# What is an electric field

1. Jul 24, 2014

### Greg Bernhardt

Definition/Summary

Electric field is electric force per charge, or electric potential energy per distance per charge.

An electric field is a vector field that permeates the space around electrical charge. It is what mediates the force between that charge and any other charge nearby. It is also caused (induced) by a changing magnetic field.

The electric field, $\mathbf{E}$, can be found from the charge producing it (using Coulomb's Law, or Gauss's Law), or from the electromagnetic potential (using $\mathbf{E}\ =\ -\nabla \phi\ -\ \frac{1}{c}\frac{\partial \mathbf{A}}{\partial t}$).

Electric field is a vector with units of newtons per coulomb (N/C) or volts per metre (V/m), and dimensions of mass.length/charge.time² (ML/QT²).

It is derived from (non-unique) vector and scalar potentials, $\mathbf{A}$ and $\phi$ (and the magnetic field $\mathbf{B}$ is derived from the same vector potential).

It transforms (between observers with different velocities) as three of the six coordinates of a 2-form, the electromagnetic field, $(\mathbf{E},\mathbf{B})$, which in turn is part of the electroweak field.

Equations

(1) $$\vec{E}=\lim_{q\rightarrow 0}\frac{\vec{F_e}}{q} =\frac{1}{4\pi \epsilon_o}\int\frac{\rho (r)}{r^2}\hat{r}d\tau$$

(2) $$\oint \vec{E}\cdot d\vec{a} =\frac{Q_{enc}}{\epsilon_0}$$

Potential equations:

(3) $$\mathbf{E}\ =\ -\nabla \phi\ -\ \frac{1}{c}\frac{\partial \mathbf{A}}{\partial t}$$

$$\mathbf{B}\ =\ \nabla\times\mathbf{A}$$

The two source-free Maxwell equations (Faraday's Law and Gauss' Law for Magnetism) follow immediately by differentiating the potential equations:

(4) $$\nabla\times\mathbf{E}\ =\ -\frac{1}{c}\frac{\partial \mathbf{B}}{\partial t}$$

$$\nabla\cdot\mathbf{B}\ =\ 0$$

Energy density:

(5) $$u_e=\frac{1}{2}\epsilon E^2$$

Total energy:

(6) $$U_e=\int_{\tau}\frac{1}{2}\epsilon E^2 d\tau$$

Extended explanation

The electric field, along with the magnetic field, were originally conceived by Michael Faraday to explain the long range nature of those forces. The mathematical development of this field theory was left to Maxwell.

Since the electric field can accelerate charged bodies, it must be able to store electrical potential energy. The energy density of an electrical field is given by equation (5). In order to find the total energy stored in the field, the density function must be integrated over all space, thus giving rise to equation (6).

Time-varying electric fields are somewhat more difficult to find due to the fact that they can be created by time-varying magnetic fields as well as a time-varying potential. This phenomenon, known as Electromagnetic Induction, is represented in the derivative of the magnetic vector potential in equation (4).

After some manipulation of (4), you will obtain Faraday's Law, a much more well known representation of induction:

$$\vec{\nabla}\times \vec{E}=-\frac{\partial \vec{B}}{\partial t}$$

Reason for definition of electric field:

Electric field $\mathbf{E}$ is defined so that multiplying it by the charge $q$ of a body gives the force $\mathbf{F}$ on that body:

$$\mathbf{F} = q\mathbf{E}$$

This is the electric part of the Lorentz force: $\mathbf{F} = q\left(\mathbf{E} + \mathbf{v}\times\mathbf{B}\right)$

So it must have dimensions of force/charge, or work/charge.length, and so can be measured in newtons/coulomb.

Since work (or energy) can be measured in electron-volts, work/charge can be measured in volts, and so electric field can also be measured in volts/metre.

By comparison, magnetic field is defined so that multiplying it by the charge of a body and cross-producting it with the velocity of the body gives the force on that body:

$\mathbf{F} = q\mathbf{v}\times\mathbf{B}$

Similarly, therefore, magnetic field must have dimensions of force/charge.velocity, and can be measured in volts/metre per metre/second, or volt.seconds/metre², which are webers/metre², or teslas.

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