In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system in coulombs per cubic meter (C⋅m−3), at any point in a volume.Surface charge density (σ) is the quantity of charge per unit area, measured in coulombs per square meter (C⋅m−2), at any point on a surface charge distribution on a two dimensional surface. Linear charge density (λ) is the quantity of charge per unit length, measured in coulombs per meter (C⋅m−1), at any point on a line charge distribution. Charge density can be either positive or negative, since electric charge can be either positive or negative.
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. A charge distribution is ultimately composed of individual charged particles separated by regions containing no charge. 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 ions on the surface of objects, and the space charge in a vacuum tube is composed of a cloud of free electrons moving randomly in space. The charge carrier density in a conductor is equal to the number of mobile charge carriers (electrons, ions, etc.) per unit volume. The charge density at any point is equal to the charge carrier density multiplied by the elementary charge on the particles. However, because the elementary charge on an electron is so small (1.6⋅10−19 C) and there are so many of them in a macroscopic volume (there are about 1022 conduction electrons in a cubic centimeter of copper) the continuous approximation is very accurate when applied to macroscopic volumes, and even microscopic volumes above the nanometer level.
At atomic scales, 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. This is the meaning of 'charge distribution' and 'charge density' used in chemistry and chemical bonding. An electron is represented by a wavefunction whose square is proportional to the probability of finding the electron at any point in space, so is proportional to the charge density of 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.
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 normaln̂, d is the dipole moment between two point charges, the volume density of these is the polarization densityP. Position vectorr is a point to calculate the electric field; r′ is a point in the charged object.
Following are the definitions for continuous charge distributions.
The linear charge density is the ratio of an infinitesimal electric charge dQ (SI unit: C) to an infinitesimal line element,
similarly the surface charge density uses a surface area element dS
and the volume charge density uses a volume element dV
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,
similarly a surface integral of the surface charge density σq(r) over a surface S,
Within the context of electromagnetism, the subscripts are usually dropped for simplicity: λ, σ, ρ. Other notations may include: ρℓ, ρs, ρv, ρL, ρS, ρV etc.
The total charge divided by the length, surface area, or volume will be the average charge densities:
Free, bound and total charge
In dielectric materials, the total charge of an object can be separated into "free" and "bound" charges.
Bound charges set up electric dipoles in response to an applied electric fieldE, 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.
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.
Total charge densities
In terms of volume charge densities, the total charge density is:
as for surface charge densities:
where subscripts "f" and "b" denote "free" and "bound" respectively.
The bound surface charge is the charge piled up at the surface of the dielectric, given by the dipole moment perpendicular to the surface:
which separates into the potential of the surface charge (surface integral) and the potential due to the volume charge (volume integral):
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 fieldD emerging from the object:
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:
The proof of this is immediate. Start with the definition of</ref></ref></ref></ref> the charge of any volume:
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:
The equivalent proofs for linear charge density and surface charge density follow the same arguments as above.
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:
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:
so the delta function ensures that when the charge density is integrated over R, the total charge in R is 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 qi at position ri, where i = 1, 2, ..., N:
The delta function for each charge qi in the sum, δ(r − ri), ensures the integral of charge density over R returns the total charge in R:
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
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
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.
^Derek F. Lawden (2012) An Introduction to Tensor Calculus: Relativity and Cosmology, page 74, Courier Corporation ISBN 0-486-13214-5
^Jack Vanderlinde (2006) Classical Electromagnetic Theory, § 11.1 The Four-potential and Coulomb's Law, page 314, Springer Science & Business Media ISBN 1-4020-2700-1
^R. J. Gillespie & P. L. A. Popelier (2001). "Chemical Bonding and Molecular Geometry". Environmental Science & Technology. Oxford University Press. 52 (7): 4108–4116. Bibcode:2018EnST...52.4108E. doi:10.1021/acs.est.7b06400. PMID29510032.
^Razi Epsztein, Evyatar Shaulsky, Nadir Dizge, David M Warsinger, Menachem Elimelech (2018). "Ionic Charge Density-Dependent Donnan Exclusion in Nanofiltration of Monovalent Anions". Environmental Science & Technology. 52 (7): 4108–4116. Bibcode:2018EnST...52.4108E. doi:10.1021/acs.est.7b06400. PMID29510032.CS1 maint: multiple names: authors list (link)
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