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Charge density

Charge density is a fundamental concept in electromagnetism that quantifies the amount of electric charge distributed per unit volume, surface area, or length within a material or space.^[1] It arises in the context of continuous charge distributions, where discrete point charges are not applicable, and serves as a key parameter for calculating electric fields and potentials through integration.^[1] There are three primary types of charge density, each corresponding to different dimensional distributions. The volume charge density, denoted by ρ, measures charge per unit volume and has units of coulombs per cubic meter (C/m³); it is defined as ρ = dq/dV, where dq is an infinitesimal charge element and dV is the corresponding volume element.^[1]^[2] The surface charge density, denoted by σ, quantifies charge per unit area with units of C/m² and is given by σ = dq/dA.^[1]^[3] Finally, the linear charge density, denoted by λ, describes charge per unit length along a line or curve, with units of C/m, expressed as λ = dq/dl.^[1]^[3] Charge density can be positive or negative, reflecting the net contribution from positively and negatively charged particles such as protons and electrons.^[2] A positive value indicates an excess of positive charge in the region, while a negative value signifies a predominance of negative charge, even if both types of particles are present.^[2] The total charge Q in a volume V is obtained by integrating the volume charge density over that volume: Q = ∭_V ρ dV.^[2] In practical applications, charge densities are essential for modeling phenomena in conductors, dielectrics, and plasmas, as well as for deriving Maxwell's equations in their differential form.^[1] They enable the use of the superposition principle to compute fields from complex distributions by breaking them into infinitesimal elements.^[1]

Fundamental Definitions

Volume Charge Density

Volume charge density, denoted \rho(\mathbf{r}), is defined as the amount of electric charge per unit volume at a point \mathbf{r} in space.^[4] In the limit of an infinitesimal volume element, this is expressed as \rho(\mathbf{r}) = \frac{dq}{dV}, where dq is the charge contained within the volume element dV.^[4] This quantity allows for the description of continuous charge distributions in three-dimensional space, extending the notion of discrete point charges to macroscopic volumes.^[5] The total electric charge Q enclosed within a finite volume V is obtained by integrating the charge density over that volume:

Q = \int_V \rho(\mathbf{r}) \, dV.

This integral extends over all space if the distribution is unbounded.^[5] For uniform distributions where \rho is constant, the total charge simplifies to Q = \rho V, with V being the total volume.^[6] In the International System of Units (SI), the unit of volume charge density is the coulomb per cubic meter (C/m³).^[7] Dimensionally, [\rho] = \frac{[\mathrm{charge}]}{[\mathrm{length}]^3}, reflecting its role as charge distributed over a volumetric measure.^[7] A representative example is a uniformly charged insulating sphere of radius R with constant volume charge density \rho. The total charge Q in this case is Q = \rho \cdot \frac{4}{3} \pi R^3, obtained by multiplying the density by the sphere's volume.^[6] The concept of volume charge density was formalized in the 19th century by James Clerk Maxwell within his development of classical electromagnetism.^[8]

Surface and Line Charge Densities

Surface charge density, denoted as \sigma(\mathbf{r}), quantifies the electric charge per unit area confined to a two-dimensional surface at position \mathbf{r}. It is formally defined as \sigma(\mathbf{r}) = \lim_{\Delta A \to 0} \frac{\Delta Q}{\Delta A} = \frac{dq}{dA}, where dq is the infinitesimal charge on the surface element dA. The SI unit for surface charge density is coulombs per square meter (C/m²).^[2] This concept is essential for modeling charge distributions on boundaries, such as the exterior of conductors, where charges accumulate due to electrostatic equilibrium.^[9] Line charge density, denoted as \lambda(\mathbf{r}), describes charge distributed along a one-dimensional line or curve at position \mathbf{r}. It is defined as \lambda(\mathbf{r}) = \lim_{\Delta l \to 0} \frac{\Delta Q}{\Delta l} = \frac{dq}{dl}, with dl being the infinitesimal length element along the line; its SI unit is coulombs per meter (C/m).^[2] To find the total charge Q associated with these distributions, one integrates over the respective geometry: for a surface, Q = \iint_S \sigma(\mathbf{r}) \, dA; for a line, Q = \int_C \lambda(\mathbf{r}) \, dl.^[2] These integrals account for variations in density across the surface or along the line.^[6] Physical applications of surface charge density include charged conducting plates, where the charge resides entirely on the surface, producing a uniform electric field in the region between parallel plates.^[10] For line charge density, an idealized example is an infinite uniformly charged wire, which generates a radially symmetric electric field decreasing with distance from the wire.^[11] These densities arise as limiting cases of volume charge density \rho: \sigma emerges from integrating \rho over a thin thickness normal to the surface when charges are confined to a negligible volume, while \lambda results from integration over a thin cross-section perpendicular to the line.^[12]

