Charge density
Encyclopedia
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.
of the charge density
, 
, 
over a line 
, surface 
, or volume 
, is equal to the total charge 
of that region, defined to be:
(line integral
)
(surface integral
)
(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
, 
, 
; or 
, 
, 
for (C/m), (C/m²), (C/m³), respectively.
charge density, that is one that is independent of position, equal to
the equation simplifies to:
The proof of this is simple. Start with the definition of the charge of any volume:
Then, by definition of homogeneity,
is a constant that we will denote 
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:
so,
The equivalent proofs for linear charge density and surface charge density follow the same arguments as above.
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:
;
where
is the test position，
is the charge of the ith charge carrier, whose position is 
.
If all charge carriers have the same charge
(for electrons 
) the charge density can be expressed through the charge carrier density
:
Again, the equivalent equations for the linear and surface charge densities follow directly from the above relations.
, the length of a segment of wire depends on velocity
of observer, so charge density is a relative concept. 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." The approach to electromagnetism
through spacetime
symmetry is called relativistic electromagnetism
.
, charge density is related to wavefunction
by the equation
when the wavefunction is normalized as
, which follows from Maxwell's Equations
in the electromagnetic theory.
Electric charge
Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...
in a line
Line (mathematics)
The notion of line or straight line was introduced by the ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects...
, surface
Surface
In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...
, or volume
Volume
Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....
, 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
Rho
Rho is the 17th letter of the Greek alphabet. In the system of Greek numerals, it has a value of 100. It is derived from Semitic resh "head"...
(ρ). Since there are positive as well as negative charges, the charge density can take on negative values. Like any density
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...
it can depend on position. It should not be confused with the charge carrier density
Charge carrier density
The charge carrier density denotes the number of charge carriers per volume. It is measured in m−3. As any density it can depend on position.It should not be confused with the charge density, which is the number of charges per volume at a given energy....
. 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.
Continuous charges
The integralIntegral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
of the charge density
, , over a line , surface , or volume , is equal to the total charge of that region, defined to be: (line integralLine integral
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve.The function to be integrated may be a scalar field or a vector field...
)
(surface integralSurface integral
In mathematics, a surface integral is a definite integral taken over a surface ; it can be thought of as the double integral analog of the line integral...
)
(volume integralVolume integral
In mathematics — in particular, in multivariable calculus — a volume integral refers to an integral over a 3-dimensional domain....
)
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
, , ; or , , for (C/m), (C/m²), (C/m³), respectively.Homogeneous charge density
For the special case of a homogeneousHomogeneity (physics)
In general, homogeneity is defined as the quality or state of being homogeneous . For instance, a uniform electric field would be compatible with homogeneity...
charge density, that is one that is independent of position, equal to
the equation simplifies to:The proof of this is simple. Start with the definition of the charge of any volume:
Then, by definition of homogeneity,
is a constant that we will denote 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:so,
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 discrete point-like charge carriers like electronElectron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
s the charge density can be expressed via the Dirac delta function
Dirac delta function
The Dirac delta function, or δ function, is a generalized function depending on a real parameter such that it is zero for all values of the parameter except when the parameter is zero, and its integral over the parameter from −∞ to ∞ is equal to one. It was introduced by theoretical...
, for example, the volume charge density is:
;where
is the test position， is the charge of the ith charge carrier, whose position is .If all charge carriers have the same charge
(for electrons ) the charge density can be expressed through the charge carrier density: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 relativityTheory of relativity
The theory of relativity, or simply relativity, encompasses two theories of Albert Einstein: special relativity and general relativity. However, the word relativity is sometimes used in reference to Galilean invariance....
, the length of a segment of wire depends on velocity
Velocity
In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...
of observer, so charge density is a relative concept. Anthony French
Anthony French
Anthony Philip French is an emeritus professor of physics at the Massachusetts Institute of Technology. He was born in Brighton, England....
has described how the magnetic field
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
force of a current-bearing wire arises from this relative charge density. He used (p 260) a Minkowski diagram
Minkowski diagram
The Minkowski diagram was developed in 1908 by Hermann Minkowski and provides an illustration of the properties of space and time in the special theory of relativity. It allows a quantitative understanding of the corresponding phenomena like time dilation and length contraction without mathematical...
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
Electromagnetism
Electromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...
through spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
symmetry is called relativistic electromagnetism
Relativistic electromagnetism
Relativistic electromagnetism is a modern teaching strategy for developing electromagnetic field theory from Coulomb’s law and Lorentz transformations. Though Coulomb’s law expresses action at a distance, it is an easily understood electric force principle...
.
Quantum charge density
In quantum mechanicsQuantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
, charge density is related to wavefunction
Wavefunction
Not to be confused with the related concept of the Wave equationA wave function or wavefunction is a probability amplitude in quantum mechanics describing the quantum state of a particle and how it behaves. Typically, its values are complex numbers and, for a single particle, it is a function of...
by the equationwhen the wavefunction is normalized as
Application
The charge density appears in the continuity equationContinuity equation
A continuity equation in physics is a differential equation that describes the transport of a conserved quantity. Since mass, energy, momentum, electric charge and other natural quantities are conserved under their respective appropriate conditions, a variety of physical phenomena may be described...
, which follows from Maxwell's Equations
Maxwell's equations
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...
in the electromagnetic theory.
See also
- Continuity equationContinuity equationA continuity equation in physics is a differential equation that describes the transport of a conserved quantity. Since mass, energy, momentum, electric charge and other natural quantities are conserved under their respective appropriate conditions, a variety of physical phenomena may be described...
relating charge density and current densityCurrent densityCurrent density is a measure of the density of flow of a conserved charge. Usually the charge is the electric charge, in which case the associated current density is the electric current per unit area of cross section, but the term current density can also be applied to other conserved... - Ionic potentialIonic potentialIonic potential is the ratio of electric charge to the radius of an ion.As such, the proportion measures the charge density at the surface of the ion; usually the denser the charge, the stronger will be the bond that the ion forms....