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Electrostatics

Electrostatics is the branch of physics that deals with electric charges at rest and the stationary electric fields they produce, encompassing the study of forces between charged particles in equilibrium.^[1] It focuses on the properties of electrical forces arising from charge distributions, where charges do not move relative to the objects they reside in, as exemplified by phenomena like static electricity.^[2] The foundational principle is Coulomb's law, which quantifies the electrostatic force between two point charges as directly proportional to the product of their charge magnitudes and inversely proportional to the square of the separation distance between them, following an inverse-square relationship analogous to gravity but much stronger at short ranges.^[3] Electric charge, the source of these forces, is a fundamental property of matter quantized in discrete units, with the elementary charge of an electron or proton being approximately $1.602 \times 10^{-19} coulombs (C), and it obeys the law of conservation, meaning the total charge in an isolated system remains constant.^[4] Key concepts include the electric field, a vector field surrounding charges that exerts forces on other charges within it, defined such that the force on a test charge q is \mathbf{F} = q\mathbf{E}, where \mathbf{E} is independent of the test charge.^[3] This leads to tools like Gauss's law, which relates the electric flux through a closed surface to the enclosed charge, enabling calculations for symmetric charge distributions: \oint \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{\text{enc}}}{\epsilon_0}, with \epsilon_0 as the vacuum permittivity.^[5] Electrostatics also involves electric potential, the scalar potential energy per unit charge, from which the electric field derives as the negative gradient, \mathbf{E} = -\nabla V, facilitating energy-based analyses of charge configurations like capacitors, which store energy in the form U = \frac{1}{2}CV^2.^[5] The field is conservative, meaning the work done by the electric force around a closed path is zero, and the principle of superposition applies, allowing complex systems to be analyzed by summing contributions from individual charges.^[3] Historically, observations of static electricity date back to ancient times, with systematic study beginning in the 18th century through experiments by figures like Charles-Augustin de Coulomb, who quantified the force law using a torsion balance.^[4] In modern contexts, electrostatics underpins diverse applications, including Van de Graaff generators for high-voltage research, xerography in photocopiers and laser printers via charged toner particles, electrostatic painting for uniform coating, and electrostatic precipitators that remove over 99% of particulate matter from industrial exhaust gases.^[6] These principles extend to natural phenomena like lightning and atmospheric electricity, highlighting electrostatics' role in both fundamental science and technology.^[6]

Fundamental Concepts

Electric Charge

Electric charge is a fundamental physical property of matter that causes it to experience a force within an electromagnetic field. It manifests in two distinct types: positive charge, primarily associated with protons, and negative charge, carried by electrons. Like charges repel one another, while opposite charges attract, forming the basis of electrostatic interactions.^[7]^[8]^[9] The modern convention of labeling charges as positive and negative originated with Benjamin Franklin's experiments in the mid-1750s, where he distinguished the two forms based on their behaviors in frictional electricity and lightning. In 1909, Robert Millikan conducted the oil-drop experiment, which demonstrated that electric charge is quantized, occurring only in discrete multiples of the elementary charge e = 1.602176634 \times 10^{-19} C.^[10]^[11]^[12] The law of conservation of electric charge states that the total electric charge in an isolated system remains constant, as charge cannot be created or destroyed, only transferred. This principle is exemplified in triboelectric charging, where friction between materials leads to electron transfer; for instance, rubbing a glass rod with silk causes the glass to gain a positive charge while the silk becomes negative, following their positions in the triboelectric series.^[13]/03%3A_Electric_Charge_and_Electric_Field/3.02%3A_Static_Electricity_and_Charge_-_Conservation_of_Charge) The international unit of electric charge is the coulomb (C), defined as the quantity of charge transported by a constant current of one ampere over one second. Early measurements of charge relied on electrometers, devices that detect and quantify charge through the mechanical deflection produced by electrostatic repulsion between similarly charged leaves or needles.^[14]^[15]^[16]

Coulomb's Law

Coulomb's law quantifies the electrostatic force between two stationary point charges q_1 and q_2 separated by a distance r in vacuum. The force \vec{F} is given by

\vec{F} = k \frac{q_1 q_2}{r^2} \hat{r},

where k = \frac{1}{4\pi\epsilon_0} is Coulomb's constant, \epsilon_0 = 8.85 \times 10^{-12} \, \mathrm{C^2/N \cdot m^2} is the vacuum permittivity, and \hat{r} is the unit vector pointing from q_1 to q_2.^[17]^[5] This vector form indicates that the force is central, acting along the line joining the charges, and its magnitude is F = k \frac{|q_1 q_2|}{r^2}.^[18] The law was empirically established through experiments by Charles-Augustin de Coulomb in 1785 using a torsion balance, an instrument he designed to measure the torsional deflection caused by repulsive or attractive forces between charged spheres.^[19]^[20] In these experiments, Coulomb suspended one charged object on a fine wire and observed the equilibrium twist when interacting with another fixed charge, confirming the inverse-square dependence of the force on distance and its proportionality to the product of the charges.^[21] The direction of the force follows from the sign of q_1 q_2: if the charges have the same sign (both positive or both negative), the force is repulsive, directed away from each other along \hat{r} or -\hat{r}; if opposite signs, it is attractive, pulling them together.^[5] This arises directly from the positive value of [k](/page/K) and the scalar product q_1 q_2, yielding a positive force magnitude that aligns with repulsion for like charges and requires reversing the direction vector for attraction in the two-charge system.^[18] Coulomb's law strictly applies to point charges at rest in vacuum, where velocities are negligible and no magnetic effects arise; it assumes non-relativistic conditions and neglects medium polarization effects present in dielectrics.^[5] For continuous charge distributions, such as lines, surfaces, or volumes, the law extends by integrating over infinitesimal charge elements: the total force on a test charge is \vec{F} = k q \int \frac{\rho(\vec{r}') (\vec{r} - \vec{r}')}{|\vec{r} - \vec{r}'|^3} dV', where \rho is the charge density, though explicit computation requires specific geometries.^[18] This inverse-square form mirrors Newton's law of universal gravitation, \vec{F} = -G \frac{m_1 m_2}{r^2} \hat{r}, where G is the gravitational constant, highlighting a structural similarity in the central, distance-dependent forces between interacting pairs, though electrostatic forces are vastly stronger and can be attractive or repulsive depending on charge signs.