Charge Distributions

Continuous Distributions

In macroscopic electrostatics, continuous charge distributions model scenarios where charge is spread over volumes much larger than atomic scales, treating the charge as a uniform "smear" rather than discrete particles. This approximation holds when variations in charge density occur over lengths far exceeding interatomic distances, typically on the order of micrometers or larger, allowing classical electrostatics to apply without resolving individual electrons or ions.^[13] The volume charge density \rho(\vec{r}) is defined as a smooth, continuous function of position \vec{r}, representing the charge per unit volume in coulombs per cubic meter (C/m³). For distributions with symmetry, \rho simplifies accordingly: in spherical symmetry, it depends only on the radial distance r (e.g., \rho(r) = \rho_0 inside a uniform sphere of radius a, and zero outside); in cylindrical symmetry, it varies with the distance from the axis, useful for modeling long wires or rods. These forms exploit geometric symmetries to facilitate analytical integrations for fields and potentials.^[13]^[14] The total charge Q of a continuous distribution is computed via the volume integral:

Q = \int_V \rho(\vec{r}) \, dV

where the integration extends over the entire volume V containing the charge. Similarly, the center of charge \vec{R}, which locates the effective average position of the distribution (analogous to the center of mass), is given by:

\vec{R} = \frac{1}{Q} \int_V \vec{r} \, \rho(\vec{r}) \, dV

This vector provides a reference point for multipole expansions and simplifies far-field approximations.^[13]^[15] The continuous model emerges from the discrete case by averaging over many point charges: for a small volume \Delta V enclosing charges q_i, the local density approximates \rho(\vec{r}) \approx \sum q_i / \Delta V, with the sum becoming an integral as \Delta V \to 0 and the number of charges grows large. This transition replaces discrete summations \sum \vec{E}_i with the continuous superposition integral \vec{E}(\vec{r}) = \frac{1}{4\pi\epsilon_0} \int \frac{\rho(\vec{r}') (\vec{r} - \vec{r}')}{|\vec{r} - \vec{r}'|^3} \, dV'.^[14]^[13] However, the continuous approximation fails at atomic or subatomic scales, where charge is inherently quantized in discrete units (electrons, protons) and quantum mechanical effects, such as wavefunction delocalization, dominate over classical densities. In such regimes, the model overlooks discrete structure and tunneling, necessitating quantum electrodynamics for accurate descriptions.^[13]

Discrete Distributions

In discrete charge distributions, electric charge is confined to a finite number of distinct locations, such as point charges, rather than being spread continuously throughout a volume. This model is particularly useful for systems where the number of charges is countable and their positions are well-defined. The charge density \rho(\mathbf{r}) for N discrete point charges q_i at positions \mathbf{r}_i is formally represented using the three-dimensional Dirac delta function as

\rho(\mathbf{r}) = \sum_{i=1}^N q_i \, \delta^3(\mathbf{r} - \mathbf{r}_i),

where the delta function \delta^3(\mathbf{r}) localizes each charge q_i precisely at \mathbf{r}_i.^[16] The Dirac delta function possesses the fundamental sifting property that its volume integral over all space equals unity:

\int \delta^3(\mathbf{r}) \, dV = 1.

This normalization ensures that the total charge Q of the distribution is simply the sum of the individual charges:

Q = \int \rho(\mathbf{r}) \, dV = \sum_{i=1}^N q_i.

Consequently, physical quantities like the electric potential or field can be computed by integrating over this density, which reduces to discrete sums: for instance, the potential at a point \mathbf{r} is \phi(\mathbf{r}) = \frac{1}{4\pi\epsilon_0} \sum_{i=1}^N \frac{q_i}{|\mathbf{r} - \mathbf{r}_i|}. Similarly, multipole moments, such as the dipole moment \mathbf{p} = \sum_{i=1}^N q_i \mathbf{r}_i, emerge directly from weighted sums over the charge positions, providing a natural framework for analyzing the distribution's electrostatic properties.^[17] Representative examples of discrete charge distributions include the ions in an ionic crystal lattice, where cations and anions act as point charges fixed at periodic lattice sites, contributing to the overall electrostatic energy via Madelung sums.^[18] Another classical example arises in simplified models of atomic structure, treating electrons as discrete negative point charges orbiting a central positive nucleus.^[17] Unlike continuous distributions, which assume a smooth, macroscopic averaging of charge over volumes, the discrete model using delta functions is exact for finite, countable collections of point charges and serves as the foundational limit from which continuous approximations are derived when the number of charges becomes large and their separations approach zero.^[16]

Classifications of Charge

Total Charge Density

The total charge density, denoted as \rho_\text{tot}(\mathbf{r}), represents the complete distribution of electric charge at a point \mathbf{r} in space and is defined as the sum of the free charge density \rho_f(\mathbf{r}) and the bound charge density \rho_b(\mathbf{r}):

\rho_\text{tot}(\mathbf{r}) = \rho_f(\mathbf{r}) + \rho_b(\mathbf{r}).

