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Charge density
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<h2 id="definitions">Definitions</h2> <h3>Continuous charges</h3> <p>Following are the definitions for continuous charge distributions.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a><a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> </p><p>The linear charge density is the ratio of an infinitesimal electric charge <i>dQ</i> (SI unit: <a href="/facts/Coulomb/WNV9ueGr">C</a>) to an infinitesimal <a href="/facts/Line_element/asquI22v">line element</a>, λ q = d Q d ℓ , {\displaystyle \lambda _{q}={\frac {dQ}{d\ell }}\,,} similarly the surface charge density uses a <a href="/facts/Surface_area/WNowzFCw">surface area</a> element <i>dS</i> σ q = d Q d S , {\displaystyle \sigma _{q}={\frac {dQ}{dS}}\,,} and the volume charge density uses a <a href="/facts/Volume/aeAXCBUd">volume</a> element <i>dV</i> ρ q = d Q d V , {\displaystyle \rho _{q}={\frac {dQ}{dV}}\,,} </p><p>Integrating the definitions gives the total charge <i>Q</i> of a region according to <a href="/facts/Line_integral/10s4B3Nn">line integral</a> of the linear charge density <i>λ</i><i>q</i>(r) over a line or 1d curve <i>C</i>, Q = ∫ L λ q ( r ) d ℓ {\displaystyle Q=\int _{L}\lambda _{q}(\mathbf {r} )\,d\ell } similarly a <a href="/facts/Surface_integral/oPk5IRI1">surface integral</a> of the surface charge density σ<i>q</i>(r) over a surface <i>S</i>, Q = ∫ S σ q ( r ) d S {\displaystyle Q=\int _{S}\sigma _{q}(\mathbf {r} )\,dS} and a <a href="/facts/Volume_integral/0fHWrQe7">volume integral</a> of the volume charge density <i>ρ</i><i>q</i>(r) over a volume <i>V</i>, Q = ∫ V ρ q ( r ) d V {\displaystyle Q=\int _{V}\rho _{q}(\mathbf {r} )\,dV} where the subscript <i>q</i> is to clarify that the density is for electric charge, not other densities like <a href="/facts/Mass_density/92p6hh9I">mass density</a>, <a href="/facts/Number_density/qtyDnLhz">number density</a>, <a href="/facts/Probability_density_function/zvfybna4">probability density</a>, and prevent conflict with the many other uses of <i>λ</i>, <i>σ</i>, <i>ρ</i> in electromagnetism for <a href="/facts/Wavelength/kcJU0ymz">wavelength</a>, <a href="/facts/Electrical_resistivity_and_conductivity/akyCKvMy">electrical resistivity and conductivity</a>. </p><p>Within the context of electromagnetism, the subscripts are usually dropped for simplicity: <i>λ</i>, <i>σ</i>, <i>ρ</i>. Other notations may include: <i>ρℓ</i>, <i>ρs</i>, <i>ρv</i>, <i>ρL</i>, <i>ρS</i>, <i>ρV</i> etc. </p><p>The total charge divided by the length, surface area, or volume will be the average charge densities: ⟨ λ q ⟩ = Q ℓ , ⟨ σ q ⟩ = Q S , ⟨ ρ q ⟩ = Q V . {\displaystyle \langle \lambda _{q}\rangle ={\frac {Q}{\ell }}\,,\quad \langle \sigma _{q}\rangle ={\frac {Q}{S}}\,,\quad \langle \rho _{q}\rangle ={\frac {Q}{V}}\,.} </p> <h2 id="free-bound-and-total-charge">Free, bound and total charge</h2> <p>In <a href="/facts/Dielectric/PnVt9P2G">dielectric</a> materials, the total charge of an object can be separated into "free" and "bound" charges. </p><p>Bound charges set up electric dipoles in response to an applied <a href="/facts/Electric_field/tt4MooBs">electric field</a> 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 <a href="/facts/Electron/L0KBo5cW">electrons</a> bound to the <a href="/facts/Atomic_nuclei/orpIm5Xl">nuclei</a>.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> </p><p>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 <a href="/facts/Electric_current/Dc1xdnUX">electric currents</a>.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> </p> <h3>Total charge densities</h3> <p>In terms of volume charge densities, the total charge density is: ρ = ρ f + ρ b . {\displaystyle \rho =\rho _{\text{f}}+\rho _{\text{b}}\,.} as for surface charge densities: σ = σ f + σ b . {\displaystyle \sigma =\sigma _{\text{f}}+\sigma _{\text{b}}\,.