] Electrostatic potential energy stored in a configuration of discrete point charges

The mutual electrostatic potential energy of two charges is equal to the potential energy of one charge in the electrostatic potential generated by the other. That is to say, if charge q1 generates an electrostatic potential \Phi_1(\mathbf r), which is a function of position \mathbf r, then U_E = q_2 \Phi_1(\mathbf r_2). Also, a similar development gives U_E = q_1 \Phi_2(\mathbf r_1).

This can be generalized to give an expression for a group of N charges, qi at positions \mathbf r_i:

U_E = \frac{1}{2}\sum_i^N q_i \Phi(\mathbf r_i)

where, for each i value, \Phi(\mathbf r_i) is the electrostatic potential due to all point charges except the one at \mathbf r_i

Note: The factor of one half accounts for the 'double counting' of charge pairs. For example, consider the case of just two charges.

Alternatively, the factor of one half may be dropped if the sum is only performed once per charge pair. This is done in the examples below to cut down on the math.
One point charge

The electrostatic potential energy of a system containing only one point charge is zero, as no work is required to move the charged particle from infinity to its location.
Two point charges

Consider bringing a second point charge into position. The electrostatic potential Φ(r1) due to charge 1 is

\Phi(r_1) = \; k_{\mathrm{e}} q_1/r_1
where ke is Coulomb's constant. In the International System of Quantities, which has been the preferred international system since the 1970s and forms the basis for the definition of SI units, the Coulomb constant is given by

k_{\mathrm{e}} = \; 1/4 \pi \epsilon_0 ,

where ε0 is the electric constant. Hence we obtain:

U_\mathrm{E} = \; \frac{q_1 q_2}{4 \pi \epsilon_0 r}

where

q1, q2 are the two charges
r is the distance between the two charges

The electrostatic potential energy will be negative if the charges have opposite sign and positive if the charges have the same sign. Negative mutual potential energy corresponds to attraction between two charges; positive mutual potential energy to repulsion between two charges
Three or more point charges

For 3 or more point charges, the electrostatic potential energy of the system may be calculated by bringing individual charges into position one after another, and taking the sum of the energies required to bring each additional charge into position. Thus

U_\mathrm{E} = \frac{1}{4 \pi \epsilon_0} \left({\frac{q_1 q_2}{r_{12}}} + {\frac{q_1 q_3}{r_{13}}} + {\frac{q_2 q_3}{r_{23}}} + ...\right)

where

q1, q2, ..., are the point charges
rmn is the distance between two point charges, m and n (e.g. r12