Here, \rho_f accounts for charges that are mobile and externally controllable, such as those on conductors, while \rho_b arises from the polarization of insulating materials; together, they encompass all charges contributing to electromagnetic fields.^[19] In classical electromagnetism, the total charge density serves as the fundamental source term in Maxwell's equations, particularly in Gauss's law, which relates the electric field to the enclosed charge. The integral form of Gauss's law states that the flux of the electric field \mathbf{E} through any closed surface S enclosing a volume V is proportional to the total enclosed charge Q_\text{encl}:

\oint_S \mathbf{E} \cdot d\mathbf{A} = \frac{Q_\text{encl}}{\epsilon_0},

where Q_\text{encl} = \int_V \rho_\text{tot}(\mathbf{r}) \, dV and \epsilon_0 is the vacuum permittivity.^[20] To derive the differential form, apply the divergence theorem, which converts the surface integral over S to a volume integral over V:

\int_V \nabla \cdot \mathbf{E} \, dV = \frac{1}{\epsilon_0} \int_V \rho_\text{tot}(\mathbf{r}) \, dV.

Since this equality holds for any arbitrary volume V, the integrands must be equal pointwise, yielding the local differential form:

\nabla \cdot \mathbf{E} = \frac{\rho_\text{tot}(\mathbf{r})}{\epsilon_0}.

This equation underscores the total charge density's role as the primary quantity governing the divergence of the electric field in all media.^[20]^[21] The total charge density in a system is calculated by directly measuring the electric field via techniques like Gauss's law applications or by integrating contributions from all known free and bound charges across the volume, surface, or line distributions. For instance, in a parallel-plate capacitor filled with a linear dielectric, the total surface charge density \sigma_\text{tot} on the plates comprises the free surface charge density \sigma_f from the applied potential and the bound surface charge density \sigma_b induced on the dielectric surfaces due to polarization; the electric field between the plates is then E = \sigma_\text{tot} / \epsilon_0.^[22] In SI units, the volume form of \rho_\text{tot} has dimensions of coulombs per cubic meter (C/m³), while the surface form is C/m² and the line form is C/m, ensuring dimensional consistency when substituting into Maxwell's equations for different geometries.

Free Charge Density

Free charge density, denoted as \rho_f(\mathbf{r}), represents the spatial distribution of electric charges that are mobile and capable of independent movement under the influence of external electric fields, distinguishing it from charges bound within atomic or molecular structures.^[23] These free charges include conduction electrons in metals, ionized carriers in semiconductors, and both electrons and ions in plasmas, enabling phenomena such as electrical conduction and responsiveness to applied voltages.^[22] In contrast, the total charge density encompasses both free and bound contributions, but free charge density specifically accounts for the manipulable component that governs macroscopic electrical behavior in materials.^[24] Common sources of free charge density vary by material type. In metals, it arises from the delocalized conduction electrons forming a degenerate electron gas, with typical densities on the order of $10^{28} to $10^{29} electrons per cubic meter, as modeled by the free electron theory.^[25] In semiconductors, free charges consist of electrons and holes generated through doping, where impurity atoms introduce excess carriers; for instance, n-type doping elevates electron density to around $10^{16} to $10^{19} per cubic meter depending on dopant concentration.^[26] Plasmas, as ionized gases, feature number densities of free charge carriers (electrons and ions) ranging from $10^{10} to $10^{20} m^{-3} in laboratory or astrophysical settings, where collective motion dominates due to long-range Coulomb interactions while maintaining quasineutrality such that net \rho_f \approx 0.^[27] Mathematically, free charge density plays a central role in the formulation of Maxwell's equations for materials, particularly in the differential form of Gauss's law for the electric displacement field \mathbf{D}: \nabla \cdot \mathbf{D} = \rho_f, where \mathbf{D} = \epsilon_0 \mathbf{E} + \mathbf{P} and \mathbf{P} is the polarization vector accounting for bound charges.^[22] This relation isolates the effect of free charges as the source of \mathbf{D}, simplifying analysis in dielectrics by separating controllable external charges from induced polarization effects.^[23] A practical example occurs in a current-carrying wire, where the free charge density of conduction electrons contributes to the current density \mathbf{J} = \rho_f \mathbf{v}_d, with \mathbf{v}_d as the drift velocity; for a typical metal like copper, \rho_f = n q where n \approx 8.5 \times 10^{28} m^{-3} is the electron number density and q = -e is the elementary charge, yielding \rho_f \approx -1.36 \times 10^{10} C/m³ under steady-state conditions.^[28] This density remains nearly uniform along the wire during ohmic conduction, as excess charges accumulate primarily at surfaces or ends rather than in the bulk.^[25] Free charge density is experimentally determined through techniques like the Hall effect, where a transverse voltage arises from charge carrier deflection in a magnetic field, allowing direct calculation of carrier density n via n = \frac{IB}{q t E_H} (with I current, B magnetic field, t thickness, and E_H Hall field), thus yielding \rho_f = n q.^[29] Conductivity measurements also infer \rho_f indirectly via \sigma = n q \mu, where \mu is mobility, providing validation in conductive materials without requiring cryogenic conditions.