} where subscripts "f" and "b" denote "free" and "bound" respectively. </p> <h3>Bound charge</h3> <p>The bound surface charge is the charge piled up at the surface of the <a href="/facts/Dielectric/PnVt9P2G">dielectric</a>, given by the dipole moment perpendicular to the surface:<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> q b = d ⋅ n ^ | s | {\displaystyle q_{b}={\frac {\mathbf {d} \cdot \mathbf {\hat {n}} }{|\mathbf {s} |}}} where s is the separation between the point charges constituting the dipole, d {\displaystyle \mathbf {d} } is the <a href="/facts/Electric_dipole_moment/ou2LDUVg">electric dipole moment</a>, n ^ {\displaystyle \mathbf {\hat {n}} } is the <a href="/facts/Unit_normal_vector/GyaXneDn">unit normal vector</a> to the surface. </p><p>Taking <a href="/facts/Infinitesimal/JKcTMIgD">infinitesimals</a>: d q b = d d | s | ⋅ n ^ {\displaystyle dq_{b}={\frac {d\mathbf {d} }{|\mathbf {s} |}}\cdot \mathbf {\hat {n}} } and dividing by the differential surface element <i>dS</i> gives the bound surface charge density: σ b = d q b d S = d d | s | d S ⋅ n ^ = d d d V ⋅ n ^ = P ⋅ n ^ . {\displaystyle \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}} \,.} where P is the <a href="/facts/Polarization_density/2oQv9EHx">polarization density</a>, i.e. density of <a href="/facts/Electric_dipole_moment/ou2LDUVg">electric dipole moments</a> within the material, and <i>dV</i> is the differential <a href="/facts/Volume_element/SD2ppCLr">volume element</a>. </p><p>Using the <a href="/facts/Divergence_theorem/pUGKF1N2">divergence theorem</a>, the bound volume charge density within the material is q b = ∫ ρ b d V = − ∮ S P ⋅ n ^ d S = − ∫ ∇ ⋅ P d V {\displaystyle q_{b}=\int \rho _{b}\,dV=-\oint _{S}\mathbf {P} \cdot {\hat {\mathbf {n} }}\,dS=-\int \nabla \cdot \mathbf {P} \,dV} hence: ρ b = − ∇ ⋅ P . {\displaystyle \rho _{b}=-\nabla \cdot \mathbf {P} \,.} </p><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. </p><p>A more rigorous derivation is given below.<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> </p> Derivation of bound surface and volume charge densities from internal dipole moments (bound charges) <p>The <a href="/facts/Electric_potential/Nmqm6hm5">electric potential</a> due to a dipole moment d is: φ = 1 4 π ε 0 ( r − r ′ ) ⋅ d | r − r ′ | 3 {\displaystyle \varphi ={\frac {1}{4\pi \varepsilon _{0}}}{\frac {(\mathbf {r} -\mathbf {r} ')\cdot \mathbf {d} }{|\mathbf {r} -\mathbf {r} '|^{3}}}} </p><p>For a continuous distribution, the material can be divided up into infinitely many <a href="/facts/Infinitesimal/JKcTMIgD">infinitesimal</a> dipoles d d = P d V = P d 3 r {\displaystyle d\mathbf {d} =\mathbf {P} dV=\mathbf {P} d^{3}\mathbf {r} } where <i>dV</i> = <i>d</i>3r′ is the volume element, so the potential is the <a href="/facts/Volume_integral/0fHWrQe7">volume integral</a> over the object: φ = 1 4 π ε 0 ∭ ( r − r ′ ) ⋅ P | r − r ′ | 3 d 3 r ′ {\displaystyle \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'} } </p><p>Since ∇ ′ ( 1 | r − r ′ | ) ≡ ( e x ∂ ∂ x ′ + e y ∂ ∂ y ′ + e z ∂ ∂ z ′ ) ( 1 | r − r ′ | ) = r − r ′ | r − r ′ | 3 {\displaystyle \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 the <a href="/facts/Gradient/TsR7uukj">gradient</a> in the r′ coordinates, φ = 1 4 π ε 0 ∭ P ⋅ ∇ ′ ( 1 | r − r ′ | ) d 3 r ′ {\displaystyle \varphi ={\frac {1}{4\pi \varepsilon _{0}}}\iiint \mathbf {P} \cdot \nabla '\left({\frac {1}{|\mathbf {r} -\mathbf {r} '|}}\right)d^{3}\mathbf {r'} } </p><p><a href="/facts/Integration_by_parts/aV6JgBNd">Integrating by parts</a> φ = 1 4 π ε 0 ∭ [ ∇ ′ ⋅ ( P | r − r ′ | ) − 1 r − r ′ ( ∇ ′ ⋅ P ) ] d 3 r ′ {\displaystyle \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: </p> φ = 1 4 π ε 0 {\displaystyle \varphi ={\frac {1}{4\pi \varepsilon _{0}}}} S {\displaystyle {\scriptstyle S}} P ⋅ n ^ d S ′ | r − r ′ | − 1 4 π ε 0 ∭ ∇ ′ ⋅ P | r − r ′ | d 3 r ′ {\displaystyle {\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'} } <p>which separates into the potential