Electric Field

Definition and Properties

The electric field \vec{E} is a vector field that describes the electric force experienced per unit positive test charge at any point in space. It is formally defined as the limit of the ratio of the electrostatic force \vec{F} on a small positive test charge q_0 to the magnitude of that charge, as q_0 approaches zero to avoid perturbing the source configuration:

\vec{E} = \lim_{q_0 \to 0} \frac{\vec{F}}{q_0}.

This definition captures the force that would act on any positive charge placed at the location, independent of the specific test charge used.^[22]^[23] For a single point charge q located at the origin, the electric field at a distance r along the radial direction is given by

\vec{E} = \frac{1}{4\pi\epsilon_0} \frac{q}{r^2} \hat{r},

where \epsilon_0 is the vacuum permittivity, and \hat{r} is the unit vector pointing away from the charge. This expression arises directly from Coulomb's law by considering the force on a test charge at distance r. The field points radially outward from a positive charge and inward toward a negative charge, with magnitude decreasing as the inverse square of the distance.^[22]^[23] Key properties of the electrostatic field include its vector nature, which means it has both magnitude and direction at every point; its additivity via the superposition principle, where the total field from multiple charges is the vector sum of individual fields; its conservative character in electrostatic equilibrium, expressed mathematically as \nabla \times \vec{E} = 0, implying that the work done by the field around any closed path is zero; and its inverse-square decay for point sources, reflecting the underlying pairwise nature of electrostatic interactions. The SI units of the electric field are newtons per coulomb (N/C), equivalently volts per meter (V/m).^[22]^[23]^[24] Electric field lines provide a visual representation of the field's direction and relative strength. These imaginary lines are drawn such that their direction at any point is tangent to the field vector there, originating from positive charges (or at infinity) and terminating on negative charges (or at infinity). The density of the lines in a region is proportional to the magnitude of the field strength, with closer spacing indicating stronger fields. Visualization techniques, such as aligning small particles like grass seeds in oil under an applied field or computational simulations, analogously reveal patterns similar to iron filings tracing magnetic fields.^[25]^[23]

Gauss's Law

Gauss's law is a fundamental principle in electrostatics that relates the electric flux through a closed surface to the total electric charge enclosed within that surface. It is expressed mathematically as

\oint_S \vec{E} \cdot d\vec{A} = \frac{Q_{\mathrm{enc}}}{\epsilon_0},

where \vec{E} is the electric field, d\vec{A} is the differential area vector on the closed surface S, Q_{\mathrm{enc}} is the net charge enclosed by S, and \epsilon_0 is the vacuum permittivity.^[26] This integral form highlights the law's reliance on symmetry, making it particularly useful for calculating electric fields in situations with high geometric symmetry, rather than directly integrating Coulomb's law for complex charge distributions.^[27] The law can be derived from Coulomb's law by considering the flux due to a point charge. For a point charge q at the origin, the electric field is radial, and the flux through a closed surface is computed using solid-angle arguments: the total solid angle subtended by any closed surface enclosing the charge is $4\pi steradians, leading to \oint \vec{E} \cdot d\vec{A} = \frac{q}{\epsilon_0}. Extending this to arbitrary charge distributions via superposition yields the general form, as the contributions from each infinitesimal charge element dq sum to the enclosed charge. Alternatively, applying the divergence theorem to the field expression from Coulomb's law transforms the integral into a volume integral of the divergence, confirming the result.^[5]^[28] In its differential form, Gauss's law is \nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0}, where \rho is the charge density; this local relation is obtained by applying the divergence theorem to the integral form over an infinitesimal volume.^[29] This version connects electrostatics to the broader framework of partial differential equations and is one of Maxwell's equations, which unify electricity, magnetism, and optics.^[30] Gauss's law excels in applications involving symmetric charge distributions, where the field magnitude is constant over the Gaussian surface. For an infinite plane with surface charge density \sigma, a cylindrical Gaussian surface perpendicular to the plane yields E = \frac{\sigma}{2\epsilon_0}, independent of distance. For an infinite line charge with linear density \lambda, a cylindrical surface gives E = \frac{\lambda}{2\pi \epsilon_0 r}. For a uniformly charged spherical shell of radius R and total charge Q, the field is zero inside (r < R) and E = \frac{Q}{4\pi \epsilon_0 r^2} outside (r > R).^[26]^[27] The law is named after Carl Friedrich Gauss, who rediscovered and formalized it in 1835 in the context of inverse-square forces, building on earlier work by Joseph-Louis Lagrange in 1773; it forms the cornerstone of electrostatic theory as part of Maxwell's equations.^[31]