Bound Charge Density

Bound charge density refers to the electric charge distribution induced in insulating materials, or dielectrics, due to the alignment of atomic or molecular dipoles in response to an applied electric field. Unlike free charges, which are mobile and can be externally controlled, bound charges are inherently tied to the material's atomic structure and arise from the polarization process. The polarization vector \mathbf{P}(\mathbf{r}), defined as the dipole moment per unit volume, quantifies this alignment, with bound charges emerging as effective sources that modify the electric field within and around the material.^[30]^[22] The volume bound charge density is given by \rho_b(\mathbf{r}) = -\nabla \cdot \mathbf{P}(\mathbf{r}), representing the divergence of the polarization field, while the surface bound charge density is \sigma_b = \mathbf{P} \cdot \hat{\mathbf{n}}, where \hat{\mathbf{n}} is the outward unit normal to the surface. These expressions capture how spatial variations in polarization lead to net charge accumulation: a decrease in \mathbf{P} along its direction produces a negative effective volume charge, as positive and negative atomic charges shift oppositely but result in an overall deficit of positive charge in regions of converging polarization. Physically, this originates from the displacement of electrons relative to positively charged nuclei in dielectric atoms under an electric field \mathbf{E}, forming temporary dipoles that collectively contribute to \mathbf{P} = N q \delta, where N is the number density of atoms, q the effective charge separation, and \delta the displacement distance; in linear dielectrics, \mathbf{P} = \chi \epsilon_0 \mathbf{E}, with \chi the susceptibility.^[30]^[22]^[30] The derivation of bound charge densities stems from the total charge density \rho_\text{tot} = \rho_f + \rho_b, where \rho_f denotes the free charge density, the independent variable in electrostatic problems involving materials. Gauss's law states \nabla \cdot \mathbf{E} = \rho_\text{tot}/\epsilon_0, but introducing the displacement field \mathbf{D} = \epsilon_0 \mathbf{E} + \mathbf{P} yields \nabla \cdot \mathbf{D} = \rho_f, implying \nabla \cdot \mathbf{E} = (\rho_f - \nabla \cdot \mathbf{P})/\epsilon_0 and thus \rho_b = -\nabla \cdot \mathbf{P}. For surfaces, integrating over a thin layer near the boundary using the divergence theorem gives the discontinuity in the normal component of \mathbf{P}, leading to \sigma_b = \mathbf{P} \cdot \hat{\mathbf{n}}. This separation shows how bound charges reduce the net field inside dielectrics, screening external fields and lowering the effective permittivity.^[30]^[22]^[30] A representative example is a uniformly polarized sphere of radius R and constant polarization \mathbf{P} = P \hat{\mathbf{z}}. Here, \rho_b = -\nabla \cdot \mathbf{P} = 0 inside the volume since \mathbf{P} is uniform, but on the surface, \sigma_b = P \cos\theta, where \theta is the polar angle, creating positive bound charge near the "north pole" and negative near the "south pole." This distribution produces an internal field \mathbf{E} = -\mathbf{P}/(3\epsilon_0), equivalent to that of a uniform dipole moment, illustrating how bound charges mimic a dipole ensemble and oppose the polarizing field.^[30]

Special Cases and Transformations

Homogeneous Charge Density

Homogeneous charge density refers to a uniform distribution of electric charge within a specified volume, where the volume charge density \rho is constant throughout the region and zero outside it. This idealization simplifies the analysis of electrostatic fields in symmetric geometries by assuming no spatial variation in charge concentration. To derive the electric field inside a uniformly charged sphere of radius R and constant charge density \rho, Gauss's law is applied, exploiting spherical symmetry. Consider a Gaussian surface as a sphere of radius r < R centered at the origin. The enclosed charge is q_\text{enc} = \rho \cdot \frac{4}{3}\pi r^3, and the flux through the surface is E(r) \cdot 4\pi r^2, since the field is radial and constant on the surface. Gauss's law states \oint \mathbf{E} \cdot d\mathbf{A} = \frac{q_\text{enc}}{\epsilon_0}, yielding E(r) \cdot 4\pi r^2 = \frac{\rho \cdot \frac{4}{3}\pi r^3}{\epsilon_0}. Solving for E(r) gives E(r) = \frac{\rho r}{3 \epsilon_0}, directed radially outward for positive \rho. This linear dependence on r highlights how the field arises solely from charge interior to the Gaussian surface.^[31] For homogeneous charge distributions, the electrostatic potential \phi satisfies Poisson's equation \nabla^2 \phi = -\frac{\rho}{\epsilon_0}, where the right-hand side is constant within the charged region. This form allows analytical solutions in symmetric cases, such as spheres or cylinders, by integrating the constant source term, often leading to quadratic potential profiles inside the volume. Outside the region, where \rho = 0, the equation reduces to Laplace's equation \nabla^2 \phi = 0.^[32] A classic example is an infinite slab of thickness $2a with uniform volume charge density \rho, extending infinitely in the transverse directions. Using a Gaussian pillbox symmetric about the midplane, the field inside (|z| < a) is E(z) = \frac{\rho z}{\epsilon_0}, perpendicular to the slab and increasing linearly from the center. Outside (|z| > a), the field saturates at E(z) = \frac{\rho a}{\epsilon_0}, independent of distance due to the infinite extent. For an infinite solid cylinder of radius R and uniform \rho, cylindrical symmetry with a coaxial Gaussian cylinder yields E(r) = \frac{\rho r}{2 \epsilon_0} inside (r < R), radial and linear in r, while outside it behaves as E(r) = \frac{\rho R^2}{2 \epsilon_0 r}, akin to a line charge. These results demonstrate how homogeneity enables exact field expressions via symmetry.^[33]^[34] Homogeneous charge density serves as an idealization, as real materials exhibit slight inhomogeneities due to atomic-scale variations or impurities, necessitating more complex models for precise applications.