of the surface charge (<a href="/facts/Surface_integral/oPk5IRI1">surface integral</a>) and the potential due to the volume charge (volume integral): </p> φ = 1 4 π ε 0 {\displaystyle \varphi ={\frac {1}{4\pi \varepsilon _{0}}}} S {\displaystyle {\scriptstyle S}} σ b d S ′ | r − r ′ | + 1 4 π ε 0 ∭ ρ b | r − r ′ | d 3 r ′ {\displaystyle {\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'} } <p>that is σ b = P ⋅ n ^ , ρ b = − ∇ ⋅ P {\displaystyle \sigma _{b}=\mathbf {P} \cdot \mathbf {\hat {n}} \,,\quad \rho _{b}=-\nabla \cdot \mathbf {P} } </p> <h3>Free charge density</h3> <p>The free charge density serves as a useful simplification in <a href="/facts/Gauss%2527s_law/Am2X21DU">Gauss's law</a> for electricity; the volume integral of it is the free charge enclosed in a charged object - equal to the net <a href="/facts/Flux/Aw3P5Bvb">flux</a> of the <a href="/facts/Electric_displacement_field/YERMcaRg">electric displacement field</a> D emerging from the object: </p> Φ D = {\displaystyle \Phi _{D}=} S {\displaystyle {\scriptstyle S}} D ⋅ n ^ d S = ∭ ρ f d V {\displaystyle \mathbf {D} \cdot \mathbf {\hat {n}} dS=\iiint \rho _{f}dV} <p>See <a href="/facts/Maxwell%2527s_equations/LfJ1drHS">Maxwell's equations</a> and <a href="/facts/Constitutive_relation/IXTjEAbL">constitutive relation</a> for more details. </p> <h2 id="homogeneous-charge-density">Homogeneous charge density</h2> <p>For the special case of a <a href="/facts/Homogeneity_(physics)/J4v4DwfF">homogeneous</a> charge density <i>ρ</i>0, independent of position i.e. constant throughout the region of the material, the equation simplifies to: Q = V ρ 0 . {\displaystyle Q=V\rho _{0}.} </p> <h3>Proof</h3> <p>Start with the definition of a continuous volume charge density: Q = ∫ V ρ q ( r ) d V . {\displaystyle Q=\int _{V}\rho _{q}(\mathbf {r} )\,dV.} </p><p>Then, by definition of homogeneity, <i>ρ</i><i>q</i>(r) is a constant denoted by <i>ρ</i><i>q</i>, 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 = ρ q , 0 ∫ V d V = ρ 0 V {\displaystyle Q=\rho _{q,0}\int _{V}\,dV=\rho _{0}V} so, Q = V ρ q , 0 . {\displaystyle Q=V\rho _{q,0}.} </p><p>The equivalent proofs for linear charge density and surface charge density follow the same arguments as above. </p> <h2 id="discrete-charges">Discrete charges</h2> <p>For a single point charge <i>q</i> at position r0 inside a region of 3d space <i>R</i>, like an <a href="/facts/Electron/L0KBo5cW">electron</a>, the volume charge density can be expressed by the <a href="/facts/Dirac_delta_function/V8PjRP8T">Dirac delta function</a>: ρ q ( r ) = q δ ( r − r 0 ) {\displaystyle \rho _{q}(\mathbf {r} )=q\delta (\mathbf {r} -\mathbf {r} _{0})} where r is the position to calculate the charge. </p><p>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 <i>shifting property</i> for any function <i>f</i>: ∫ R d 3 r f ( r ) δ ( r − r 0 ) = f ( r 0 ) {\displaystyle \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 <i>R</i>, the total charge in <i>R</i> is <i>q</i>: Q = ∫ R d 3 r ρ q = ∫ R d 3 r q δ ( r − r 0 ) = q ∫ R d 3 r δ ( r − r 0 ) = q {\displaystyle 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} </p><p>This can be extended to <i>N</i> 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 <i>qi</i> at position r<i>i</i>, where <i>i</i> = 1, 2, ..., <i>N</i>: ρ q ( r ) = ∑ i = 1 N q i δ ( r − r i ) {\displaystyle \rho _{q}(\mathbf {r} )=\sum _{i=1}^{N}\ q_{i}\delta (\mathbf {r} -\mathbf {r} _{i})} </p><p>The delta function for each charge <i>qi</i> in the sum, <i>δ</i>(r − r<i>i</i>), ensures the integral of charge density over <i>R</i> returns the total charge in <i>R</i>: Q = ∫ R d 3 r ∑ i = 1 N q i δ ( r − r i ) = ∑ i = 1 N q i ∫ R d 3 r δ ( r − r i ) = ∑ i = 1 N q i {\displaystyle 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}} </p><p>If all charge carriers have the same charge <i>q</i> (for electrons <i>q</i> = −<i>e</i>, the <a href="/facts/Electron_charge/lPcj6CoQ">electron charge</a>) the charge density can be expressed through the number of charge carriers per unit volume, <i>n</i>(r), by ρ q ( r ) = q n ( r ) . {\displaystyle \rho _{q}(\mathbf {r} )=qn(\mathbf {r} )\,.