Superposition and Field Lines

The superposition principle in electrostatics states that the total electric field at any point due to multiple charges is the vector sum of the electric fields produced by each charge individually.^[5] This principle arises from the linearity of Coulomb's law, which describes the force between two point charges as proportional to the product of their charges and inversely proportional to the square of their separation distance, allowing forces (and thus fields) to add linearly without interference.^[32] Furthermore, the electrostatic limit of Maxwell's equations—specifically, \nabla \cdot \vec{E} = \rho / \epsilon_0 and \nabla \times \vec{E} = 0—is linear in the fields and charge density, ensuring that solutions superpose for arbitrary charge distributions.^[33] For example, consider two point charges: the electric field at a point equidistant from equal positive charges points away from both along the perpendicular bisector, resulting in a net field twice the magnitude of one charge's contribution in that direction. In the case of an electric dipole—two equal and opposite charges separated by a small distance—the fields near the charges are complex, but far from the dipole (at distances much larger than the separation), the opposing contributions partially cancel, yielding a net field that falls off as $1/r^3 and aligns with the dipole axis.^[34] This approximation simplifies analysis of neutral systems like molecules, where the dipole moment \vec{p} = q \vec{d} (with q the charge magnitude and \vec{d} the displacement vector) dominates the far-field behavior.^[35] Electric field lines provide a visual representation of the electric field's direction and relative strength, defined as imaginary lines tangent to the field vector at every point.^[25] Key rules for drawing these lines include: they originate from positive charges (or extend to infinity for isolated positives) and terminate on negative charges (or from infinity for isolated negatives); they never cross, as the field has a unique direction at each point; the density of lines is proportional to field strength, with more lines indicating stronger fields; and lines are closer together in regions of higher field magnitude.^[36] Qualitative sketches illustrate these for common configurations. For a single positive point charge (monopole), lines radiate symmetrically outward in all directions, becoming sparser with distance to reflect the $1/r^2 field decay.^[37] In a dipole, lines emerge from the positive charge, curve around to enter the negative charge, forming closed loops that bulge outward along the equatorial plane, with the tightest spacing midway between the charges where the field is strongest.^[38] For a uniform field, such as between parallel oppositely charged plates, lines are straight, parallel, and equally spaced, indicating constant magnitude and direction.^[39] Practically, the superposition principle enables prediction of field cancellation in neutral systems, such as atoms or molecules, where positive and negative charges balance to produce weak external fields, facilitating the study of induced dipoles in external fields.^[5] It also allows qualitative predictions without full calculations, such as estimating field patterns in complex charge arrangements by mentally overlaying individual contributions.^[40] However, field lines are merely illustrative tools for visualization and do not represent actual paths of charged particles, which follow parabolic trajectories under constant fields due to inertia.^[36]

Electrostatic Potential

Definition and Gradient Relation

The electrostatic potential V at a position \vec{[r](/page/R)} due to an electrostatic field is the amount of work done per unit positive test charge to bring it slowly from infinity to \vec{r}, expressed through the line integral V(\vec{r}) = -\int_{\infty}^{\vec{r}} \vec{E} \cdot d\vec{[l](/page/L')}, where \vec{E} is the electric field and the integral is taken along any path from infinity to \vec{r}.^[41] This definition assumes the potential vanishes at infinity for localized charge distributions, a standard convention in electrostatics.^[42] The electrostatic field \vec{E} is conservative, meaning the line integral is path-independent, as the work done by the field depends only on the endpoints and not the route taken.^[43]^[44] For a single point charge q at the origin, the potential simplifies to V(r) = \frac{1}{4\pi\epsilon_0} \frac{q}{r}, where r = |\vec{r}| is the distance from the charge and \epsilon_0 is the vacuum permittivity; this follows directly from integrating the Coulomb field along a radial path.^[45]^[46] The relation between the potential and the electric field is given by \vec{E} = -\nabla V, the negative gradient of the scalar potential, which in Cartesian coordinates yields the components E_x = -\frac{\partial V}{\partial x}, E_y = -\frac{\partial V}{\partial y}, and E_z = -\frac{\partial V}{\partial z}.^[47]^[48] This vector calculus relation highlights how the directional field \vec{E} derives from the scalar V, facilitating computations in symmetric systems. Surfaces of constant potential, known as equipotential surfaces, are everywhere perpendicular to the electric field lines, since the tangential component of \vec{E} vanishes on such surfaces, implying no work is done moving a charge along them.^[49]^[50] The unit of electric potential is the volt (V), defined such that 1 V = 1 J/C, representing the potential difference across which 1 C of charge experiences 1 J of work.^[51] Voltmeters measure potential differences by connecting in parallel to the points of interest, drawing negligible current through a high internal resistance to approximate the open-circuit voltage without significantly perturbing the field.^[52] Due to the linearity of electrostatics, the superposition principle applies to potentials as scalars: the total potential at any point from multiple charges is the algebraic sum of the individual potentials, simplifying calculations compared to vector addition for fields.^[53]^[46]

Poisson's and Laplace's Equations

In electrostatics, Poisson's equation governs the scalar potential V in the presence of a charge density distribution \rho. It is derived by combining Gauss's law in differential form, \nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0}, with the relation between the electric field and potential, \vec{E} = -\nabla V. Substituting yields \nabla \cdot (-\nabla V) = \frac{\rho}{\epsilon_0}, or equivalently,

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

where \nabla^2 is the Laplacian operator and \epsilon_0 is the vacuum permittivity.^[54] This second-order partial differential equation describes how the potential varies spatially due to localized charges. In regions devoid of charge, where \rho = 0, the equation simplifies to Laplace's equation,

\nabla^2 V = 0.