Relativistic Transformations

In special relativity, the charge density \rho and current density \vec{j} are combined into the four-current J^\mu = (c\rho, \vec{j}), a contravariant four-vector that transforms under Lorentz transformations to maintain the invariance of physical laws.^[35] For a boost along the direction of velocity \vec{v} between frames, the components transform according to the Lorentz matrix, yielding the charge density in the primed frame as

\rho' = \gamma \left( \rho - \frac{\vec{v} \cdot \vec{j}}{c^2} \right),

where \gamma = 1 / \sqrt{1 - v^2/c^2} is the Lorentz factor.^[35] This formula reflects how motion affects the observed density through both length contraction and the contribution from current, ensuring consistency across inertial frames. The parallel components of \vec{j}' follow a similar form, j_\parallel' = \gamma \left( j_\parallel - v \rho \right), while perpendicular components remain unchanged, \vec{j}_\perp' = \vec{j}_\perp.^[35] For a charge distribution at rest in the original frame (\vec{j} = 0), the transformation simplifies to \rho' = \gamma \rho, where \rho is the proper charge density.^[36] This apparent increase in density results from Lorentz contraction, which shortens distances in the direction of motion, thereby concentrating the charges into a smaller volume without altering the total charge. As an example, consider a spherical cloud of uniform charge density \rho_0 in its rest frame, with total charge Q = \rho_0 V_0 and volume V_0 = (4/3)\pi R_0^3. In a frame where the cloud moves with speed v along the x-axis, the dimension parallel to \vec{v} contracts to $2R_0 / \gamma, reducing the volume to V' = V_0 / \gamma and yielding \rho' = \gamma \rho_0.^[36] The total charge Q = \int \rho' \, dV' remains invariant, as the volume element transforms inversely to the density.^[36] This framework guarantees the Lorentz covariance of Maxwell's equations, since the four-divergence \partial_\mu J^\mu = 0, expressing local charge conservation, is a scalar invariant under boosts.^[37] The transformation of \rho thus preserves the structure of electromagnetic field equations across frames, linking charge distributions to the observed fields in relativistic contexts.^[37]

Quantum Mechanical Description

In quantum mechanics, the charge density for a single particle carrying charge q is defined as the product of the charge and the probability density given by the square of the absolute value of its wavefunction \psi(\mathbf{r}), such that \rho(\mathbf{r}) = q |\psi(\mathbf{r})|^2, where the wavefunction is normalized according to \int |\psi(\mathbf{r})|^2 dV = 1.^[38] This formulation arises from the Born interpretation, which associates |\psi(\mathbf{r})|^2 with the probability of locating the particle at position \mathbf{r}, thereby interpreting the charge distribution as probabilistic rather than deterministic.^[38] A representative example is the electron charge density in the ground state of the hydrogen atom, where the wavefunction is \psi_{100}(r) = \frac{1}{\sqrt{\pi a_0^3}} e^{-r/a_0} with a_0 the Bohr radius, yielding \rho(r) = -e |\psi_{100}(r)|^2 \propto e^{-2r/a_0}, which decays exponentially from the nucleus and integrates to the total electron charge -e.^[39] For many-body systems, the charge density is the expectation value of the charge density operator, expressed as \rho(\mathbf{r}) = q \sum_i \langle \psi | \delta(\mathbf{r} - \mathbf{r}_i) | \psi \rangle, where the sum is over particles and |\psi\rangle is the many-body wavefunction; in second quantization, this corresponds to \rho(\mathbf{r}) = q \psi^\dagger(\mathbf{r}) \psi(\mathbf{r}).^[40] Approximations for practical computation often employ density functional theory (DFT), where the ground-state electron density n(\mathbf{r}) (and thus charge density \rho(\mathbf{r}) = -e n(\mathbf{r})) uniquely determines the external potential via the Hohenberg-Kohn theorems, enabling self-consistent solutions through the Kohn-Sham equations. This quantum charge density enters quantum electrostatics via the self-consistent Schrödinger-Poisson equations, where the wavefunction satisfies the time-independent Schrödinger equation -\frac{\hbar^2}{2m} \nabla^2 \psi + V(\mathbf{r}) \psi = E \psi with potential V(\mathbf{r}) from Poisson's equation \nabla^2 V(\mathbf{r}) = -\rho(\mathbf{r})/\epsilon_0, iteratively coupling the quantum density to the electrostatic field it generates.^[41] Unlike classical charge density, which describes definite point-like or continuous distributions, the quantum version is inherently probabilistic due to the wave nature of particles, and incorporates zero-point fluctuations even in the ground state arising from the uncertainty principle.^[38]^[42]

Practical Applications

In Classical Electromagnetism

In classical electrostatics, the relationship between the electric potential \phi and the charge density \rho is governed by Poisson's equation,