} </p><p>Similar equations are used for the linear and surface charge densities. </p> <h2 id="charge-density-in-special-relativity">Charge density in special relativity</h2> <p class="note">Further information: <a href="/facts/Classical_electromagnetism_and_special_relativity/0UEQ840t">classical electromagnetism and special relativity</a> and <a href="/facts/Relativistic_electromagnetism/MqKhJPy2">relativistic electromagnetism</a></p> <p>In <a href="/facts/Special_relativity/vsFFc63E">special relativity</a>, the length of a segment of wire depends on <a href="/facts/Velocity/Q3o1LVDe">velocity</a> of observer because of <a href="/facts/Length_contraction/4of503Dg">length contraction</a>, so charge density will also depend on velocity. <a href="/facts/Anthony_French/Lr9JYIWb">Anthony French</a><a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> has described how the <a href="/facts/Magnetic_field/Ta8Ev588">magnetic field</a> force of a current-bearing wire arises from this relative charge density. He used (p 260) a <a href="/facts/Minkowski_diagram/YF2SeXRL">Minkowski diagram</a> 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 <a href="/facts/Frame_of_reference/15yYfB9M">frame of reference</a> it is called proper charge density.<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a><a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a><a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a> </p><p>It turns out the charge density <i>ρ</i> and <a href="/facts/Current_density/IlUKSe2B">current density</a> J transform together as a <a href="/facts/Four-current/C54racI8">four-current</a> vector under <a href="/facts/Lorentz_transformations/jotJL06G">Lorentz transformations</a>. </p> <h2 id="charge-density-in-quantum-mechanics">Charge density in quantum mechanics</h2> <p class="note">See also: <a href="/facts/Density_functional_theory/wwz4Gqdg">Density functional theory</a> and <a href="/facts/Hartree%25E2%2580%2593Fock_method/U9mlRWVR">Hartree–Fock method</a></p> <p>In <a href="/facts/Quantum_mechanics/BRc3Mzgr">quantum mechanics</a>, charge density <i>ρ</i><i>q</i> is related to <a href="/facts/Wavefunction/KgM5ykjY">wavefunction</a> <i>ψ</i>(r) by the equation ρ q ( r ) = q | ψ ( r ) | 2 {\displaystyle \rho _{q}(\mathbf {r} )=q|\psi (\mathbf {r} )|^{2}} where <i>q</i> is the charge of the particle and |<i>ψ</i>(r)|2 = <i>ψ</i>*(r)<i>ψ</i>(r) is the <a href="/facts/Probability_density_function/zvfybna4">probability density function</a> i.e. probability per unit volume of a particle located at r. When the wavefunction is normalized - the average charge in the region r ∈ <i>R</i> is Q = ∫ R q | ψ ( r ) | 2 d 3 r {\displaystyle Q=\int _{R}q|\psi (\mathbf {r} )|^{2}\,d^{3}\mathbf {r} } where <i>d</i>3r is the <a href="/facts/List_of_integration_and_measure_theory_topics/LSd8Fuqt">integration measure</a> over 3d position space. </p><p>For system of identical fermions, the number density is given as sum of probability density of each particle in : </p><p> n ( r ) = ∑ i ⟨ ψ | δ 3 ( r − r i ′ ) | ψ ⟩ {\displaystyle n(\mathbf {r} )=\sum _{i}\langle \psi |\delta ^{3}(\mathbf {r} -\mathbf {r} _{i}')|\psi \rangle } </p><p> n ( r ) = ∑ i ∫ d 3 r 2 ⋯ ∫ d 3 r N Ψ ∗ ( r , r 2 , … , r i = r ′ , … , r N ) Ψ ( r , r 2 , … , r i = r ′ , … , r N ) . {\displaystyle 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}).