Charge density

"Charge distribution" redirects here. This article is about thephysical quantityinelectromagnetism.For other uses seechargeanddensity.

Inelectromagnetism,charge densityis the amount ofelectric chargeper unitlength,surface area,orvolume.Volume charge density(symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in theSIsystem incoulombsper cubicmeter(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 asurface charge distributionon 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.

Likemass density,charge density can vary with position. Inclassical electromagnetic theorycharge density is idealized as acontinuousscalarfunction of position${\boldsymbol {x}}$,like a fluid, and$\rho ({\boldsymbol {x}})$,$\sigma ({\boldsymbol {x}})$,and$\lambda ({\boldsymbol {x}})$are usually regarded ascontinuous charge distributions,even though all real charge distributions are made up of discrete charged particles. Due to theconservation of electric charge,the charge density in any volume can only change if anelectric currentof charge flows into or out of the volume. This is expressed by acontinuity equationwhich links the rate of change of charge density$\rho ({\boldsymbol {x}})$and thecurrent density${\boldsymbol {J}}({\boldsymbol {x}})$.

Since all charge is carried bysubatomic particles,which can be idealized as points, the concept of acontinuouscharge 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 ofconduction electronsmoving randomly in the metal'scrystal lattice.Static electricityis caused by surface charges consisting of electrons andionsnear the surface of objects, and thespace chargein avacuum tubeis composed of a cloud of free electrons moving randomly in space. Thecharge carrier densityin a conductor is equal to the number of mobilecharge 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 theelementary chargeon an electron is so small (1.6⋅10⁻¹⁹C) and there are so many of them in a macroscopic volume (there are about 10²²conduction 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 even smaller scales, of atoms and molecules, due to theuncertainty principleofquantum mechanics,a charged particle does nothavea precise position but is represented by aprobability 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 inchemistryandchemical bonding.An electron is represented by awavefunction$\psi ({\boldsymbol {x}})$whose square is proportional to the probability of finding the electron at any point${\boldsymbol {x}}$in space, so$|\psi ({\boldsymbol {x}})|^{2}$is proportional to the charge density of the electron at any point. Inatomsandmoleculesthe charge of the electrons is distributed in clouds calledorbitalswhich surround the atom or molecule, and are responsible forchemical bonds.

Definitions

Continuous charges

Continuous charge distribution. The volume charge densityρis the amount of charge per unit volume (three dimensional), surface charge densityσis amount per unit surface area (circle) with outwardunit normaln̂,dis thedipole momentbetween two point charges, the volume density of these is thepolarization densityP.Position vectorris a point to calculate theelectric 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 chargedQ(SI unit:C) to an infinitesimalline element, $$\lambda _{q}={\frac {dQ}{d\ell }}\,,$$ similarly the surface charge density uses asurface areaelementdS $$\sigma _{q}={\frac {dQ}{dS}}\,,$$ and the volume charge density uses avolumeelementdV $$\rho _{q}={\frac {dQ}{dV}}\,,$$

Integrating the definitions gives the total chargeQof a region according toline integralof the linear charge densityλ_q(r) over a line or 1d curveC, $$Q=\int _{L}\lambda _{q}(\mathbf {r} )\,d\ell$$ similarly asurface integralof the surface charge density σ_q(r) over a surfaceS, $$Q=\int _{S}\sigma _{q}(\mathbf {r} )\,dS$$ and avolume integralof the volume charge densityρ_q(r) over a volumeV, $$Q=\int _{V}\rho _{q}(\mathbf {r} )\,dV$$ where the subscriptqis to clarify that the density is for electric charge, not other densities likemass density,number density,probability density,and prevent conflict with the many other uses ofλ,σ,ρin electromagnetism forwavelength,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_Vetc.

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

Indielectricmaterials, the total charge of an object can be separated into "free" and "bound" charges.

Bound chargesset up electric dipoles in response to an appliedelectric 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 theelectronsbound to thenuclei.

Free chargesare the excess charges which can move intoelectrostatic equilibrium,i.e. when the charges are not moving and the resultant electric field is independent of time, or constituteelectric currents.