\nabla^2 \phi = -\frac{\rho}{\epsilon_0},

where \epsilon_0 is the vacuum permittivity. This equation arises from combining Gauss's law with the definition of the electric field as \mathbf{E} = -\nabla \phi. For arbitrary charge distributions, the solution can be obtained using Green's functions, where the potential at a point \mathbf{r} is given by

\phi(\mathbf{r}) = \frac{1}{4\pi\epsilon_0} \int \frac{\rho(\mathbf{r}')}{|\mathbf{r} - \mathbf{r}'|} dV' + \text{surface terms},

with the Green's function G(\mathbf{r}, \mathbf{r}') = 1/(4\pi |\mathbf{r} - \mathbf{r}'|) satisfying \nabla^2 G = -\delta(\mathbf{r} - \mathbf{r}') in free space. This integral form directly extends Coulomb's law to continuous charge distributions. Gauss's law in integral form, \oint \mathbf{E} \cdot d\mathbf{A} = Q_{\text{encl}} / \epsilon_0, provides a powerful method to compute the electric field for charge densities with high symmetry, such as uniform distributions. By choosing a Gaussian surface that exploits the symmetry, the flux simplifies, allowing direct calculation of E. For instance, consider a sphere of radius R with uniform volume charge density \rho. To find the field inside (r < R), apply Gauss's law to a concentric spherical Gaussian surface of radius r. The enclosed charge is Q_{\text{encl}} = \rho \cdot (4/3 \pi r^3), and by symmetry, \mathbf{E} is radial and constant on the surface, so E \cdot 4\pi r^2 = [\rho (4/3 \pi r^3)] / \epsilon_0. Solving yields E(r) = \rho r / (3 \epsilon_0). For the exterior field (r > R), the Gaussian surface encloses the total charge Q = \rho (4/3 \pi R^3), giving E(r) = \rho R^3 / (3 \epsilon_0 r^2), equivalent to a point charge at the center. A similar approach applies to an infinite cylinder with uniform volume charge density \rho and radius R. For the radial field inside (r < R), use a cylindrical Gaussian surface of radius r and length L: Q_{\text{encl}} = \rho \pi r^2 L, and the flux is E \cdot 2\pi r L = Q_{\text{encl}} / \epsilon_0, so E(r) = \rho r / (2 \epsilon_0). Outside (r > R), Q_{\text{encl}} = \rho \pi R^2 L, yielding E(r) = \rho R^2 / (2 \epsilon_0 r). In magnetostatics, charge density connects to magnetic fields through the current density \mathbf{j}, defined for steady-state motion of charges as \mathbf{j} = \rho \mathbf{v}, where \mathbf{v} is the drift velocity. This leads to Ampère's law, \oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{\text{encl}}, or in differential form \nabla \times \mathbf{B} = \mu_0 \mathbf{j}, enabling field calculations for steady currents analogous to electrostatics. For complex charge distributions lacking symmetry, numerical methods such as the finite element method (FEM) are employed to solve Poisson's equation. FEM discretizes the domain into elements, approximates \phi with basis functions, and reduces the problem to solving a sparse linear system K \phi = f, where K incorporates the Laplacian operator and boundary conditions.

In Materials and Condensed Matter

In solids, particularly semiconductors, charge density arises primarily from donor or acceptor impurities introduced through doping. In n-type semiconductors, donor impurities such as phosphorus in silicon provide extra electrons to the conduction band, resulting in a free charge density approximated as \rho_f \approx e N_D, where e is the elementary charge and N_D is the donor concentration, typically on the order of $10^{15} to $10^{18} cm^{-3} for practical devices.^[43] This excess electron density dominates the electrical conductivity and is balanced by ionized donors in charge neutrality. In p-type semiconductors, acceptor impurities like boron create holes, leading to a positive free charge density from the absence of electrons.^[44] In dielectrics, bound charge density originates from local variations in polarization \mathbf{P} induced by an external electric field, given by \rho_b = -\nabla \cdot \mathbf{P}. These variations contribute to the material's permittivity \varepsilon(\mathbf{r}), which describes the local dielectric response and screens free charges, with \varepsilon(\mathbf{r}) = \varepsilon_0 (1 + \chi(\mathbf{r})) where \chi is the susceptibility related to \mathbf{P}. For example, in heterogeneous dielectrics like polymers or ceramics, spatial inhomogeneities in \rho_b lead to position-dependent permittivity, influencing capacitance and energy storage.^[45]^[30] Charge density waves (CDWs) represent a periodic modulation of the electron charge density \rho(\mathbf{r}) in low-dimensional materials, often accompanied by lattice distortion. In transition metal dichalcogenides such as monolayer 1T-TaSe₂, a prototypical CDW system, the charge density exhibits a \sqrt{13} \times \sqrt{13} superlattice below approximately 530 K, driven by electron-phonon coupling and Fermi surface nesting, with modulation amplitudes up to 0.1 electrons per unit cell. This phenomenon suppresses metallic conductivity and can coexist with superconductivity at low temperatures.^[46]^[47] Modern computational approaches, such as density functional theory (DFT), enable ab initio simulations of ground-state charge density in materials by minimizing the total energy functional E[\rho] with respect to \rho(\mathbf{r}), as per the Hohenberg-Kohn theorems. In the Kohn-Sham formulation, the electron density is obtained from self-consistent solution of single-particle equations, providing insights into bonding and electronic structure in complex solids like alloys or nanomaterials, with typical accuracies for density profiles on the order of 0.01 e/Å³.^[48]^[49] Experimental mapping of surface charge density is achieved through scanning tunneling microscopy (STM), which probes local density of states with atomic resolution, typically ~1 Å laterally and ~0.01 Å vertically. In materials exhibiting CDWs, such as TaSe₂, STM reveals periodic density modulations directly, confirming theoretical predictions and visualizing defects or domain walls.^[50]^[51]

