/m³), respectively. Since there are positive as well as negative charges, the charge density can take on negative values. Like any density
Density

The density of a material is defined as its mass per unit volume. The symbol of density is ....
 it can depend on position. It should not be confused with the charge carrier density
Charge carrier density

The charge carrier density denotes the number of charge carriers per volume. It is measured in m-3. As any density it can depend on position....
. As related to chemistry, it can refer to the charge distribution over the volume of a particle, molecule, or atom.






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The linear, surface, or volume charge density is the amount of electric charge
Electric charge

Electric 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....
 in a line
Line (mathematics)

In geometry, a line is a Curvature curve. When geometry is used to model the real world, lines are used to represent straight objects with negligible width and height....
, surface
Surface

In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space E3....
, or volume
Volume

The volume of any solid, liquid, plasma, vacuum or theoretical object is how much three-dimensional space it occupies, often quantified numerically....
. It is measured in coulomb
Coulomb

The coulomb is the SI unit of electric charge. It is named after Charles-Augustin de Coulomb....
s per metre (C/m), square metre (C/m²), or cubic metre (C/m³), respectively. Since there are positive as well as negative charges, the charge density can take on negative values. Like any density
Density

The density of a material is defined as its mass per unit volume. The symbol of density is ....
 it can depend on position. It should not be confused with the charge carrier density
Charge carrier density

The charge carrier density denotes the number of charge carriers per volume. It is measured in m-3. As any density it can depend on position....
. As related to chemistry, it can refer to the charge distribution over the volume of a particle, molecule, or atom. Therefore, a lithium cation will carry a higher charge density than a sodium cation due to its smaller ionic radius.

Classical charge density


Continuous charges

The integral
Integral

Integration is an important concept in mathematics, specifically in the field of calculus and, more broadly, mathematical analysis. Given a function ƒ of a Real number variable x and an interval [a, b] of the real line, the integral...
 of the charge density , , over a line , surface , or volume , is equal to the total charge of that region, defined to be:

, ,

This relation defines the charge density mathematically. Note that the symbols used to denote the various dimensions of charge density vary between fields of studies. Other commonly used notations are , , ; or , , for (C/m), (C/m²), (C/m³) and respectively.

Homogeneous charge density


For the special case of a homogeneous
Homogeneity (physics)

In physics, homogeneous mixtures are mixtures that have definite, consistent composition and properties. Particles are uniformly spread. For example, any amount of a given mixture has the same composition and properties....
 charge density, that is one that is independent of position, equal to the equation simplifies to:

The proof of this is simple. Start with the definition of the charge of any volume:

Then, by definition of homogeneity, is a constant that we will denote to differentiate between the constant and non-constant forms, and thus by the properties of an integral can be pulled outside of the integral resulting in: so,

The equivalent proofs for linear charge density and surface charge density follow the same arguments as above.

Discrete charges

If the charge in a region consists of discrete point-like charge carriers like electron
Electron

The electron is a subatomic particle that carries a negative electric charge. It has elementary particle and is believed to be a point particle....
s the charge density can be expressed via the Dirac delta function
Dirac delta function

The Dirac delta or Dirac's delta is a mathematics construct introduced by theoretical physicist Paul Dirac. Informally, it is a function representing an infinitely sharp peak bounding unit area: a function d that has the value 0 everywhere except at x = 0 where its value is infinity in such a way that its total integral is 1....
, for example, the volume charge density is: Here, and the position of the th charge carrier. If all charge carriers have the same charge (for electrons ) the charge density can be expressed through the charge carrier density
Again, the equivalent equations for the linear and surface charge densities follow directly from the above relations.

Quantum charge density


In quantum mechanics
Quantum mechanics

Quantum mechanics is a set of principles underlying the most fundamental known description of all physical systems at the microscopic scale . Notable amongst these principles are both a dual wave-like and particle-like behavior of matter and radiation, and prediction of probabilities in situations where classical physics predicts certaintie...
, charge density is related to wavefunction
Wavefunction

A wave function or wavefunction is a mathematical tool used in quantum mechanics to describe any physical system. It is a function from a mathematical space that maps the possible states of the system into the complex numbers....
  by the equation

when the wavefunction is normalized as

Application

The charge density appears in the continuity equation
Continuity equation

A continuity equation is a differential equation that describes the conservative transport of some kind of quantity. Since mass, energy, momentum, and other natural quantities are conserved, a vast variety of physics may be described with continuity equations....
 which follows from Maxwell's Equations
Maxwell's equations

In electromagnetism, James Clerk Maxwell equations are a set of four partial differential equations that describe the properties of the electric field and magnetic field fields and relate them to their sources, charge density and current density....
 in the electromagnetic theory.

See also

  • Density
    Density

    The density of a material is defined as its mass per unit volume. The symbol of density is ....
  • Continuity equation
    Continuity equation

    A continuity equation is a differential equation that describes the conservative transport of some kind of quantity. Since mass, energy, momentum, and other natural quantities are conserved, a vast variety of physics may be described with continuity equations....
     relating charge density and current density
    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 quantities....