Charge density

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The charge density is the amount of electric charge in a volume. It is measured in coulombs per cubic metre (C/m³). 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.

1 Definition
2 Calculation
3 Application
4 See also

[edit] Definition

The integral of the charge density <math>\rho_q(\mathbf r)</math> over a volume <math>V</math> is equal to the total charge <math>Q</math> of that volume

<math>Q=\int\limits_V \rho_q(\mathbf r) \,\mathrm{d}V.</math>

This relation defines the charge density mathematically. For a homogeneous charge density, that is one that is independent of position, equal to <math>\rho_{q,0}</math> the equation simplifies to

Two special cases of charge density are obtained by replacing in the above definition "volume" with either "area" or "length". The densities thus defined are called area charge density <math>\sigma_q</math> and linear charge density <math>\alpha_q</math>

<math>Q=\int\limits_A \sigma_q(\mathbf r) dA</math>,

<math>Q=\int\limits_L \alpha_q(\mathbf r) dl</math>

They are measured in coulombs per square metre (C/m²) and coulombs per metre (C/m) respectively and are particularly relevant in two and one-dimensional systems. Note that the symbols used to denote the various dimensions of charge density vary between fields of studies. Other commonly used notations are <math>\rho</math>, <math>\sigma</math>, <math>\lambda</math>; or <math>\rho_l</math>, <math>\rho_s</math>, <math>\rho_v</math> for (C/m³), (C/m²), and (C/m) respectively.

[edit] Calculation

If the charge in a volume consists of <math>N</math> discrete point-like charge carriers like electrons the charge density can be expressed via the Dirac delta function

<math>\rho_q(\mathbf r) =\sum_{i=1}^N q_i\cdot \delta(\mathbf r - \mathbf r_i).</math>

Here, <math>q_i</math> is the charge and <math>\mathbf r_i</math> the position of the <math>i</math>th charge carrier. If all charge carriers have the same charge <math>q</math> (for electrons <math>q=-e</math>) the charge density can be expressed through the charge carrier density <math>n(\mathbf r)</math>:

<math>\rho_q(\mathbf r)=q\cdot\sum_{i=1}^N \delta(\mathbf r - \mathbf r_i)=q\cdot n(\mathbf r)</math>

In quantum mechanics, charge density is related to wavefunction <math> \psi(\mathbf r)</math> by the equation

<math>\rho_q(\mathbf r) = q\cdot|\psi(\mathbf r)|^2 </math>

when the wavefunction is normalized as

<math>Q= q\cdot \int |\psi(\mathbf r)|^2 \, d\mathbf r </math>