Electric potential energy

Electric potential energy is a potential energy (measured in joules) that results from conservative Coulomb forces and is associated with the configuration of a particular set of point charges within a defined system.

In the CGS system the erg is the unit of energy, being equal to 10−7 Joules.

The following outline of proof states the derivation from the definition of electric potential energy and Coulomb's law to this formula.

The electrostatic force F acting on a charge q can be written in terms of the electric field E as

When the curl ∇ × E is zero, the line integral above does not depend on the specific path C chosen but only on its endpoints.

When talking about electrostatic potential energy, time-invariant electric fields are always assumed so, in this case, the electric field is conservative and Coulomb's law can be used.

Using Coulomb's law, it is known that the electrostatic force F and the electric field E created by a discrete point charge Q are radially directed from Q.

The electrostatic potential energy UE stored in a system of N charges q1, q2, …, qN at positions r1, r2, …, rN respectively, is:

That is to say, if charge q1 generates an electrostatic potential V1, which is a function of position r, then

The electrostatic potential energy is mutually shared by

This can be generalized to say that the electrostatic potential energy UE stored in a system of n charges q1, q2, …, qn at positions r1, r2, …, rn respectively, is:

The electrostatic potential energy of a system containing only one point charge is zero, as there are no other sources of electrostatic force against which an external agent must do work in moving the point charge from infinity to its final location.

A common question arises concerning the interaction of a point charge with its own electrostatic potential.

The electrostatic potential energy of a system of three charges should not be confused with the electrostatic potential energy of Q1 due to two charges Q2 and Q3, because the latter doesn't include the electrostatic potential energy of the system of the two charges Q2 and Q3.

The electrostatic potential energy stored in the system of three charges is:

Using the formula given in (1), the electrostatic potential energy of the system of the three charges will then be:

Finally, we get that the electrostatic potential energy stored in the system of three charges:

Since Gauss's law for electrostatic field in differential form states

For example, a resistor converts electrical energy to heat.

The total electrostatic potential energy stored in a capacitor is given by

where C is the capacitance, V is the electric potential difference, and Q the charge stored in the capacitor.

, such that the amount of work done to assemble each increment to its final location may be expressed as

The total work done to fully charge the capacitor in this way is then

This work is stored as electrostatic potential energy, hence,

, which holds for many-charge systems such as large capacitors having metallic electrodes.

For few-charge systems the discrete nature of charge is important.

The total energy stored in a few-charge capacitor is

The total electrostatic potential energy may also be expressed in terms of the electric field in the form

These latter two expressions are valid only for cases when the smallest increment of charge is zero (

Note that a virtual experiment based on the energy transfert between capacitor plates reveals that an additional term should be taken into account when dealing with semiconductors for instance [3].