Electrostatics

Electrostatics is the branch of electromagnetism concerned with stationary electric charges and the fields, forces, and potentials they produce. Its central problem is to determine the electric field generated by a specified charge distribution, including charges redistributed by surrounding materials. Electrostatic theory explains familiar static-electricity effects and provides the foundation for analyzing charged conductors, capacitors, and many electrical technologies. Its defining condition is a time-independent charge distribution, rather than the complete absence of electric forces or stored energy. (ocw.mit.edu)

Charge and electrostatic force

Electric charge has two signs, conventionally called positive and negative. Like charges repel and unlike charges attract. Ordinary matter contains positive protons and negative electrons; a body becomes electrically charged when these contributions no longer balance. Net charge is conserved, so charging involves transferring or separating charge rather than creating it. In the International System of Units, charge is measured in coulombs. (openstax.org)

For two point charges in vacuum, Coulomb’s law gives the force magnitude as

where $r$ is their separation and $\varepsilon_0$ is the vacuum permittivity. The force acts along the line joining the charges. The superposition principle states that the total force from several source charges is their vector sum. Continuous charge distributions are treated by dividing them into infinitesimal elements and integrating their contributions. (openstax.org)

The field separates the source configuration from the charge experiencing its influence: a test charge $q$ experiences $\mathbf F=q\mathbf E$. Ideally, the test charge is sufficiently small that it does not appreciably alter the source distribution. (openstax.org)

Field equations and potential

Gauss’s law relates the electric flux through a closed surface to the enclosed charge:

Its differential form is $\nabla\cdot\mathbf E=\rho/\varepsilon_0$, where $\rho$ is the volume charge density. Although valid for arbitrary distributions, the integral form is especially convenient for spherical, cylindrical, or planar symmetry. Charges outside the surface can affect the local field but contribute zero net flux through that closed surface. (openstax.org)

An electrostatic field is conservative: its circulation around a closed path vanishes. Consequently, it can be expressed through the scalar electric potential $V$:

Here $\nabla V$ is the gradient of the potential. Potential differences are path-independent, and adding a constant to $V$ leaves the field unchanged. The potential energy of a test charge in a prescribed external potential is $U=qV$. (ocw.mit.edu)

Combining the potential relation with Gauss’s law yields Poisson’s equation,

In a charge-free region it reduces to Laplace’s equation, $\nabla^2V=0$. Electrostatic calculations therefore often become a boundary-value problem: finding a potential consistent with specified charges and boundary conditions on surrounding surfaces. (ocw.mit.edu)

Conductors and electrostatic equilibrium

An electrical conductor contains mobile charge carriers. If an electric field exists within its conducting material, these carriers move. In electrostatic equilibrium, their redistribution makes the macroscopic field inside the conducting material zero. Each connected conductor is therefore an equipotential, and excess charge resides on its surfaces. The field immediately outside a smooth surface has no tangential component. (openstax.org)

An external charged object can redistribute a conductor’s charges without changing its net charge, a process called electrostatic induction. For a conductor surrounded by vacuum, the normal field just outside its surface satisfies $E_n=\sigma/\varepsilon_0$, with sign determined by the outward normal and surface charge density $\sigma$. A closed conducting enclosure shields an empty internal cavity from external static electric fields; this is the electrostatic basis of a Faraday cage. The zero-field statement applies to the conducting material and does not imply that every cavity containing internal charges is field-free. (openstax.org)

Dielectrics and polarization

In a dielectric, charges cannot move freely through the material as they do in a conductor. Nevertheless, an applied field can displace bound charges or orient permanent electric dipoles. This produces electric polarization and associated bound charge, which must be included when calculating the total field. (openstax.org)

Macroscopic theory distinguishes free charge from bound charge by introducing the electric displacement field $\mathbf D=\varepsilon_0\mathbf E+\mathbf P$, where $\mathbf P$ is the polarization density. Gauss’s law then becomes $\nabla\cdot\mathbf D=\rho_{\mathrm{free}}$. For a linear, isotropic dielectric, $\mathbf D=\varepsilon\mathbf E$; the permittivity $\varepsilon$ describes the material’s electric response. This simple relation is a material model, not a universal property of all insulators. (ocw.mit.edu)

Capacitance, energy, and applications

A capacitor consists of separated conductors carrying opposite charges. Its capacitance is $C=Q/\Delta V$, where $Q$ is the magnitude of either conductor’s charge and $\Delta V$ is their potential difference. For parallel plates of area $A$, separation $d$, and a uniform dielectric, neglecting edge effects,

Capacitance depends on geometry and the intervening material. (openstax.org)

Charging a capacitor requires work, stored as electrostatic energy:

In vacuum, the field has energy density $u=\tfrac12\varepsilon_0|\mathbf E|^2$. These expressions connect electrostatic forces with energy storage in electrical devices. (openstax.org)

Practical applications include photocopiers and laser printers, which use charge patterns to control toner deposition; electrostatic precipitators, which collect charged airborne particles; and Van de Graaff generators, which accumulate charge to produce high potentials. Such devices may involve moving particles while still using electrostatic field calculations to determine their forces. (openstax.org)