Laplace's equation arises naturally in charge-free spaces, such as between conductors or outside charge distributions, and its solutions represent harmonic functions that model equilibrium potentials.^[54] The Laplacian operator \nabla^2 takes different explicit forms depending on the coordinate system, which is essential for solving these equations in problems with specific symmetries. In Cartesian coordinates (x, y, z), it is

\nabla^2 V = \frac{\partial^2 V}{\partial x^2} + \frac{\partial^2 V}{\partial y^2} + \frac{\partial^2 V}{\partial z^2}.

This form is straightforward for rectangular geometries but less convenient for cylindrical or spherical symmetries common in electrostatic problems. In spherical coordinates (r, \theta, \phi), where r is the radial distance, \theta the polar angle, and \phi the azimuthal angle, Laplace's equation becomes

\frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial V}{\partial r} \right) + \frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta} \left( \sin \theta \frac{\partial V}{\partial \theta} \right) + \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 V}{\partial \phi^2} = 0.

Poisson's equation in spherical coordinates follows the same left-hand side equal to -\rho / \epsilon_0. This form facilitates solutions for problems like point charges or spherical conductors using separation of variables.^[55] For cylindrical coordinates (\rho, \phi, z), with \rho the radial distance from the axis, \phi the azimuthal angle, and z along the axis, the equation is

\frac{1}{\rho} \frac{\partial}{\partial \rho} \left( \rho \frac{\partial V}{\partial \rho} \right) + \frac{1}{\rho^2} \frac{\partial^2 V}{\partial \phi^2} + \frac{\partial^2 V}{\partial z^2} = 0

for Laplace's case, and similarly for Poisson's with the right-hand side -\rho / \epsilon_0. This is particularly useful for infinite line charges or coaxial geometries.^[56] Solutions to these equations require appropriate boundary conditions to ensure physical relevance. The potential V is continuous across interfaces, such as between dielectrics or at conductor surfaces. The normal component of the electric field, -\partial V / \partial n, exhibits a discontinuity at surfaces with surface charge density \sigma, jumping by \sigma / \epsilon_0. For conductors, V is constant on the surface, and the tangential field vanishes. At infinity, V typically approaches zero for localized charges.^[57] A key result is the uniqueness theorem, which guarantees that the solution to Poisson's or Laplace's equation in a given volume is uniquely determined by the charge distribution inside and the boundary values of V (or its normal derivative) on the enclosing surface. To see this, suppose two solutions V_1 and V_2 satisfy the same equation and boundaries; their difference u = V_1 - V_2 obeys Laplace's equation with zero boundary values. Integrating \nabla \cdot (u \nabla u) = |\nabla u|^2 = 0 over the volume and applying the divergence theorem yields \int |\nabla u|^2 d\tau = 0, implying \nabla u = 0 and thus u = constant, which is zero by boundaries. This theorem underpins computational methods and symmetry arguments in electrostatics.^[57] Historically, Laplace's equation emerged in the late 1700s from Pierre-Simon Laplace's work on gravitational potentials in Mécanique Céleste, where he recognized that the Newtonian potential satisfies \nabla^2 V = 0 in source-free regions around 1782. Poisson extended this in the 1810s, deriving the inhomogeneous form for electrostatics in memoirs on charge distributions, with the equation formalized in his 1823 paper on magnetism, \nabla^2 V = -4\pi k \rho (in cgs units). These contributions laid the foundation for potential theory.^[58]^[59]

Electrostatic Approximation and Phenomena

The electrostatic approximation is valid when electric charges are either at rest or moving with velocities much smaller than the speed of light, such that time-dependent magnetic fields and electromagnetic radiation effects can be neglected. Under this approximation, the electric field is irrotational (curl-free) and conservative, allowing the use of static equations like Coulomb's law and Gauss's law without considering retardation or inductive effects.^[60] This framework enables the analysis of various electrostatic phenomena discussed below.

Energy and Capacitance

The electrostatic potential energy associated with a pair of point charges q_1 and q_2 separated by a distance r is given by

U = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r},

where \epsilon_0 is the vacuum permittivity.^[61] This expression represents the work required to assemble the charges from infinite separation, assuming they are stationary.^[62] For a system of multiple point charges, the total potential energy is the sum over all unique pairs:

U = \frac{1}{4\pi\epsilon_0} \sum_{i < j} \frac{q_i q_j}{r_{ij}}.

This pairwise summation accounts for the interactions without double-counting.^[62] For a continuous charge distribution with density \rho(\mathbf{r}), the electrostatic potential energy generalizes to the integral form

U = \frac{1}{2} \int \rho(\mathbf{r}) V(\mathbf{r}) \, d\tau,

where V(\mathbf{r}) is the electric potential at position \mathbf{r}, and the integral is over all space.^[63] The factor of $1/2 arises from avoiding double-counting of interactions in the continuous limit, analogous to the discrete case.^[62] This formulation connects the stored energy directly to the charge density and potential, providing a basis for analyzing complex configurations. An equivalent expression for the total energy can be derived in terms of the electric field \mathbf{E}, revealing the energy density in the electrostatic field as

u = \frac{1}{2} \epsilon_0 E^2.

