Electric Field and Changing Electric Potential

Any charge will create a vector field around itself (known as an electric field). Electric field is the gradient of potential, which depends inversely upon distance of a given point of interest from a charge. Placing a second charge in the system (a "test charge") results in the two charges experiencing a force (the field's units are Newtons, a measure of force per Coulomb), causing the charges to move relative to one another. It is easiest to model interactions between two charges such that one is considered stationary while the test charge moves.

As the test charge moves, the potential between it and another charge changes, as does the electric field. The relationship between potential and field (E) is a differential: electric field is the gradient of potential (V) in the x direction. This can be represented as:

$E_x=-\frac {dV}{dx}$.

Thus, as the test charge is moved in the x direction, the rate of the its change in potential is the value of the electric field.

The instant before the test charge moves, its potential energy is at a maximum, and its kinetic energy is 0. For any charge of constant value (Q), the potential at a certain distance from it (r) can be calculated from the following equation:

$V_E=\frac {1}{4 \pi \epsilon_0} \frac {Q}{r}$,

where ε₀ is the electric constant, otherwise known as permittivity of free space. Moving towards and away from the charge results in change of potential; the relationship between distance and potential is inverse.

For one point charge, potential will be constant for all points a certain radial distance away. Multiple points of the same potential are known as equipotential. In the case of fields created by a single point charge, all points on any circle centered around the point charge will be equipotential, as illustrated in .

Equipotential Lines

An isolated point charge Q with its electric field lines (blue) and equipotential lines (green)

shows that when multiple charges create a field, the equipotential lines become irregularly shaped. This is due to the fact that the fields created by each charge overlap, thus potential is increased at any point relative to that which would have arisen from one or the other charge.