Charge density


Charge density

The linear, surface, or volume charge density is the amount of electric charge in a line, surface, or volume, respectively. It is measured in coulombs per meter (C/m), square meter (C/m²), or cubic meter (C/m³), respectively, and represented by the lowercase Greek letter Rho (ρ). 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. As related to chemistry, it can refer to the charge distribution over the volume of a particle, molecule, or atom. Therefore, a lithium cation will carry a higher charge density than a sodium cation due to its smaller ionic radius.

Contents

Classical charge density

Continuous charges

The integral of the charge density \alpha_q(\mathbf r), \sigma_q(\mathbf r), \rho_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:[1]

Q=\int\limits_L \alpha_q(\mathbf r) \,\mathrm{d}l (line integral)
Q=\int\limits_S \sigma_q(\mathbf r) \,\mathrm{d}S (surface integral)
Q=\int\limits_V \rho_q(\mathbf r) \,\mathrm{d}V (volume integral)

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 λ, σ, ρ; or ρl, ρs, ρv for (C/m), (C/m²), (C/m³), respectively.

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:

Q=V\cdot \rho_{q,0}.

The proof of this is simple. Start with the definition of the charge of any volume:

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

Then, by definition of homogeneity, \rho_q(\mathbf r) 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:

Q=\rho_{q,0} \int\limits_V \,\mathrm{d}V = \rho_{q,0} \cdot V

so,

Q=V \cdot \rho_{q,0}.

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:

\rho(\mathbf{r})=\sum_{i=1}^N\ q_i\delta(\mathbf{r} - \mathbf{r}_i)\,\! ;

where \mathbf{r}\,\! is the test position,q_i\,\! is the charge of the ith charge carrier, whose position is \mathbf{r}_i\,\! .

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): Again, the equivalent equations for the linear and surface charge densities follow directly from the above relations.

Relative charge density

From the perspective of the theory of relativity, the length of a segment of wire depends on velocity of observer, so charge density is a relative concept. Anthony French[2] 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." The approach to electromagnetism through spacetime symmetry is called relativistic electromagnetism.

Quantum charge density

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

\rho_q(\mathbf r) = q\cdot|\psi(\mathbf r)|^2

when the wavefunction is normalized as

Q= q\cdot \int |\psi(\mathbf r)|^2 \, d\mathbf r.

Application

The charge density appears in the continuity equation, which follows from Maxwell's Equations in the electromagnetic theory.

See also

References

  1. ^ Spacial Charge Distributions - http://faculty.wwu.edu/vawter/PhysicsNet/Topics/Gauss/SpacialCharge.html
  2. ^ A. French (1968) Special Relativity, chapter 8 Relativity and electricity, pp 229–65, W. W. Norton.

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