The total energy is then the volume integral of this density:

U = \frac{1}{2} \epsilon_0 \int E^2 \, d\tau.

This field-based perspective highlights how energy is distributed throughout space, independent of the specific charge arrangement, and follows from vector calculus identities relating \rho V to \mathbf{E}.^[62]^[63] Capacitors are devices designed to store electrostatic energy by maintaining separated charges on conductors. The capacitance C of a capacitor is defined as the ratio of the magnitude of charge Q on each conductor to the potential difference V between them:

C = \frac{Q}{V}.

The SI unit of capacitance is the farad (F), equivalent to one coulomb per volt.^[64] A common example is the parallel-plate capacitor, consisting of two conducting plates of area A separated by a small distance d in vacuum, with capacitance

C = \epsilon_0 \frac{A}{d}.

This formula assumes d \ll \sqrt{A} for uniform field approximation and neglects edge effects.^[65] The energy stored in a capacitor follows from the work done to charge it and can be expressed as

U = \frac{1}{2} C V^2 = \frac{1}{2} \frac{Q^2}{C}.

These equivalent forms derive from integrating the incremental work dU = V \, dq during charging, yielding a quadratic dependence on voltage or charge.^[66] For the parallel-plate case, substituting the capacitance gives U = \frac{1}{2} \epsilon_0 \frac{A}{d} V^2, which matches the field energy integral with E = V/d.^[67] Introducing a dielectric material between the plates increases the capacitance by a factor \kappa > 1, known as the dielectric constant, without altering the charge or plate geometry.^[64] This enhancement allows greater energy storage for the same voltage, as C' = \kappa C. The effect stems from the material's response to the field, though detailed mechanisms are beyond this scope. The Leyden jar, invented independently in 1745 by Ewald Jürgen von Kleist and Pieter van Musschenbroek, served as the first capacitor, consisting of a glass jar with conductive coatings inside and outside to store charge.^[68] This device enabled early experiments in electrostatics by providing a means to accumulate and discharge electrical energy on demand.

Forces and Pressure on Conductors

In electrostatic equilibrium, a conductor exhibits zero electric field throughout its interior volume, with any excess charge residing exclusively on its surface. This configuration ensures that the entire conductor maintains a constant electrostatic potential, as free charges within the material redistribute to cancel any internal fields.^[69] The mechanical force exerted on a conductor in an electrostatic field can be derived using the principle of virtual work, which relates the force to variations in the system's electrostatic energy U. For a displacement along a coordinate x, the force component is given by F_x = -\frac{\partial U}{\partial x} evaluated at constant charge Q or constant potential V, depending on the boundary conditions of the setup.^[69]^[33] Just outside the surface of a conductor, the perpendicular component of the electric field E_n relates directly to the local surface charge density \sigma via Gauss's law applied to a Gaussian pillbox straddling the surface: \sigma = \epsilon_0 E_n. This discontinuity in the field arises because the internal field is zero, confining the flux to the external side.^[69] The electrostatic pressure on the conductor's surface, representing the outward force per unit area due to the repulsion of like charges, is derived by considering the force on a small surface element. This yields P = \frac{\sigma^2}{2\epsilon_0} = \frac{1}{2} \epsilon_0 E_n^2, directed normal to the surface and away from the conductor.^[70] A classic example is the attractive force between the oppositely charged plates of a parallel-plate capacitor. For plates of area A separated by distance d with total charge \pm Q, the uniform field between them is E = Q/(\epsilon_0 A), leading to a force magnitude F = \frac{Q^2}{2\epsilon_0 A} pulling the plates together, independent of d for small separations where fringing is negligible.^[71] For an isolated, uniformly charged spherical conductor, the net self-force is zero due to the symmetry of the charge distribution, which produces a radial field outside but no unbalanced tangential components to drive translation.^[72] When a point charge q is placed near an infinite grounded conducting plane, it induces an opposite surface charge distribution on the plane to maintain zero potential. The method of images models this by replacing the plane with a mirror image charge -q at the symmetric position across the plane, allowing calculation of the field in the region of interest and the resulting force on q as if interacting with the image.^[73]

