Electric Field

An electric field is a physical field that specifies the electric force experienced by an electric charge at a particular position and time. Represented by the vector E, it has both magnitude and direction. Electric fields arise from charges and can also accompany changing magnetic fields. Together with the magnetic field, the electric field forms the electromagnetic field described by electromagnetism. A field can exist in a region even when no test charge is present there. (openstax.org)

Definition and units

Operationally, the electric field is defined through the electric force on a small, stationary test charge:

The test charge must be sufficiently small that it does not appreciably alter the source charges. The field points in the direction of the force on a positive charge; a negative charge, such as an electron, experiences an electric force in the opposite direction. In the International System of Units, field strength is measured in newtons per coulomb, equivalently volts per metre. (openstax.org)

For a moving charged particle, the complete electromagnetic force is the Lorentz force:

where v is particle velocity and B is the magnetic field. The electric contribution is $q\mathbf E$; the magnetic contribution depends explicitly on velocity. (feynmanlectures.caltech.edu)

Fields produced by charges

For a stationary point charge $Q$ in vacuum, Coulomb’s law gives

where $r$ is distance from the charge, $\hat{\mathbf r}$ points outward from it, and $\varepsilon_0$ is the vacuum permittivity. The field is outward for a positive source and inward for a negative source. Its magnitude decreases with the square of distance. (openstax.org)

Electric fields obey the superposition principle: the total field is the vector sum of the fields produced by individual sources. For a continuous, stationary charge distribution, this sum becomes an integral over its charge density. Oppositely directed contributions may cancel even where individual source fields are substantial. An electric dipole, consisting of equal and opposite charges separated by a distance, illustrates how source arrangement determines field geometry. (openstax.org)

Field lines and electric flux

Field lines visualize direction and relative strength. At each point, the field is tangent to the line; greater line density represents a stronger field. Electrostatic lines begin on positive charges or at infinity and end on negative charges or at infinity. They do not cross where the field has a definite, nonzero direction. These lines are representations, not physical threads or necessarily particle trajectories. (openstax.org)

Electric flux measures the field passing through a surface. Gauss’s law states

where $S$ is a closed surface and $Q_{\mathrm{enc}}$ is its enclosed net charge. External charges can affect the field on the surface but contribute zero net flux through it. The law holds generally, although extracting the field directly from it is especially convenient for spherical, cylindrical, or planar symmetry. (feynmanlectures.caltech.edu)

Electric potential

In electrostatics, the electric field is related to the scalar electric potential $V$ by

The gradient expresses how potential changes with position. The field points toward decreasing potential, and potential differences satisfy

The integral is path-independent for an electrostatic field. Equipotential surfaces are perpendicular to the field wherever it is nonzero. Potential is therefore not identical to field strength: the field depends on spatial variation, rather than the absolute value of potential. (openstax.org)

For a uniform field between ideal parallel plates, the magnitude is $E=|\Delta V|/d$, where $d$ is plate separation. This approximation applies away from plate edges, where fringing alters the field. (openstax.org)

Fields in materials

Within the conducting material of an electrical conductor at electrostatic equilibrium, the macroscopic electric field is zero. Mobile charges redistribute until their field cancels the interior field. The conductor is equipotential, and the field immediately outside its surface is perpendicular to that surface. These statements concern equilibrium, not every conductor carrying a current. (openstax.org)

In a dielectric, bound charges respond by shifting or reorienting, producing electric polarization P. The electric displacement field is defined as

It separates free charge from the contribution of bound polarization charge. For a linear, isotropic dielectric, $\mathbf D=\varepsilon\mathbf E$. This simple proportionality is not universal; material response can depend on direction, field strength, and frequency. (feynmanlectures.caltech.edu)

Time dependence and relativity

Maxwell’s equations connect electric and magnetic fields. In particular,

A changing magnetic field can therefore produce a circulating electric field whose closed-loop integral is nonzero. Such a field cannot generally be described solely by an electrostatic scalar potential. Coupled electric and magnetic disturbances propagate as electromagnetic radiation, including light, at the speed of light in vacuum. (feynmanlectures.caltech.edu)

Electric and magnetic components also depend on the observer’s inertial reference frame. Under a Lorentz transformation, components of the two fields mix; a purely electric field in one frame may have a magnetic component in another. (feynmanlectures.caltech.edu)

Field energy

An electric field stores energy. In vacuum, its contribution to electromagnetic energy density is

Integrating this density over space gives the electric-field energy when the integral is finite. An ideal capacitor stores energy in the field between its charged conductors. For an ideal point charge, the classical field-energy integral diverges near the charge, exposing a limitation of treating a charged particle as a structureless point within classical electrostatics. (feynmanlectures.caltech.edu)