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Charge density

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Electric charge per unit length, area or volume

"Charge distribution" redirects here. This article is about thephysical quantity inelectromagnetism. For other uses seecharge anddensity.

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Inelectromagnetism,charge density is the amount ofelectric charge per unitlength,surface area, orvolume.Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in theSI system incoulombs per cubicmeter (C⋅m⁻³), at any point in a volume.^[1]^[2]^[3]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 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.

Likemass density, charge density can vary with position. Inclassical electromagnetic theory charge density is idealized as acontinuousscalar function 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 current of charge flows into or out of the volume. This is expressed by acontinuity equation which 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 acontinuous 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.^[4] For example, the charge in an electrically charged metal object is made up ofconduction electrons moving randomly in the metal'scrystal lattice.Static electricity is caused by surface charges consisting of electrons andions near the surface of objects, and thespace charge in avacuum tube is composed of a cloud of free electrons moving randomly in space. Thecharge carrier density in 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 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 even smaller scales, of atoms and molecules, due to theuncertainty principle ofquantum mechanics, a charged particle does nothave a 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.^[4] This is the meaning of 'charge distribution' and 'charge density' used inchemistry andchemical 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. Inatoms andmolecules the charge of the electrons is distributed in clouds calledorbitals which surround the atom or molecule, and are responsible forchemical bonds.

Definitions

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Continuous charges

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Continuous charge distribution. The volume charge densityρ is the amount of charge per unit volume (three dimensional), surface charge densityσ is amount per unit surface area (circle) with outwardunit normaln̂,d is thedipole moment between two point charges, the volume density of these is thepolarization densityP.Position vectorr is a point to calculate theelectric field;r′ is a point in the charged object.

Following are the definitions for continuous charge distributions.^[5]^[6]

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 area elementdS $\sigma _{q}={\frac {dQ}{dS}}\,,$ and the volume charge density uses avolume elementdV $\rho _{q}={\frac {dQ}{dV}}\,,$

Integrating the definitions gives the total chargeQ of a region according toline integral of the linear charge densityλ_q(r) over a line or 1d curveC, $Q=\int _{L}\lambda _{q}(\mathbf {r} )\,d\ell$ similarly asurface integral of the surface charge density σ_q(r) over a surfaceS, $Q=\int _{S}\sigma _{q}(\mathbf {r} )\,dS$ and avolume integral of the volume charge densityρ_q(r) over a volumeV, $Q=\int _{V}\rho _{q}(\mathbf {r} )\,dV$ where the subscriptq is 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,ρ in 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_V etc.

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

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Indielectric 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 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 theelectrons bound to thenuclei.^[6]

Free charges are 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.^[5]

Total charge densities

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In terms of volume charge densities, thetotal charge 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

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The bound surface charge is the charge piled up at the surface of thedielectric, given by the dipole moment perpendicular to the surface:^[6] $q_{b}={\frac {\mathbf {d} \cdot \mathbf {\hat {n}} }{|\mathbf {s} |}}$ wheres is the separation between the point charges constituting the dipole, $\mathbf {d}$ is theelectric dipole moment, $\mathbf {\hat {n}}$ is theunit normal vector to the surface.

Takinginfinitesimals: $dq_{b}={\frac {d\mathbf {d} }{|\mathbf {s} |}}\cdot \mathbf {\hat {n}}$ and dividing by the differential surface elementdS gives the bound surface charge density: $\sigma _{b}={\frac {dq_{b}}{dS}}={\frac {d\mathbf {d} }{|\mathbf {s} |dS}}\cdot \mathbf {\hat {n}} ={\frac {d\mathbf {d} }{dV}}\cdot \mathbf {\hat {n}} =\mathbf {P} \cdot \mathbf {\hat {n}} \,.$ whereP is thepolarization density, i.e. density ofelectric dipole moments within the material, anddV is 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.^[6]

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

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

For a continuous distribution, the material can be divided up into infinitely manyinfinitesimal dipoles $d\mathbf {d} =\mathbf {P} dV=\mathbf {P} d^{3}\mathbf {r}$ wheredV =d³r′ is the volume element, so the potential is thevolume integral over 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 thegradient in 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}}}

$\oiint$

{\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}}}

$\oiint$

{\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

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The free charge density serves as a useful simplification inGauss's law for electricity; the volume integral of it is the free charge enclosed in a charged object - equal to the netflux of theelectric displacement fieldD emerging from the object:

\Phi _{D}=

$\oiint$

{\scriptstyle S}

\mathbf {D} \cdot \mathbf {\hat {n}} dS=\iiint \rho _{f}dV

SeeMaxwell's equations andconstitutive relation for more details.

