Charge density

In electromagnetism, charge density is a measure of electric charge per unit volume of space, in one, two or three dimensions. More specifically: the linear, surface, or volume charge density is the amount of electric charge per unit length, surface area, or volume, respectively. The respective SI units are C·m⁻¹, C·m⁻² or C·m⁻³.^[1]

Like any density, charge density can depend on position, but charge and thus charge density can be negative. It should not be confused with the charge carrier density, the number of charge carriers (e.g. electrons, ions) in a material per unit volume, not including the actual charge on the carriers

In chemistry, it can refer to the charge distribution over the volume of a particle; such as a molecule, atom or ion. Therefore, a lithium cation will carry a higher charge density than a sodium cation due to the lithium cation's having a smaller ionic radius, even though sodium has more protons (11) than lithium (3).

Definitions

Continuous charges

Continuous charge distribution. The volume charge density ρ is the amount of charge per unit volume (cube), 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.

Following are the definitions for continuous charge distributions.^[2][3]

The linear charge density is the ratio of an infinitesimal electric charge dQ (SI unit: C) to an infinitesimal line element,

$\lambda_q = \frac{d Q}{d \ell}\,,\quad$

similarly the surface charge density uses a surface area element dS

$\sigma_q = \frac{d Q}{d S}\,,\quad$

and the volume charge density uses a volume element dV

$\rho_q =\frac{d Q}{d V}\,,\quad$

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

$Q=\int\limits_L \lambda_q(\bold{r}) \,d\ell$

similarly a surface integral of the surface charge density σ_q(r) over a surface S,

$Q=\int\limits_S \sigma_q(\bold{r}) \,dS$

and a volume integral of the volume charge density ρ_q(r) over a volume V,

$Q=\int\limits_V \rho_q(\bold{r}) \,dV$

where the subscript q is to clarify that the density is for electric charge, not other densities like mass density, number density, probability density, and prevent conflict with the many other uses of λ, σ, ρ in electromagnetism for wavelength, electrical resistivity and conductivity.

Within the context of electromagnetism, the subscripts are usually dropped for simplicity: λ, σ, ρ. Other notations may include: ρ_ℓ, ρ_s, ρ_v, ρ_L, ρ_S, ρ_V etc.

Average charge densities

The total charge divided by the length, surface area, or volume will be the average charge densities:

$\langle\lambda_q \rangle = \frac{Q}{\ell}\,,\quad \langle\sigma_q\rangle = \frac{Q}{S}\,,\quad\langle\rho_q\rangle = \frac{Q}{V}\,.$

Free, bound and total charge

In dielectric materials, the total charge of an object can separate into "free" and "bound" charges.

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. They are called bound because they cannot be removed: in the dielectric material the charges are the electrons bound to the nuclei.^[3]

Free charges are the excess charges which can move into electrostatic equilibrium, i.e. when the charges are not moving and the resultant electric field is independent of time, or constitute electric currents.^[2]

Total charge densities

In terms of volume charges densities, the total charge density is:

$\rho = \rho_f + \rho_b\,.$

as for surface charge densities:

$\sigma = \sigma_f + \sigma_b\,.$

where subscripts "f" and "b" denote "free" and "bound" respectively.

Bound charge

The bound surface charge is the charge piled-up at the surface of the dielectric, given by the dipole moment perpendicular to the surface:^[3]

$q_b = \frac{\bold{d} \cdot\mathbf{\hat{n}}}{|\bold{s}|}$

where s is the separation between the point charges constituting the dipole. Taking infinitesimals:

$d q_b = \frac{d\bold{d}}{|\bold{s}|}\cdot\mathbf{\hat{n}}$

and dividing by the differential surface element dS gives the bound surface charge density:

$\sigma_b = \frac{d q_b}{d S} = \frac{d\bold{d}}{|\bold{s}| dS} \cdot\mathbf{\hat{n}} = \frac{d\bold{d}}{dV} \cdot\mathbf{\hat{n}} = \bold{P} \cdot\mathbf{\hat{n}}\,.$

where P is the polarization density, i.e. density of electric dipole moments within the material, and dV is the differential volume element.