} </p><p>Using symmetrization condition: n ( r ) = N ∫ d 3 r 2 ⋯ ∫ d 3 r N Ψ ∗ ( r , r 2 , … , r N ) Ψ ( r , r 2 , … , r N ) . {\displaystyle 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 ρ q ( r ) = q ⋅ n ( r ) {\textstyle \rho _{q}(\mathbf {r} )=q\cdot n(\mathbf {r} )} is considered as the charge density. </p><p>The potential energy of a system is written as: ⟨ ψ | U | ψ ⟩ = ∫ V ( r ) n ( r ) δ 3 r {\displaystyle \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 e e [ n ] = J [ n ] = 1 2 ∫ δ 3 r ′ ∫ δ 3 r ( ( e n ( r ) ) ( e n ( r ′ ) ) | r − r ′ | ) = 1 2 ∫ δ 3 r ′ ∫ δ 3 r ( ρ ( r ) ρ ( r ′ ) | r − r ′ | ) {\displaystyle 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. </p><p>Then, the energy is given using Hartree-Fock method as: </p><p> E [ n ] = I + J − K {\displaystyle E[n]=I+J-K} </p><p>Where <i>I</i> is the kinetic and potential energy of electrons due to positive charges, <i>J</i> is the electron electron interaction energy and <i>K</i> is the exchange energy of electrons.<a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a><a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a> </p> <h2 id="application">Application</h2> <p>The charge density appears in the <a href="/facts/Continuity_equation/GVav2Vet">continuity equation</a> for electric current, and also in <a href="/facts/Maxwell%2527s_Equations/LfJ1drHS">Maxwell's Equations</a>. It is the principal source term of the <a href="/facts/Electromagnetic_field/MacraS8y">electromagnetic field</a>; when the charge distribution moves, this corresponds to a <a href="/facts/Current_density/IlUKSe2B">current density</a>. The charge density of molecules impacts chemical and separation processes. For example, charge density influences metal-metal bonding and <a href="/facts/Hydrogen_bonding/85V86IIg">hydrogen bonding</a>.<a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a> For separation processes such as <a href="/facts/Nanofiltration/JVfSAtgg">nanofiltration</a>, the charge density of ions influences their rejection by the membrane.<a class="footnote-ref" id="fnref:19" href="#fn:19"><sup>19</sup></a> </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Continuity_equation/GVav2Vet">Continuity equation</a> relating charge density and <a href="/facts/Current_density/IlUKSe2B">current density</a></li> <li><a href="/facts/Ionic_potential/2coN6dHE">Ionic potential</a></li> <li><a href="/facts/Charge_density_wave/RaoynuXH">Charge density wave</a></li></ul> <h2 id="further-reading">Further reading</h2> <ul><li>A. Halpern (1988). <i>3000 Solved Problems in Physics</i>. Schaum Series, Mc Graw Hill. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-07-025734-4.</li> <li>G. Woan (2010). <a href="https://archive.org/details/cambridgehandboo0000woan"><i>The Cambridge Handbook of Physics Formulas</i></a>. Cambridge University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-57507-2.</li> <li>P. A. Tipler, G. Mosca (2008). <i>Physics for Scientists and Engineers - with Modern Physics</i> (6th ed.). Freeman. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-7167-8964-2.</li> <li><a href="/facts/Rita_G._Lerner/tbrHyaqv">R.G. Lerner</a>, G.L. Trigg (1991). <a href="https://archive.org/details/encyclopediaofph00lern"><i>Encyclopaedia of Physics</i></a> (2nd ed.). VHC publishers. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-89573-752-6.</li> <li>C.B. Parker (1994). <a href="https://archive.org/details/mcgrawhillencycl1993park"><i>McGraw Hill Encyclopaedia of Physics</i></a> (2nd ed.). VHC publishers. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-07-051400-3.</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="https://web.archive.org/web/20101129201012/http://faculty.wwu.edu/vawter/PhysicsNet/Topics/Gauss/SpacialCharge.html">[1]</a> - Spatial charge distributions</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>P.M. Whelan, M.J. Hodgson (1978). Essential Principles of Physics (2nd ed.). John Murray. ISBN 0-7195-3382-1. <a href="0-7195-3382-1" target="_blank">0-7195-3382-1</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>"Physics 2: Electricity and Magnetism, Course Notes, Ch. 2, p. 15-16" (PDF). MIT OpenCourseware. Massachusetts Institute of Technology. 