References

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Definitions of charge density: λ ≡ charge per unit length (linear charge density); units are coulombs per meter (C/m); σ ≡ charge per unit area (surface charge ...<|control11|><|separator|>
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[5]
[PDF] Electromagnetism - Galileo and Einstein
to consider the charge density, ρ(x,t) defined as charge per unit volume. The total charge Q in a given region V is simply. Q = RV d3x ρ(x,t). In most ...
[6]
Total Charge - BOOKS
If we know the charge density ρ in some volume of space, we can find the total charge by chopping up the volume into lots of small (actually infinitesimal) ...
[7]
[PDF] Guide for the Use of the International System of Units (SI)
Feb 3, 1975 · electric charge density coulomb per cubic meter C/m3 m−3 · s · A surface charge density coulomb per square meter C/m2 m−2 · s · A electric ...
[8]
'…a paper …I hold to be great guns': a commentary on Maxwell ...
Apr 13, 2015 · Maxwell's great paper of 1865 established his dynamical theory of the electromagnetic field. The origins of the paper lay in his earlier papers of 1856.
[9]
Conductors - Richard Fitzpatrick
The charge enclosed by the box is simply $\sigma A$ , from the definition of a surface charge density. Thus, Gauss' law yields. \begin{displaymath} E_\perp ...
[10]
7.5 Equipotential Surfaces and Conductors - UCF Pressbooks
Two large conducting plates carry equal and opposite charges, with a surface charge density σ of magnitude 6.81 × 10 − 7 C/m 2 , as shown in Figure 7.37. The ...
[11]
PHYS 201 - Lecture 3 - Gauss's Law I | Open Yale Courses
The electric field is discussed in greater detail and field due an infinite line charge is computed. The concepts of charge density and electric flux are ...Missing: implication | Show results with:implication
[12]
[PDF] M ρ
Since the charges are confined to the surface, we should use the surface-charge-density ( ). ( ) bound bound m m h σ ρ. = instead of the volume charge-density.
[13]
5.5 Calculating Electric Fields of Charge Distributions - University Physics Volume 2 | OpenStax
### Summary of Volume Charge Density for Continuous Distributions
[14]
[PDF] Problem Solving 2: Continuous Charge Distributions
In order to calculate the electric field created by a continuous charge distribution we must break the charge into a number of small pieces dq, ...
[15]
(PDF) Scope of Center of Charge in Electrostatics - Academia.edu
CENTER OF CHARGE We propose the 'definition' of the “Center of charge” of a charge distribution almost in the same way center of mass is defined in ...
[16]
[PDF] Classical Electrodynamics
A discrete set of point charges can be described with a charge density by means of delta functions. For example,. N p(x) = Σ qi d(x − x¿). =1. (1.6).
[17]
[PDF] introduction to electrodynamics
... Dirac Delta Function 45. 1.5.1. The Divergence of r/r2. 45. 1.5.2. The One-Dimensional Dirac Delta Function 46. 1.5.3. The Three-Dimensional Delta Function 50 v ...
[18]
Electric Fields Produced by a Charge Density in Ionic Crystals - PMC
We shall consider an ionic crystal which contains one molecule M n + X ... point charges occupying the lattice sites. The anion ionicity is Za and the ...
[19]
[PDF] 7. Electromagnetism in Matter - DAMTP
(⇢free + ⇢bound). = 1. ✏0. (⇢free r · P). We define the electric ... total charge density is simply given by. ⇢(EF ) = q Z. EF. 0. dE g(E) where g(E) is ...
[20]
Differential Form of Gauss' Law - BOOKS
The differential form of Gauss’ Law states that the divergence of the electric field at any point is just a measure of the charge density there.
[21]
[PDF] Gauss's Law - UTK-EECS
Differential form (“small picture”) of. Gauss's law: The divergence of electric field at each point is proportional to the local charge density. holds for any ...
[22]
The Feynman Lectures on Physics Vol. II Ch. 10: Dielectrics - Caltech
The charge ρfree was considered to be the entire charge density. In order to write Maxwell's equations in a simple form, a new vector D was defined to be equal ...
[23]
Maxwell's Equations in Matter - Ocean Optics Web Book
Oct 13, 2021 · In addition to his work on electromagnetism, Maxwell made major contributions to thermodynamics (the Maxwell relations connecting temperature, ...
[24]
[PDF] Basic Plasma Physics - DESCANSO
where P is the charge density in the plasma, J is the current density in the plasma, and o and μo are the permittivity and permeability of free space,.
[25]
[PDF] Current, continuity equation, resistance, Ohm's law. - MIT
Feb 24, 2005 · the same velocity u, the current density is just the charge density ρ ≡ nq times that velocity. The current is then given by I = ~J ·. ~A.Missing: wire | Show results with:wire
[26]
[PDF] DIELECTRIC POLARIZATION AND BOUND CHARGES - UT Physics