Dielectrics and Polarization

Dielectrics are insulating materials that do not conduct electricity freely but respond to an applied electric field by developing polarization, which alters the local electric field within the material.^[74] This polarization arises from the displacement or reorientation of bound charges, such as electrons and ions, within the dielectric's atomic or molecular structure, leading to a net dipole moment per unit volume known as the polarization vector \vec{P}.^[75] The magnitude of \vec{P} quantifies the extent of this alignment and is crucial for understanding how dielectrics modify electrostatic fields compared to vacuum. Polarization in dielectrics occurs through several mechanisms, each dominant in specific materials or conditions. Electronic polarization involves the distortion of electron clouds around atoms, shifting negative charge relative to the positive nucleus without permanent dipoles. Ionic polarization, prevalent in ionic crystals like NaCl, results from the relative displacement of oppositely charged ions in a lattice.^[76] Orientational polarization occurs in materials with permanent molecular dipoles, such as water, where thermal motion randomizes dipoles in the absence of a field, but an applied field aligns them partially.^[77] These processes collectively contribute to \vec{P}, typically on the order of $10^{-8} to $10^{-6} C/m² in common dielectrics.^[74] The presence of polarization introduces bound charges that affect the electric field. Volume bound charge density is given by \rho_b = -\nabla \cdot \vec{P}, arising from spatial variations in polarization, while surface bound charge density is \sigma_b = \vec{P} \cdot \hat{n}, where \hat{n} is the outward normal to the surface.^[75] These bound charges oppose the applied field, reducing the net field inside the dielectric. To account for this, the electric displacement field \vec{D} is defined as \vec{D} = \epsilon_0 \vec{E} + \vec{P}, where \epsilon_0 is the vacuum permittivity and \vec{E} is the electric field. In linear dielectrics, where polarization is proportional to the field, this simplifies to \vec{D} = \epsilon_0 \epsilon_r \vec{E}, with \epsilon_r the relative permittivity (dielectric constant).^[78] Linear dielectrics are classified as isotropic if \vec{P} is parallel to \vec{E} with a scalar \epsilon_r > 1, or anisotropic if the response depends on direction, as in crystals, requiring a tensor description.^[78] For example, water exhibits \epsilon_r \approx 80 at room temperature due to strong orientational polarization from its polar molecules, while glass has \epsilon_r \approx 5-10, primarily from electronic and ionic contributions.^[79]^[80] This variation in \epsilon_r explains why dielectrics enhance capacitance and store more energy in electrostatic configurations. In the presence of dielectrics, Gauss's law is reformulated using \vec{D} to focus on free charges: \nabla \cdot \vec{D} = \rho_f, where \rho_f denotes the density of free (unbound) charges.^[81] This form simplifies calculations by excluding bound charges, which are incorporated into \vec{P}. The energy stored in the electrostatic field within a dielectric is then U = \frac{1}{2} \int \vec{D} \cdot \vec{E} \, d\tau, reflecting the work done to establish the field against the material's response.^[82] For linear dielectrics, this reduces to \frac{1}{2} \epsilon_0 \epsilon_r \int E^2 \, d\tau, highlighting how higher \epsilon_r increases energy density.^[83]