2007. Retrieved December 3, 2017. <a href="https://ocw.mit.edu/courses/physics/8-02-physics-ii-electricity-and-magnetism-spring-2007/readings/summary_w01d2.pdf" target="_blank">https://ocw.mit.edu/courses/physics/8-02-physics-ii-electricity-and-magnetism-spring-2007/readings/summary_w01d2.pdf</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Serway, Raymond A.; Jewett, John W. (2013). Physics for Scientists and Engineers, Vol. 2, 9th Ed. Cengage Learning. p. 704. ISBN 9781133954149. <a href="9781133954149" target="_blank">9781133954149</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Purcell, Edward (2011-09-22). Electricity and Magnetism. Cambridge University Press. ISBN 9781107013605. <a href="9781107013605" target="_blank">9781107013605</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Purcell, Edward (2011-09-22). Electricity and Magnetism. Cambridge University Press. ISBN 9781107013605. <a href="9781107013605" target="_blank">9781107013605</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>I.S. Grant; W.R. Phillips (2008). Electromagnetism (2nd ed.). Manchester Physics, John Wiley & Sons. ISBN 978-0-471-92712-9. <a href="978-0-471-92712-9" target="_blank">978-0-471-92712-9</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>D.J. Griffiths (2007). Introduction to Electrodynamics (3rd ed.). Pearson Education, Dorling Kindersley. ISBN 978-81-7758-293-2. <a href="978-81-7758-293-2" target="_blank">978-81-7758-293-2</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>D.J. Griffiths (2007). Introduction to Electrodynamics (3rd ed.). Pearson Education, Dorling Kindersley. ISBN 978-81-7758-293-2. <a href="978-81-7758-293-2" target="_blank">978-81-7758-293-2</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>I.S. Grant; W.R. Phillips (2008). Electromagnetism (2nd ed.). Manchester Physics, John Wiley & Sons. ISBN 978-0-471-92712-9. <a href="978-0-471-92712-9" target="_blank">978-0-471-92712-9</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>D.J. Griffiths (2007). Introduction to Electrodynamics (3rd ed.). Pearson Education, Dorling Kindersley. ISBN 978-81-7758-293-2. <a href="978-81-7758-293-2" target="_blank">978-81-7758-293-2</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>D.J. Griffiths (2007). Introduction to Electrodynamics (3rd ed.). Pearson Education, Dorling Kindersley. ISBN 978-81-7758-293-2. <a href="978-81-7758-293-2" target="_blank">978-81-7758-293-2</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>French, A. (1968). "8:Relativity and electricity". Special Relativity. W. W. Norton. pp. 229–265. <a href="/wiki/W._W._Norton" target="_blank">/wiki/W._W._Norton</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Mould, Richard A. (2001). "Lorentz force". Basic Relativity. Springer Science & Business Media. ISBN 0-387-95210-1. <a href="0-387-95210-1" target="_blank">0-387-95210-1</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>Lawden, Derek F. (2012). An Introduction to Tensor Calculus: Relativity and Cosmology. Courier Corporation. p. 74. ISBN 978-0-486-13214-3. <a href="978-0-486-13214-3" target="_blank">978-0-486-13214-3</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>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. <a href="1-4020-2700-1" target="_blank">1-4020-2700-1</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> <li id="fn:16"><p>Sakurai, Jun John; Napolitano, Jim (2021). Modern quantum mechanics (3rd ed.). Cambridge: Cambridge University Press. pp. 443–453. ISBN 978-1-108-47322-4. <a href="978-1-108-47322-4" target="_blank">978-1-108-47322-4</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></p></li> <li id="fn:17"><p>Littlejohn, Robert G. "The Hartree-Fock Method in Atoms" (PDF). <a href="https://bohr.physics.berkeley.edu/classes/221/1112/notes/hartfock.pdf" target="_blank">https://bohr.physics.berkeley.edu/classes/221/1112/notes/hartfock.pdf</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></p></li> <li id="fn:18"><p>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. 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