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:
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 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:
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 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:
- ;
where is the test position, is the charge of the ith charge carrier, whose position is .
If all charge carriers have the same charge q (for electrons q = − e) 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, charge density is related to wavefunction by the equation
when the wavefunction is normalized as