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CLASSICAL CHARGE DENSITY



Continuous charges

The Integral of the charge density \alpha_q(\mathbf r), \sigma_q(\mathbf r), ho_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: Spacial Charge Distributions - http://www.ac.wwu.edu/~vawter/PhysicsNet/Topics/Gauss/SpacialCharge.html

:Q=\int\limits_L \alpha_q(\mathbf r) dl,
:Q=\int\limits_S \sigma_q(\mathbf r) dS,
:Q=\int\limits_V ho_q(\mathbf r) \,\mathrm{d}V.

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 \lambda, \sigma, ho; or ho_l, ho_s, ho_v for (C/m), (C/m&2), (C/m³) and respectively.


Homogeneous charge density


For the special case of a Homogeneous charge density, that is one that is independent of position, equal to ho_{q,0} the equation simplifies to:
:Q=V\cdot ho_{q,0}

The proof of this is simple. Start with the definition of the charge of any volume:
:Q=\int\limits_V ho_q(\mathbf r) \,\mathrm{d}V

Then, by definition of homogeneity, ho_q(\mathbf r) is a constant that we will denote ho_{q,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= ho_{q,0} \int\limits_V \,\mathrm{d}V

Again, by the properties of integrals:
:\int\limits_V \,\mathrm{d}V = V

Therefore by substitution:
: ho_{q,0} \int\limits_V \,\mathrm{d}V = V\cdot ho_{q,0}

Which leads to:
:Q=V\cdot ho_{q,0}

Which is precisely the result mentioned above for volume charge density. 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 Electron s the charge density can be expressed via the Dirac Delta Function , for example, the volume charge density is:

: ho_q(\mathbf r) =\sum_{i=1}^N q_i\cdot \delta(\mathbf r - \mathbf r_i).

Here, q_i is the charge and \mathbf r_i the position of the ith charge carrier. 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):

: ho_q(\mathbf r)=q\cdot\sum_{i=1}^N \delta(\mathbf r - \mathbf r_i)=q\cdot n(\mathbf r)

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 \psi(\mathbf r) by the equation

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