Total charge densities

In terms of volume charge densities, thetotalcharge density is: $$\rho =\rho _{\text{f}}+\rho _{\text{b}}\,.$$ as for surface charge densities: $$\sigma =\sigma _{\text{f}}+\sigma _{\text{b}}\,.$$ 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 thedielectric,given by the dipole moment perpendicular to the surface: $$q_{b}={\frac {\mathbf {d} \cdot \mathbf {\hat {n}} }{|\mathbf {s} |}}$$ wheresis the separation between the point charges constituting the dipole,$\mathbf {d}$is theelectric dipole moment,$\mathbf {\hat {n}}$is theunit normal vectorto the surface.

Takinginfinitesimals: $$dq_{b}={\frac {d\mathbf {d} }{|\mathbf {s} |}}\cdot \mathbf {\hat {n}}$$ and dividing by the differential surface elementdSgives the bound surface charge density: $$\sigma _{b}={\frac {dq_{b}}{dS}}={\frac {d\mathbf {d} }{|\mathbf {s} |dS}}\cdot \mathbf {\hat {n}} ={\frac {d\mathbf {d} }{dV}}\cdot \mathbf {\hat {n}} =\mathbf {P} \cdot \mathbf {\hat {n}} \,.$$ wherePis thepolarization density,i.e. density ofelectric dipole momentswithin the material, anddVis the differentialvolume element.

Using thedivergence theorem,the bound volume charge density within the material is $$q_{b}=\int \rho _{b}\,dV=-\oint _{S}\mathbf {P} \cdot {\hat {\mathbf {n} }}\,dS=-\int \nabla \cdot \mathbf {P} \,dV$$ 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.

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

Theelectric potentialdue to a dipole momentdis: $$\varphi ={\frac {1}{4\pi \varepsilon _{0}}}{\frac {(\mathbf {r} -\mathbf {r} ')\cdot \mathbf {d} }{|\mathbf {r} -\mathbf {r} '|^{3}}}$$

For a continuous distribution, the material can be divided up into infinitely manyinfinitesimaldipoles $$d\mathbf {d} =\mathbf {P} dV=\mathbf {P} d^{3}\mathbf {r}$$ wheredV=d³r′is the volume element, so the potential is thevolume integralover the object: $$\varphi ={\frac {1}{4\pi \varepsilon _{0}}}\iiint {\frac {(\mathbf {r} -\mathbf {r} ')\cdot \mathbf {P} }{|\mathbf {r} -\mathbf {r} '|^{3}}}d^{3}\mathbf {r'}$$

Since $$\nabla '\left({\frac {1}{|\mathbf {r} -\mathbf {r} '|}}\right)\equiv \left(\mathbf {e} _{x}{\frac {\partial }{\partial x'}}+\mathbf {e} _{y}{\frac {\partial }{\partial y'}}+\mathbf {e} _{z}{\frac {\partial }{\partial z'}}\right)\left({\frac {1}{|\mathbf {r} -\mathbf {r} '|}}\right)={\frac {\mathbf {r} -\mathbf {r} '}{|\mathbf {r} -\mathbf {r} '|^{3}}}$$ where ∇′ is thegradientin ther′coordinates, $$\varphi ={\frac {1}{4\pi \varepsilon _{0}}}\iiint \mathbf {P} \cdot \nabla '\left({\frac {1}{|\mathbf {r} -\mathbf {r} '|}}\right)d^{3}\mathbf {r'}$$

Integrating by parts $$\varphi ={\frac {1}{4\pi \varepsilon _{0}}}\iiint \left[\nabla '\cdot \left({\frac {\mathbf {P} }{|\mathbf {r} -\mathbf {r} '|}}\right)-{\frac {1}{\mathbf {r} -\mathbf {r} '}}(\nabla '\cdot \mathbf {P} )\right]d^{3}\mathbf {r'}$$ using the divergence theorem:

$\varphi ={\frac {1}{4\pi \varepsilon _{0}}}$${\scriptstyle S}$${\frac {\mathbf {P} \cdot \mathbf {\hat {n}} dS'}{|\mathbf {r} -\mathbf {r} '|}}-{\frac {1}{4\pi \varepsilon _{0}}}\iiint {\frac {\nabla '\cdot \mathbf {P} }{|\mathbf {r} -\mathbf {r} '|}}d^{3}\mathbf {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 \varepsilon _{0}}}$${\scriptstyle S}$${\frac {\sigma _{b}dS'}{|\mathbf {r} -\mathbf {r} '|}}+{\frac {1}{4\pi \varepsilon _{0}}}\iiint {\frac {\rho _{b}}{|\mathbf {r} -\mathbf {r} '|}}d^{3}\mathbf {r'}$

that is $$\sigma _{b}=\mathbf {P} \cdot \mathbf {\hat {n}} \,,\quad \rho _{b}=-\nabla \cdot \mathbf {P}$$

