# Charge density

Charge density

The linear, surface, or volume charge density is the amount of electric charge in a line, surface, or volume, respectively. It is measured in coulombs per meter (C/m), square meter (C/m²), or cubic meter (C/m³), respectively, and represented by the lowercase Greek letter Rho (ρ). Since there are positive as well as negative charges, the charge density can take on negative values. Like any density it can depend on position. It should not be confused with the charge carrier density. 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 of the charge density $\alpha_q(\mathbf r)$, $\sigma_q(\mathbf r)$, $\rho_q(\mathbf r)$ over a line l, surface S, or volume V, is equal to the total charge Q of that region, defined to be:[1]

$Q=\int\limits_L \alpha_q(\mathbf r) \,\mathrm{d}l$ (line integral)
$Q=\int\limits_S \sigma_q(\mathbf r) \,\mathrm{d}S$ (surface integral)
$Q=\int\limits_V \rho_q(\mathbf r) \,\mathrm{d}V$ (volume integral)

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 λ, σ, ρ; or ρl, ρs, ρv for (C/m), (C/m²), (C/m³), respectively.

### Homogeneous charge density

For the special case of a homogeneous charge density, that is one that is independent of position, equal to ρq,0 the equation simplifies to:

$Q=V\cdot \rho_{q,0}.$

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

$Q=\int\limits_V \rho_q(\mathbf r) \,\mathrm{d}V.$

Then, by definition of homogeneity, $\rho_q(\mathbf r)$ is a constant that we will denote ρq,0 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:

$Q=\rho_{q,0} \int\limits_V \,\mathrm{d}V = \rho_{q,0} \cdot V$

so,

$Q=V \cdot \rho_{q,0}.$

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 N discrete point-like charge carriers like electrons the charge density can be expressed via the Dirac delta function, for example, the volume charge density is:

$\rho(\mathbf{r})=\sum_{i=1}^N\ q_i\delta(\mathbf{r} - \mathbf{r}_i)\,\!$ ;

where $\mathbf{r}\,\!$ is the test position，$q_i\,\!$ is the charge of the ith charge carrier, whose position is $\mathbf{r}_i\,\!$ .

If all charge carriers have the same charge q (for electrons q = − e) the charge density can be expressed through the charge carrier density $n(\mathbf r)$: Again, the equivalent equations for the linear and surface charge densities follow directly from the above relations.

## Relative charge density

From the perspective of the theory of relativity, the length of a segment of wire depends on velocity of observer, so charge density is a relative concept. Anthony French[2] has described how the magnetic field force of a current-bearing wire arises from this relative charge density. He used (p 260) a Minkowski diagram to show "how a neutral current-bearing wire appears to carry a net charge density as observed in a moving frame." The approach to electromagnetism through spacetime symmetry is called relativistic electromagnetism.

## Quantum charge density

In quantum mechanics, charge density is related to wavefunction $\psi(\mathbf r)$ by the equation

$\rho_q(\mathbf r) = q\cdot|\psi(\mathbf r)|^2$

when the wavefunction is normalized as

$Q= q\cdot \int |\psi(\mathbf r)|^2 \, d\mathbf r.$

## Application

The charge density appears in the continuity equation, which follows from Maxwell's Equations in the electromagnetic theory.

## References

1. ^ Spacial Charge Distributions - http://faculty.wwu.edu/vawter/PhysicsNet/Topics/Gauss/SpacialCharge.html
2. ^ A. French (1968) Special Relativity, chapter 8 Relativity and electricity, pp 229–65, W. W. Norton.

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### Look at other dictionaries:

• charge density — noun The electric charge per unit volume of a field or body, or per unit area of a surface • • • Main Entry: ↑charge …   Useful english dictionary

• charge density —  Charge Density  Плотность зарядов   В коллоидных системах – величина заряда на поверхности, отнесенная к единице ее площади …   Толковый англо-русский словарь по нанотехнологии. - М.

• charge density — tūrinis krūvio tankis statusas T sritis Standartizacija ir metrologija apibrėžtis Elektros krūvis, padalytas iš tūrio. Matavimo vienetas: C/m³. atitikmenys: angl. charge density; volume density of charge; volumic charge vok. Raumladungsdichte, f; …   Penkiakalbis aiškinamasis metrologijos terminų žodynas

• charge density — krūvio tankis statusas T sritis fizika atitikmenys: angl. charge density; electric charge density vok. elektrische Ladungsdichte, f; Ladungsdichte, f rus. плотность заряда, f; плотность электрических зарядов, f pranc. densité de charge, f;… …   Fizikos terminų žodynas

• charge density — tūrinis krūvio tankis statusas T sritis fizika atitikmenys: angl. charge density; volume density of charge; volumic charge vok. Raumladungsdichte, f; Volumladung, f rus. объемная плотность электрического заряда, f; объемный заряд, m;… …   Fizikos terminų žodynas

• charge density — krūvio tankis statusas T sritis chemija apibrėžtis Fizikinis dydis, skaitine verte lygus elektros kiekiui vienetiniame tūryje, paviršiuje arba ilgyje. atitikmenys: angl. charge density rus. плотность заряда …   Chemijos terminų aiškinamasis žodynas

• charge density operator — krūvio tankio operatorius statusas T sritis fizika atitikmenys: angl. charge density operator vok. Ladungsdichteoperator, m rus. оператор плотности заряда, m pranc. opérateur de densité de charge, m …   Fizikos terminų žodynas

• charge density — noun The amount of electric charge per unit volume of space or unit area of a surface …   Wiktionary

• space charge density — tūrinio krūvio tankis statusas T sritis fizika atitikmenys: angl. space charge density; spatial charge density; volume charge density vok. Raumladungsdichte, f; räumliche Ladungsdichte, f rus. объёмная плотность заряда, f; плотность объёмного… …   Fizikos terminų žodynas

• spatial charge density — tūrinio krūvio tankis statusas T sritis fizika atitikmenys: angl. space charge density; spatial charge density; volume charge density vok. Raumladungsdichte, f; räumliche Ladungsdichte, f rus. объёмная плотность заряда, f; плотность объёмного… …   Fizikos terminų žodynas