Charge density

In classical electromagnetic theory charge density is idealized as a continuous scalar function of position

This is expressed by a continuity equation which links the rate of change of charge density

Since all charge is carried by subatomic particles, which can be idealized as points, the concept of a continuous charge distribution is an approximation, which becomes inaccurate at small length scales.

[4] For example, the charge in an electrically charged metal object is made up of conduction electrons moving randomly in the metal's crystal lattice.

Static electricity is caused by surface charges consisting of electrons and ions near the surface of objects, and the space charge in a vacuum tube is composed of a cloud of free electrons moving randomly in space.

At even smaller scales, of atoms and molecules, due to the uncertainty principle of quantum mechanics, a charged particle does not have a precise position but is represented by a probability distribution, so the charge of an individual particle is not concentrated at a point but is 'smeared out' in space and acts like a true continuous charge distribution.

[4] This is the meaning of 'charge distribution' and 'charge density' used in chemistry and chemical bonding.

whose square is proportional to the probability of finding the electron at any point

In atoms and molecules the charge of the electrons is distributed in clouds called orbitals which surround the atom or molecule, and are responsible for chemical bonds.

Integrating the definitions gives the total charge Q of a region according to line integral of the linear charge density λq(r) over a line or 1d curve C,

Within the context of electromagnetism, the subscripts are usually dropped for simplicity: λ, σ, ρ.

Bound charges set up electric dipoles in response to an applied electric field E, and polarize other nearby dipoles tending to line them up, the net accumulation of charge from the orientation of the dipoles is the bound charge.

Using the divergence theorem, the bound volume charge density within the material is

For a continuous distribution, the material can be divided up into infinitely many infinitesimal dipoles

using the divergence theorem: which separates into the potential of the surface charge (surface integral) and the potential due to the volume charge (volume integral): that is

The free charge density serves as a useful simplification in Gauss's law for electricity; the volume integral of it is the free charge enclosed in a charged object - equal to the net flux of the electric displacement field D emerging from the object: See Maxwell's equations and constitutive relation for more details.

For the special case of a homogeneous charge density ρ0, independent of position i.e. constant throughout the region of the material, the equation simplifies to:

Start with the definition of a continuous volume charge density:

Then, by definition of homogeneity, ρq(r) is a constant denoted by ρq, 0 (to differ between the constant and non-constant densities), and so by the properties of an integral can be pulled outside of the integral resulting in:

For a single point charge q at position r0 inside a region of 3d space R, like an electron, the volume charge density can be expressed by the Dirac delta function:

Similar equations are used for the linear and surface charge densities.

In special relativity, the length of a segment of wire depends on velocity of observer because of length contraction, so charge density will also depend on velocity.

Anthony French[7] 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."

In quantum mechanics, charge density ρq is related to wavefunction ψ(r) by the equation

where q is the charge of the particle and |ψ(r)|2 = ψ*(r)ψ(r) is the probability density function i.e. probability per unit volume of a particle located at r. When the wavefunction is normalized - the average charge in the region r ∈ R is

Note that this is excluding the exchange energy of the system, which is a purely quantum mechanical phenomenon, has to be calculated separately.

It is the principal source term of the electromagnetic field; when the charge distribution moves, this corresponds to a current density.

The charge density of molecules impacts chemical and separation processes.

[13] For separation processes such as nanofiltration, the charge density of ions influences their rejection by the membrane.

Continuous charge distribution. The volume charge density ρ is the amount of charge per unit volume (three dimensional), surface charge density σ is amount per unit surface area (circle) with outward unit normal n̂ , d is the dipole moment between two point charges, the volume density of these is the polarization density P . Position vector r is a point to calculate the electric field ; r′ is a point in the charged object.