The

**electric field** is a concept from

physics. It has two sources, (1) electric charges and (2) the

magnetic field. The relation of the electric field to its two sources is expressed by two of

Maxwell’s Equations:

∇ • **E** = ρ / ε_{o}

and

∇ x **E** = - ∂**B**/∂t

where where **E** is the Electric field, **B** is the Magnetic field, ρ is the position dependent density of electric charge, ε_{o} is a constant called the *permettivity of free space* to be determined experimentally, ∇ • **V** is the divergence of a vector field, and ∇ x **V** is the curl of a vector field.^{1} The units of the electric field are force per unit charge, which in SI units is Newtons per Coulomb.

The first equation relates the divergence of the electric field to a distribution of charges. It is the integral formulation of the more familiar Gauss’ Law^{2}

∫_{surf}**E** • d**a** = Q_{enc}/ ε_{o}.

This relates the component of the electric field perpendicular to the surface of a boundary added over the entire boundary in terms of the total charge contained within the boundary. In the case of a point charge and a boundary of constant radius, we have the even more familiar Coulomb Force law:

**E** = 1/4πε_{o} q* **r**^{unit}/r^{2}

where r is the distance from the source to the point at which the field is being evaluated and q is the charge.

For a distribution of point charges, the Coulomb forces are summed over all the charges:

**E** = 1/4πε_{o} ∫ &rho***r**^{unit}/r^{2}

where ρ is now the position dependent charge density. This is an equivalent statement to the first Maxwell Equation above.

The other Maxwell Equation above relates the curl of the electric field to the change in the flux of the magnetic field. It is the integral formulation of the more familiar Faraday’s Law,^{2}

∫ **E** • d**l** = - dΦ/dt, where Φ is the magnetic flux Φ = ∫_{surf} **B** • d**a**. This is why electric generators work.

If there are only stationary charges, then there are no currents and therefore no magnetic field, so the electric field is given by the first equation only. This situation is called electrostatics. In this case, ∇ x **E** = 0 and the electric field is curl free, and can therefore be expressed in terms of a scalar potential **E** = -∇V. This potential is the more familiar voltage.

In the presence of non-stationary charges, then there will in general be a magnetic field and the electric field will be determined by both equations.

Analysis of Maxwell’s Equations show that the electric field is physically real, and that alternate electric and magnetic fields are responsible for electromagnetic radiation.

^{1} The

**divergence** and

**curl** are quantities from

vector calculus. The

divergence is can be thought of as a measure of how closely a vector field radiates from a single point, and the

curl is in a way a measure of how much a vector field curls around a point. The character ∇ is a directional derivative operator, in symbols (i ∂/∂x, j ∂/∂y, k ∂/∂z), and is pronounced ‘

del,’ or ‘

nabla’ by some freaks.

^{2}These integral formulations are equivalent to the original formulations by the Divergence Theorem and Stokes’ Theorem, two results from vector calculus. For more information, see my w/u under Maxwell’s Equations.