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

$q_b = \iiint \rho_b dV = -$${\scriptstyle S}$$\bold{P} \cdot \mathbf{\hat{n}} dS = -\iiint\nabla\cdot\mathbf{P}dV$

hence:

$\rho_b = - \nabla\cdot\mathbf{P}\,.$

The negative sign arises due to the opposite signs on the charges in the dipoles, one end is within the volume of the object, the other at the surface.

A more rigorous derivation is given below.^[3]

Derivation of bound surface and volume charge densities from internal dipole moments (bound charges)

The electric potential due to a dipole moment d is: $\varphi = \frac{1}{4\pi\epsilon_0}\frac{(\bold{r}-\bold{r}')\cdot\bold{d}}{|\bold{r}-\bold{r}'|^3}$ For a continuous distribution, the material can be divided up into infinitely many infinitesimal dipoles $d\bold{d}=\bold{P}dV=\bold{P}d^3\bold{r}$ where dV = d^3r′ is the volume element, so the potential is the volume integral over the object: $\varphi = \frac{1}{4\pi\epsilon_0}\iiint\frac{(\bold{r}-\bold{r}')\cdot\bold{P}}{|\bold{r}-\bold{r}'|^3}d^3\bold{r'}$ Since $\nabla'\left(\frac{1}{|\bold{r}-\bold{r}'|}\right) \equiv \left(\bold{e}_x \frac{\partial }{\partial x'} + \bold{e}_y\frac{\partial }{\partial y'} + \bold{e}_z\frac{\partial }{\partial z'}\right)\left(\frac{1}{|\bold{r}-\bold{r}'|}\right) = \frac{\bold{r}-\bold{r}'}{|\bold{r}-\bold{r}'|^3}$ where ∇′ is the gradient in the r′ coordinates, $\varphi = \frac{1}{4\pi\epsilon_0}\iiint\bold{P}\cdot\nabla'\left(\frac{1}{|\bold{r}-\bold{r}'|}\right)d^3\bold{r'}$ integrating by parts $\varphi = \frac{1}{4\pi\epsilon_0}\iiint\left[\nabla'\cdot\left(\frac{\bold{P}}{|\bold{r}-\bold{r}'|}\right)-\frac{1}{\bold{r}-\bold{r}'}(\nabla'\cdot\bold{P})\right]d^3\bold{r'}$ using the divergence theorem: $\varphi = \frac{1}{4\pi\epsilon_0}$${\scriptstyle S}$$\frac{\bold{P}\cdot \mathbf{\hat{n}}dS'}{|\bold{r}-\bold{r}'|}- \frac{1}{4\pi\epsilon_0} \iiint\frac{\nabla'\cdot\bold{P}}{|\bold{r}-\bold{r}'|} d^3\bold{r'}$ which separates into the potential of the surface charge (surface integral) and the potential due to the volume charge (volume integral): $\varphi = \frac{1}{4\pi\epsilon_0}$${\scriptstyle S}$$\frac{\sigma_b dS'}{|\bold{r}-\bold{r}'|}+ \frac{1}{4\pi\epsilon_0} \iiint\frac{\rho_b}{|\bold{r}-\bold{r}'|} d^3\bold{r'}$ that is $\sigma_b=\bold{P}\cdot\bold{\hat{n}}\,,\quad \rho_b = -\nabla\cdot\bold{P}$

Free charge density

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:

$\Phi_D =$${\scriptstyle S}$$\bold{D} \cdot\bold{\hat{n}}dS = \iiint \rho_f dV$

See Maxwell's equations and constitutive relation for more details.

Homogeneous charge density

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:

$Q=V\cdot \rho_0.$

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

$Q=\int\limits_V \rho_q(\bold{r}) \,dV.$

Then, by definition of homogeneity, ρ_q(r) is a constant denoted by ρ_{q, 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:

$Q=\rho_{q,0} \int\limits_V \,dV = \rho_0 V$

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

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:

$\rho_q(\bold{r}) = q \delta(\mathbf{r} - \mathbf{r}_0)$

where r is the position to calculate the charge.

