What is the symbol of charge density

Charge density

The electrical charge density is a physical quantity from electrodynamics that describes a charge distribution. Since there are both positive and negative charges, both positive and negative values ​​are possible for the charge density.

The surface charge density σ (Sigma) on the right half of the metal ball is negative because the electrons escape there due to the repulsion by the negative charge shown on the left; on the left hemisphere the surface charge density is positive because electrons are now missing there.

Since charges can also be distributed on surfaces or along a thin wire, the charge density can describe:

  • the charge per volume (Space charge density ρ)
  • the charge per area (Surface charge density σ)
  • the load per length (Line charge density λ).

The achievable surface charge density is limited by corona discharge into the surrounding air if the maximum field strength of 105 V / m is exceeded:

$ \ sigma_ \ mathrm {max} = 2 \ cdot E_ \ mathrm {max} \ cdot \ varepsilon_0 \ cdot \ varepsilon_r \ approx 1.8 \ cdot 10 ^ {- 6} \ mathrm {As / m ^ 2}. $

So every negatively charged square centimeter carries the excess charge 1.8 · 10-10 As, which is 1.1 · 109 corresponds to freely moving electrons. About a million times more electrons are bound to the atomic cores of the metal surface (see also Influence # number of electrons involved).

Similar sizes

A quantity corresponding to the surface charge density σ is the electrical flux density $ \ vec D $ (also called electrical excitation, dielectric displacement or displacement density), a vector that is perpendicular to the surface in question; on the other hand, σ is a scalar (and under certain circumstances it is equal to the amount $ | \ vec D | $).

In addition, the charge should not be confused with the charge densitycarrierdensity, i.e. the number of protons, electrons, etc. per unit of space, area or length, as well as the electron density calculated in density functional theory.

definition

The definition of the space charge density is similar to that of the mass density:

$ \ rho (\ vec r) = \ frac {\ mathrm d Q} {\ mathrm d V} \ Leftrightarrow Q = \ int_V \ rho (\ vec r) \, \ mathrm d V $,

in which Q the electric charge and V. the volume is.

For the area and line charge density, the following is derived according to the area A or the length l:

$ \ sigma (\ vec r) = \ frac {\ mathrm d Q} {\ mathrm d A} \ Leftrightarrow Q = \ int_A \ sigma (\ vec r) \, \ mathrm d A $
$ \ lambda (\ vec r) = \ frac {\ mathrm d Q} {\ mathrm d l} \ Leftrightarrow Q = \ int_l \ lambda (\ vec r) \, \ mathrm d l. $

Discrete charge distribution

If the charge in a volume consists of $ N $ discrete charge carriers (such as electrons), the charge density can be expressed using the delta distribution:

$ \ rho (\ vec r) = \ sum_ {i = 1} ^ N q_i \ cdot \ delta (\ vec r - \ vec r_i) $

With

  • the charge $ q_i $ and
  • the location $ \ vec r_i $ of the $ i $ -th load carrier.

If all charge carriers carry the same charge $ q $ (with electrons the same as the negative elementary charge: $ q = -e $), the above formula can be simplified with the help of the charge carrier density $ n (\ vec r) $:

$ \ begin {align} \ rho (\ vec r) & = q \ cdot \ sum_ {i = 1} ^ N \ delta (\ vec r - \ vec r_i) \ & = q \ cdot n (\ vec r ). \ end {align} $

Electrical potential

The electrical potential depends according to the Poisson equation of electrostatics

$ \ Delta \ Phi (\ vec r) = - \ frac {\ rho (\ vec r)} {\ varepsilon} $

only depends on the charge density. Here $ \ varepsilon $ denotes the permittivity.