Homogeneous charge density

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

Proof

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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

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For a single point chargeq at positionr₀ inside a region of 3d spaceR, like anelectron, the volume charge density can be expressed by theDirac delta function: $\rho _{q}(\mathbf {r} )=q\delta (\mathbf {r} -\mathbf {r} _{0})$ wherer 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 theshifting property for 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 inR isq: $Q=\int _{R}d^{3}\mathbf {r} \,\rho _{q}=\int _{R}d^{3}\mathbf {r} \,q\delta (\mathbf {r} -\mathbf {r} _{0})=q\int _{R}d^{3}\mathbf {r} \,\delta (\mathbf {r} -\mathbf {r} _{0})=q$

This can be extended toN discrete point-like charge carriers. The charge density of the system at a pointr is a sum of the charge densities for each chargeq_i at 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_i in the sum,δ(r −r_i), ensures the integral of charge density overR returns 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

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Further information:classical electromagnetism and special relativity andrelativistic electromagnetism

Inspecial relativity, the length of a segment of wire depends onvelocity of observer because oflength contraction, so charge density will also depend on velocity.Anthony French^[7]has described how themagnetic field force of a current-bearing wire arises from this relative charge density. He used (p 260) aMinkowski 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 movingframe of reference it is calledproper charge density.^[8]^[9]^[10]

It turns out the charge densityρ andcurrent densityJ transform together as afour-current vector underLorentz transformations.

Charge density in quantum mechanics

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Inquantum mechanics, charge densityρ_q is related towavefunctionψ(r) by the equation $\rho _{q}(\mathbf {r} )=q|\psi (\mathbf {r} )|^{2}$ whereq is the charge of the particle and|ψ(r)|² =ψ*(r)ψ(r) is theprobability density function i.e. probability per unit volume of a particle located atr.When the wavefunction is normalized - the average charge in the regionr ∈R is $Q=\int _{R}q|\psi (\mathbf {r} )|^{2}\,d^{3}\mathbf {r}$ whered³r is theintegration measure over 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:

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

WhereI is the kinetic and potential energy of electrons due to positive charges,J is the electron electron interaction energy andK is the exchange energy of electrons.^[11]^[12]

Application

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The charge density appears in thecontinuity equation for 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.^[13] For separation processes such asnanofiltration, the charge density of ions influences their rejection by the membrane.^[14]

References

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^P.M. Whelan, M.J. Hodgson (1978).Essential Principles of Physics (2nd ed.). John Murray.ISBN 0-7195-3382-1.
^"Physics 2: Electricity and Magnetism, Course Notes, Ch. 2, p. 15-16"(PDF).MIT OpenCourseware. Massachusetts Institute of Technology. 2007. RetrievedDecember 3, 2017.
^Serway, Raymond A.; Jewett, John W. (2013).Physics for Scientists and Engineers, Vol. 2, 9th Ed. Cengage Learning. p. 704.ISBN 9781133954149.
^^a ^bPurcell, Edward (2011-09-22).Electricity and Magnetism. Cambridge University Press.ISBN 9781107013605.
^^a ^bI.S. Grant; W.R. Phillips (2008).Electromagnetism (2nd ed.). Manchester Physics, John Wiley & Sons.ISBN 978-0-471-92712-9.
^^a ^b ^c ^dD.J. Griffiths (2007).Introduction to Electrodynamics (3rd ed.). Pearson Education, Dorling Kindersley.ISBN 978-81-7758-293-2.
^French, A. (1968). "8:Relativity and electricity".Special Relativity.W. W. Norton. pp. 229–265.
^Mould, Richard A. (2001). "Lorentz force".Basic Relativity.Springer Science & Business Media.ISBN 0-387-95210-1.
^Lawden, Derek F. (2012).An Introduction to Tensor Calculus: Relativity and Cosmology. Courier Corporation. p. 74.ISBN 978-0-486-13214-3.
^Vanderlinde, Jack (2006). "11.1:The Four-potential and Coulomb's Law".Classical Electromagnetic Theory. Springer Science & Business Media. p. 314.ISBN 1-4020-2700-1.
^Sakurai, Jun John; Napolitano, Jim (2021).Modern quantum mechanics (3rd ed.). Cambridge: Cambridge University Press. pp. 443–453.ISBN 978-1-108-47322-4.
^Littlejohn, Robert G."The Hartree-Fock Method in Atoms"(PDF).
^R. J. Gillespie & P. L. A. Popelier (2001). "Chemical Bonding and Molecular Geometry".Environmental Science & Technology.52 (7). Oxford University Press:4108–4116.Bibcode:2018EnST...52.4108E.doi:10.1021/acs.est.7b06400.PMID 29510032.
^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.PMID 29510032.{{cite journal}}: CS1 maint: multiple names: authors list (link)