As always, the integral of the charge density over a region of space is the charge contained in that region. The delta function has the sifting property for any function f:

$\int_R d^3 \mathbf{r} f(\mathbf{r})\delta(\mathbf{r} - \mathbf{r}_0) = f(\mathbf{r}_0)$

so the delta function ensures that when the charge density is integrated over R, the total charge in R is q:

$Q =\int_R d^3 \mathbf{r} \, \rho_q =\int_R d^3 \mathbf{r} \, q \delta(\mathbf{r} - \mathbf{r}_0) = q \int_R d^3 \mathbf{r} \, \delta(\mathbf{r} - \mathbf{r}_0) = q$

This can be extended to N discrete point-like charge carriers. The charge density of the system at a point r is a sum of the charge densities for each charge q_i at position r_i, where i = 1, 2, ..., N:

$\rho_q(\bold{r})=\sum_{i=1}^N\ q_i\delta(\mathbf{r} - \mathbf{r}_i)\,\!$

The delta function for each charge q_i in the sum, δ(r − r_i), ensures the integral of charge density over R returns the total charge in R:

$Q=\int_R d^3 \mathbf{r} \sum_{i=1}^N\ q_i\delta(\mathbf{r} - \mathbf{r}_i) = \sum_{i=1}^N\ q_i \int_R d^3 \mathbf{r} \delta(\mathbf{r} - \mathbf{r}_i) = \sum_{i=1}^N\ q_i$

If all charge carriers have the same charge q (for electrons q = −e, the electron charge) the charge density can be expressed through the number of charge carriers per unit volume, n(r), by

$\rho_q(\bold{r})= q n(\mathbf{r})\,.$

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

Charge density in special relativity

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^[4] 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." When a charge density is measured in a moving frame of reference it is called proper charge density.^[5][6][7]

It turns out the charge density ρ and current density J transform together as a four current vector under Lorentz transformations.

Charge density in quantum mechanics

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

$\rho_q(\bold{r}) = q |\psi(\mathbf r)|^2$

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

$Q= \int_R q |\psi(\mathbf r)|^2 \, {\rm d}^3 \bold{r}$

where d³r is the integration measure over 3d position space.

Application

The charge density appears in the continuity equation for electric current, also in Maxwell's Equations. It is the principal source term of the electromagnetic field, when the charge distribution moves this corresponds to a current density.

See also

• Continuity equation relating charge density and current density
• Ionic potential
• charge density wave

References

1. P.M. Whelan, M.J. Hodgeson (1978). Essential Principles of Physics (2nd ed.). John Murray. ISBN 0-7195-3382-1.
2. I.S. Grant, W.R. Phillips (2008). Electromagnetism (2nd ed.). Manchester Physics, John Wiley & Sons. ISBN 9-780471-927129.
3. D.J. Griffiths (2007). Introduction to Electrodynamics (3rd ed.). Pearson Education, Dorling Kindersley. ISBN 81-7758-293-3.
4. A. French (1968) Special Relativity, chapter 8 Relativity and electricity, pp 229–65, W. W. Norton.
5. Richard A. Mould (2001) Basic Relativity, §62 Lorentz force, Springer Science & Business Media ISBN 0387952101
6. Derek F. Lawden (2012) An Introduction to Tensor Calculus: Relativity and Cosmology, page 74, Courier Corporation ISBN 0486132145
7. Jack Vanderlinde (2006) Classical Electromagnetic Theory, § 11.1 The Four-potential and Coulomb’s Law, page 314, Springer Science & Business Media ISBN 1402027001

• A. Halpern (1988). 3000 Solved Problems in Physics. Schaum Series, Mc Graw Hill. ISBN 978-0-07-025734-4.
• G. Woan (2010). The Cambridge Handbook of Physics Formulas. Cambridge University Press. ISBN 978-0-521-57507-2.
• P. A. Tipler, G. Mosca (2008). Physics for Scientists and Engineers - with Modern Physics (6th ed.). Freeman. ISBN 978-0-7167-8964-2.
• R.G. Lerner, G.L. Trigg (1991). Encyclopaedia of Physics (2nd ed.). VHC publishers. ISBN 978-0-89573-752-6.
• C.B. Parker (1994). McGraw Hill Encyclopaedia of Physics (2nd ed.). VHC publishers. ISBN 978-0-07-051400-3.

External links

• - Spatial charge distributions