Free charge density

The free charge density serves as a useful simplification inGauss's lawfor electricity; the volume integral of it is the free charge enclosed in a charged object - equal to the netfluxof theelectric displacement fieldDemerging from the object:

$\Phi _{D}=$${\scriptstyle S}$$\mathbf {D} \cdot \mathbf {\hat {n}} dS=\iiint \rho _{f}dV$

SeeMaxwell's equationsandconstitutive relationfor more details.

Homogeneous charge density

For the special case of ahomogeneouscharge densityρ₀,independent of position i.e. constant throughout the region of the material, the equation simplifies to: $$Q=V\rho _{0}.$$

Proof

Start with the definition of a continuous volume charge density: $$Q=\int _{V}\rho _{q}(\mathbf {r} )\,dV.$$

Then, by definition of homogeneity,ρ_q(r) is a constant denoted byρ_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: $$Q=\rho _{q,0}\int _{V}\,dV=\rho _{0}V$$ so, $$Q=V\rho _{q,0}.$$

The equivalent proofs for linear charge density and surface charge density follow the same arguments as above.

Discrete charges

For a single point chargeqat positionr₀inside a region of 3d spaceR,like anelectron,the volume charge density can be expressed by theDirac delta function: $$\rho _{q}(\mathbf {r} )=q\delta (\mathbf {r} -\mathbf {r} _{0})$$ whereris 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 theshifting propertyfor any functionf: $$\int _{R}d^{3}\mathbf {r} f(\mathbf {r} )\delta (\mathbf {r} -\mathbf {r} _{0})=f(\mathbf {r} _{0})$$ so the delta function ensures that when the charge density is integrated overR,the total charge inRisq: $$Q=\int _{R}d^{3}\mathbf {r} \,\rho _{q}=\int _{R}d^{3}\mathbf {r} \,q\delta (\mathbf {r} -\mathbf {r} _{0})=q\int _{R}d^{3}\mathbf {r} \,\delta (\mathbf {r} -\mathbf {r} _{0})=q$$

This can be extended toNdiscrete point-like charge carriers. The charge density of the system at a pointris a sum of the charge densities for each chargeq_iat positionr_i,wherei= 1, 2,...,N: $$\rho _{q}(\mathbf {r} )=\sum _{i=1}^{N}\ q_{i}\delta (\mathbf {r} -\mathbf {r} _{i})$$

The delta function for each chargeq_iin the sum,δ(r−r_i), ensures the integral of charge density overRreturns the total charge inR: $$Q=\int _{R}d^{3}\mathbf {r} \sum _{i=1}^{N}\ q_{i}\delta (\mathbf {r} -\mathbf {r} _{i})=\sum _{i=1}^{N}\ q_{i}\int _{R}d^{3}\mathbf {r} \delta (\mathbf {r} -\mathbf {r} _{i})=\sum _{i=1}^{N}\ q_{i}$$

If all charge carriers have the same chargeq(for electronsq= −e,theelectron charge) the charge density can be expressed through the number of charge carriers per unit volume,n(r), by $$\rho _{q}(\mathbf {r} )=qn(\mathbf {r} )\,.$$

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

Charge density in special relativity

Further information:classical electromagnetism and special relativityandrelativistic electromagnetism

Inspecial relativity,the length of a segment of wire depends onvelocityof observer because oflength contraction,so charge density will also depend on velocity.Anthony French has described how themagnetic fieldforce of a current-bearing wire arises from this relative charge density. He used (p 260) aMinkowski diagramto 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 movingframe of referenceit is calledproper charge density.

It turns out the charge densityρandcurrent densityJtransform together as afour-currentvector underLorentz transformations.