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elementary charge - CODATA Value
Concise form, 1.602 176 634 x 10-19 C ; Click here for correlation coefficient of this constant with other constants ; Source: 2022 CODATA
[13]
18.1 Electrical Charges, Conservation of Charge, and Transfer of ...
Mar 26, 2020 · By the end of this section, you will be able to do the following: Describe positive and negative electric charges; Use conservation of ...
[14]
Defining the standard electrical units (article) | Khan Academy
The size of a coulomb is derived from the ampere. One coulomb is defined as the amount of charge flowing when the current is 1 ampere.Missing: electrometers | Show results with:electrometers
[15]
Coulomb - (Intro to Electrical Engineering) - Fiveable
A coulomb is the unit of electric charge in the International System of Units (SI), defined as the amount of charge transported by a constant current of one ...
[16]
J. J. Thomson 1897 - Cathode Rays - Le Moyne
This experiment proves that something charged with negative electricity is shot off from the cathode, travelling at right angles to it, and that this something ...
[17]
vacuum electric permittivity - CODATA Value
vacuum electric permittivity $\varepsilon_0$ ; Numerical value, 8.854 187 8188 x 10-12 F m ; Standard uncertainty, 0.000 000 0014 x 10-12 F m ...
[18]
[PDF] Chapter 2 Coulomb's Law - MIT
There are two types of observed electric charge, which we designate as positive and negative. The convention was derived from Benjamin Franklin's ...Missing: limitations extensions
[19]
[PDF] Charles-Augustin Coulomb First Memoir on Electricity and Magnetism
In a memoir presented to the Academy, in 1784, I have determined from experiments the laws governing the torsional resistance of a filament of metal and I.
[20]
This Month in Physics History | American Physical Society
Charles Augustin Coulomb (top) used a calibrated torsion balance (bottom) to measure the force between electric charges. Around 600 BC, the Greek ...
[21]
Torsion Balance – 1785 - Magnet Academy - National MagLab
According to Coulomb's law, the electric force between objects is inversely proportional to the distance between the objects. A similar inverse square law ...Missing: paper | Show results with:paper
[22]
18.3 Electric Field - Physics | OpenStax
Mar 26, 2020 · Mathematically, saying that electric field is the force per unit charge is written as. E → = F → q test E → = F → q test. 18.15.Missing: primary | Show results with:primary
[23]
11.3: Electric Field - Physics LibreTexts
Feb 14, 2023 · The electric field has a fixed magnitude for a given radial distance away from the charge, with vectors pointing away from a positive source.Missing: primary | Show results with:primary
[24]
[PDF] CHAPTER 2 ELECTROSTATICS - Physics
Oct 5, 2015 · For electrostatic, there is no time-dependent terms, therefore the curl of a static is zero everywhere. The above result can be obtained ...
[25]
[PDF] Chapter 2 Electric Fields
It follows from Coulomb's law that the electric field at point r due to a charge q located at the origin is given by. E = k q r2 r. (2.3) where r is the unit ...Missing: primary | Show results with:primary
[26]
[PDF] Chapter 4 Gauss's Law
The following steps may be useful when applying Gauss's law: (1) Identify the symmetry associated with the charge distribution. (2) Determine the direction of ...
[27]
[PDF] Chapter 3 Gauss'(s) Law
Gauss'(s) Law is used to find the electric field for charge distributions which have a symmetry which we can exploit in calculating both sides of the equation: ...
[28]
[PDF] Lectures on Electromagnetic Field Theory - Purdue Engineering
Mar 5, 2025 · ... Coulomb's Law (Statics) ... Gauss's Law for Electric Flux (Statics) . . . . . . . . . . . . . . . . . . 13. 1.3.4 Derivation of Gauss's ...
[29]
[PDF] gauss-law.pdf
We are left with the differential form of Gauss's law. We have shown that the triad of electrostatics (1) Coulomb's Law (2) Gauss's integral form and (3) Gauss ...
[30]
[PDF] Section 1: Maxwell Equations
Eq. (1.16) is Gauss's law, which is the direct consequence of Coulomb's law and is the first of four Maxwell's equations. Eq. (1.17) is valid for ...
[31]
Gauss's Law
Gauss's law applies to any inverse-square central force. This law was first discovered by Joseph-Louis Lagrange in 1773, and was later rediscovered by Carl ...
[32]
[PDF] Chapter 2 Coulomb's Law
2.3 Principle of Superposition ... The electrostatic force, like the gravitational force, is a force that acts at a distance, even when the objects are not ...
[33]
[PDF] Introduction to Electrostatics
Dec 23, 2000 · The electric field has the property of being independent of the 'test' charge q; it is a function of the charge distribution which gives rise to.<|control11|><|separator|>
[34]
6 The Electric Field in Various Circumstances - Feynman Lectures
A “dipole” antenna can often be approximated by two charges separated by a small distance—if we don't ask about the field too close to the antenna. (We are ...
[35]
[PDF] Ph501 Electrodynamics Problem Set 8 - Kirk T. McDonald
We now consider the superposition of the fields (223)-(226) inside a conducting sphere ... where in the dipole approximation, the far-zone scattered electric ...
[36]
18.5 Electric Field Lines: Multiple Charges - UCF Pressbooks
Example 1: Adding Electric Fields · Field lines must begin on positive charges and terminate on negative charges, or at infinity in the hypothetical case of ...
[37]
Electric Fields
Field lines originate on positive charges and terminate on negative charges. A field line represents the path through space that a small positive test charge ...
[38]
[PDF] Electric Field Lines - De Anza College
Rules for drawing electric field lines: 1. The lines must begin on a positive charge and terminate on a negative charge. In the case of an excess of one type ...
[39]
[PDF] Electric Field
• Field of an Electric Dipole. • Electric Dipole in an External Field: Torque ... Use superposition to calculate net electric field at each point due to ...
[40]
[PDF] 02. Coulomb force in 2D. Electric field. Superposition principle
Sep 11, 2020 · Electric fields from different sources add up without affecting each other. The superposition principle applies.
[41]
[PDF] How To Find the Electric Potential for a Given Charge Distribution
Often, it's convenient to choose V = 0 at infinity. 3. Now, set up a line integral which gives the potential difference ∆V between two points A and B. ∆V ...
[42]
19.3 Electrical Potential Due to a Point Charge - UCF Pressbooks
The electric potential due to a point charge is, thus, a case we need to consider. Using calculus to find the work needed to move a test charge q from a ...
[43]
Electric Potential Energy - Richard Fitzpatrick
We call a force-field which stores energy without loss a conservative field. Thus, an electric field, or rather an electrostatic field (i.e., an electric field ...
[44]
[PDF] The Electrostatic Field Is Conservative - UTK-EECS
The Electrostatic Field Is Conservative. Let's first consider the field of a point ... Since E = E(r) (recall. Coulomb's law), W is independent of the path.<|control11|><|separator|>
[45]
[PDF] Chapter 4 Electric Potential - MIT Open Learning Library