the nucleus and the electron cloud's center-of-charge, an d hence a dipole ... so in a thermal distribution there are more molecules with p ↑↑ E (and ...
[27]
5 Application of Gauss' Law - Feynman Lectures - Caltech
Gauss' law can be used to find the field inside a uniformly charged sphere. Suppose that we have a sphere of radius R filled uniformly with charge. Let ρ be ...
[28]
6 The Electric Field in Various Circumstances - Feynman Lectures
The normal component of the electric field just outside a conductor is equal to the density of surface charge σ divided by ϵ0. We can obtain the density of ...
[29]
[PDF] Electric Field and Potential for an infinite slab with uniform charge ...
Suggest using Gauss's law. First we find the field OUTSIDE using a pillbox wider than 2a and both ends of area A placed OUTSIDE of the charge slab.
[30]
[PDF] Chapter 6: Gauss's Law II
Example 6.1: Calculate the electric field in all regions r > 0 for a solid charged sphere with uniform charge density, a total charge of Q, and a radius of a.
[31]
[PDF] Section 2: Electrostatics
The only change necessary is in the surface charge density (2.48), where the normal derivative is now outward, implying a change of sign. Conducting Sphere in a ...
[32]
[PDF] Classical Electrodynamics - Duke Physics
function is a delta function with the same arguments d dτ θ(x0 − r0(τ)) = δ[x0 − r0(τ)]. (20.27) which constrains the other delta function to be δ(−R2).
[33]
The current density 4-vector - Richard Fitzpatrick
Clearly, charge density and current density transform as the time-like and space-like components of the same 4-vector.
[34]
Electrodynamics in Relativistic Notation - Feynman Lectures - Caltech
We have already seen (Section 13-7) that the electric charge density ρ and the current density j form a four-vector jμ=(ρ,j).
[35]
[PDF] Probability Current and Current Operators in Quantum Mechanics
The squared wave function gives the probability density, so the charge density is defined to be ρe = e|ψ|2. One can get the current by looking at changes of ...
[36]
Hydrogen Ground State Properties
### Summary of Radial Probability Density for Hydrogen Ground State
[37]
[PDF] Chap 1 Manybody wave function and Second quantization
Naturally, one can define the particle-density operator as ρ(r) = ψ†(r)ψ(r). (67). This looks like the usual particle density in quantum mechanics. But ...
[38]
[PDF] Efficient Solution of the Schroedinger-Poisson Equations In Layered ...
Jul 24, 2008 · The use of these equations assumes that the electrons interact solely through the electrostatic field and spin effects are ignored.
[39]
[PDF] Electron Charge Density: - arXiv
Electron charge density is the spread-out electron charge surrounding a nucleus, obtained by multiplying the squared amplitude of a wave function by the ...
[40]
[PDF] n = density of conduction (or free) electrons - [n] = è /cm³
Donor Impurities - one extre è in outer shell: donate é's. Acceptor ... Due to doping impurities: if nap: material is called n-type it pin: material is ...
[41]
[PDF] Lecture-2 - Electrical Engineering
Such impurities donate excess electron carriers and are referred to as donor, or n-type, impurities. If intrinsic semiconductor material is doped with n-type ...
[42]
[PDF] Section 4: Electrostatics of Dielectrics
charges characterized by the charge density ρ and bound charges characterized by polarization P. We can build up the potential and the field by linear ...
[43]
Robust charge-density wave strengthened by electron correlations ...
Oct 7, 2021 · In other words, the robust CDW in monolayer TaSe2 and NbSe2 is derived from a combination of electron correlations, a strong lattice distortion, ...
[44]
Modulating Charge-Density Wave Order and Superconductivity from ...
Mar 6, 2023 · Bulk 2H-TaSe2 is a model charge d. wave (CDW) metal with supercond. emerging at extremely low temp. (Tc = 0.1 K). Here, by first-principles ...
[45]
[PDF] An Introduction to Density Functional Theory
Density functional theory provides us with a relatively efficient and unbiased tool with which to compute the ground state energy in realistic models of bulk ...
[46]
[PDF] Density Functional Theory (DFT) - Rutgers Physics
DFT views system properties as a functional of ground state density, using a scalar function instead of a complex wave function. Kohn and Sham replaced the ...
[47]
Applications of scanning tunneling microscopy to the study of charge ...
Scanning tunneling microscopy (STM) studies of the surfaces of transition metal di- and tri-chalcogenides have been used to detect a variety of charge-density ...
[48]
Giant surface charge density of graphene resolved from scanning ...
Oct 24, 2011 · Scanning tunneling microscopy (STM) has also proven to be a powerful tool for studying the electronic and geometric structure of graphene.Missing: mapping | Show results with:mapping