Charge density in quantum mechanics

See also:Density functional theoryandHartree–Fock method

Inquantum mechanics,charge densityρ_qis related towavefunctionψ(r) by the equation$$\rho _{q}(\mathbf {r} )=q|\psi (\mathbf {r} )|^{2}$$whereqis the charge of the particle and|ψ(r)|²=ψ*(r)ψ(r)is theprobability density functioni.e. probability per unit volume of a particle located atr. When the wavefunction is normalized - the average charge in the regionr∈Ris$$Q=\int _{R}q|\psi (\mathbf {r} )|^{2}\,d^{3}\mathbf {r}$$whered³ris theintegration measureover 3d position space.

For system of identical fermions, the number density is given as sum of probability density of each particle in:

$n(\mathbf {r} )=\sum _{i}\langle \psi |\delta ^{3}(\mathbf {r} -\mathbf {r} _{i}')|\psi \rangle$

$n(\mathbf {r} )=\sum _{i}\int {\mathrm {d} }^{3}\mathbf {r} _{2}\cdots \int {\mathrm {d} }^{3}\mathbf {r} _{N}\,\Psi ^{*}(\mathbf {r},\mathbf {r} _{2},\dots,\mathbf {r} _{i}=\mathbf {r} ',\dots,\mathbf {r} _{N})\Psi (\mathbf {r},\mathbf {r} _{2},\dots,\mathbf {r} _{i}=\mathbf {r} ',\dots,\mathbf {r} _{N}).$

Using symmetrization condition:$$n(\mathbf {r} )=N\int {\mathrm {d} }^{3}\mathbf {r} _{2}\cdots \int {\mathrm {d} }^{3}\mathbf {r} _{N}\,\Psi ^{*}(\mathbf {r},\mathbf {r} _{2},\dots,\mathbf {r} _{N})\Psi (\mathbf {r},\mathbf {r} _{2},\dots,\mathbf {r} _{N}).$$where${\textstyle \rho _{q}(\mathbf {r} )=q\cdot n(\mathbf {r} )}$is considered as the charge density.

The potential energy of a system is written as:$$\langle \psi |U|\psi \rangle =\int V(\mathbf {r} )n(\mathbf {r} )\delta ^{3}\mathbf {r}$$The electron-electron repulsion energy is thus derived under these conditions to be:$$U_{ee}[n]=J[n]={\frac {1}{2}}\int \delta ^{3}\mathbf {r} '\int \delta ^{3}\mathbf {r} \left({\frac {(en(\mathbf {r} ))(en(\mathbf {r} '))}{|\mathbf {r} -\mathbf {r} '|}}\right)={\frac {1}{2}}\int \delta ^{3}\mathbf {r} '\int \delta ^{3}\mathbf {r} \left({\frac {\rho (\mathbf {r} )\rho (\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}\right)$$Note that this is excluding the exchange energy of the system, which is a purely quantum mechanical phenomenon, has to be calculated separately.

Then, the energy is given using Hartree-Fock method as:

$E[n]=I+J-K$

WhereIis the kinetic and potential energy of electrons due to positive charges,Jis the electron electron interaction energy andKis the exchange energy of electrons.

Application

The charge density appears in thecontinuity equationfor electric current, and also inMaxwell's Equations.It is the principal source term of theelectromagnetic field;when the charge distribution moves, this corresponds to acurrent density.The charge density of molecules impacts chemical and separation processes. For example, charge density influences metal-metal bonding andhydrogen bonding.For separation processes such asnanofiltration,the charge density of ions influences their rejection by the membrane.

See also

• Continuity equationrelating charge density andcurrent density
• Ionic potential
• Charge density wave

Further reading

• A. Halpern (1988).3000 Solved Problems in Physics.Schaum Series, Mc Graw Hill.ISBN978-0-07-025734-4.
• G. Woan (2010).The Cambridge Handbook of Physics Formulas.Cambridge University Press.ISBN978-0-521-57507-2.
• P. A. Tipler, G. Mosca (2008).Physics for Scientists and Engineers - with Modern Physics(6th ed.). Freeman.ISBN978-0-7167-8964-2.
• R.G. Lerner,G.L. Trigg (1991).Encyclopaedia of Physics(2nd ed.). VHC publishers.ISBN978-0-89573-752-6.
• C.B. Parker (1994).McGraw Hill Encyclopaedia of Physics(2nd ed.). VHC publishers.ISBN978-0-07-051400-3.

External links

• [1]- Spatial charge distributions