Jan 4, 2013 · • The electric potential due to a point charge Q at a distance r away from the charge is. V = 1. 4πε0. Q r . For a collection of charges, using ...
[46]
[PDF] Electric Potential of Many Charges
Electric potential due to a point charge. □ V is the electric potential [units are volts, V]. □ K = 1. 4πϵ0. = 8.99 × 109 N · m2/C2 is the electrostatic ...
[47]
7.4 Determining Field from Potential – University Physics Volume 2
We can calculate the electric field from the potential with E → = − ∇ → V , a process we call calculating the gradient of the potential.
[48]
[PDF] THE ELECTRIC POTENTIAL - Particle Physics
Thus if we identify E with - 'Vep,Eqs. 11 and 12 become identical. So the electric field is the negative of the gradient of the potential: E = -'Vep. (13).
[49]
7.5 Equipotential Surfaces and Conductors - UCF Pressbooks
Equipotential lines are always perpendicular to electric field lines. No work is required to move a charge along an equipotential, since Δ V = 0.
[50]
Equipotential surfaces
Equipotential surfaces are lines where every point has the same potential. Electric field lines are perpendicular to these surfaces, and conductors form them. ...
[51]
THE ELECTROSTATIC POTENTIAL - Home Page of Frank LH Wolfs
The electrostatic potential V at a given position is defined as the potential energy of a test particle divided by the charge q of this object.
[52]
PHYS345 Laboratory: Introduction to Electrical Measurements
Sep 2, 1998 · The Voltmeter The potential difference, or change in electric potential, between two points is measured with a voltmeter. Current flows through ...
[53]
[PDF] Chapter 3 Electric Potential
Unlike electric field, electric potential is a scalar quantity. For the discrete distribution, we apply the superposition principle and sum over individual ...
[54]
5.15: Poisson's and Laplace's Equations - Engineering LibreTexts
Sep 12, 2022 · This alternative approach is based on Poisson's Equation, which we now derive. We begin with the differential form of Gauss' Law (Section 5.7):.
[55]
Laplace's Equation--Spherical Coordinates -- from Wolfram MathWorld
To solve Laplace's equation in spherical coordinates, attempt separation of variables by writing F(r,theta,phi)=R(r)Theta(theta)Phi(phi).
[56]
Laplace's Equation in Cylindrical Coordinates
We wish to solve Laplace's equation, within a cylindrical volume of radius $ a$ and height $ L$. Let us adopt the standard cylindrical coordinates, $ r$, $ \ ...
[57]
The uniqueness theorem - Richard Fitzpatrick
The uniqueness theorem is to prove that the electric field inside an empty cavity in a conductor is zero.
[58]
Introduction - Potential Theory in Gravity and Magnetic Applications
Oct 10, 2009 · Pierre Simon, Marquis de Laplace, showed in 1782 that Newtonian potential obeys a simple differential equation. Laplace's equation, as it ...
[59]
[PDF] Seven Concepts Attributed to Siméon-Denis Poisson - arXiv
Nov 29, 2022 · Abstract. Siméon-Denis Poisson was 25 years old when he was appointed Professor of. Mathematics at the École Polytechnique in 1806.
[60]
Electrostatic Energy - Richard Fitzpatrick
This expression specifies the electrostatic potential energy of a collection of point charges. We can think of this energy as the work required to bring ...
[61]
The Feynman Lectures on Physics Vol. II Ch. 8: Electrostatic Energy
We evaluate the surface integral in the case that the surface goes to infinity ... E⋅EdV. We see that it is possible for us to represent the energy of any ...
[62]
Electrostatic energy - Richard Fitzpatrick
This is the potential energy (ie, the difference between the total energy and the kinetic energy) of a collection of charges.
[63]
19.5 Capacitors and Dielectrics – College Physics - UCF Pressbooks
Capacitance of a Parallel Plate Capacitor ... C = ε 0 A d . A is the area of one plate in square meters, and d is the distance between the plates in meters. The ...
[64]
Capacitor
The capacitance of a parallel plate capacitor with two plates of area A separated by a distance d and no dielectric material between the plates is C = ε0A/d.Missing: definition | Show results with:definition
[65]
8.3 Energy Stored in a Capacitor – University Physics Volume 2
The total work W needed to charge a capacitor is the electrical potential energy U C stored in it, or U C = W . When the charge is expressed in coulombs, ...
[66]
Energy Stored on a Capacitor - HyperPhysics
A capacitor = = x 10^ F which is charged to voltage V= V will have charge Q = x10^ C and will have stored energy E = x10^ J.
[67]
[PDF] The Art of Making Leyden Jars and Batteries According to Benjamin ...
Devised in 1745 independently by Ewald Jürgen von Kleist in Pomerania and Pieter van. Musschenbroek in Leiden, the Leyden jar was a device that stored and ...
[68]
Conductors - Richard Fitzpatrick
This technique for calculating a force given an expression for the energy of a system as a function of some adjustable parameter is called the principle of ...
[69]
Surface Charge and the Force on a Conductor - Jean-Sébastien Caux
Feb 27, 2024 · f = σ 2 2 ε 0 n ^ amounting to an outward electrostatic pressure.<|separator|>
[70]
5.12: Force Between the Plates of a Plane Parallel Plate Capacitor
Mar 5, 2022 · We connect a battery across the plates, so the plates will attract each other. The upper plate will move down, but only so far, because the ...
[71]
Electric field - Physics
Jul 7, 1999 · Charges are distributed uniformly along both conductors. With the circular shape, each charge has no net force on it, because there is the ...
[72]
1.8: Method of Images - Physics LibreTexts
Aug 7, 2024 · For a point charge, this trick involves introducing an imaginary image charge reflected across the conducting surface, and using that charge to ...Familiar-Looking Field · Clever Trick · Charge on the Conductor
[73]
[PDF] DIELECTRIC POLARIZATION AND BOUND CHARGES - UT Physics
In a dielectric, bound charges cannot move separately. An electric field pushes them, creating tiny dipole moments. Macroscopically, this polarization can ...
[74]
Polarisation mechanisms - DoITPoMS
There are three main polarisation mechanisms that can occur within a dielectric material: electronic polarisation, ionic polarisation (sometimes referred to as ...
[75]
[PDF] 1 Fundamentals of Dielectrics - Wiley-VCH
Figure 1.2 Polarization processes: (a) electronic polarization, (b) ionic polarization, (c) orientational polarization, and (d) space charge polarization. Page ...
[76]
[PDF] A Database for the Static Dielectric Constant of Water and Steam
Oct 15, 2009 · All reliable sources of data for the static dielectric constant or relative pennittivity of water and steam, many of them unpublished or ...
[77]
Relative Permittivity - the Dielectric Constant
Common materials and their relative permittivity. ; Glass, 3.7 - 10 ; Glass, Pyrex, 4.8 ; Glycerin, Liquid, 47-68 ; Glycerol (77°F), 42.5.<|separator|>
[78]
Polarization - Richard Fitzpatrick
$$ \rho_f$ , due to free charges, which represents a net surplus or deficit ... Gauss' law takes the differential form. $\displaystyle \nabla \cdot{\bf E} ...
[79]
Energy density within a dielectric medium - Richard Fitzpatrick
The electrostatic energy density inside a dielectric medium is given by \begin{displaymath} U = \frac{1}{2} {\bf E}\!\cdot\!{\bf D}. \end{displaymath}