Question

What is meant by the electric potential energy of the system of point charges? Compute the total electric potential energy of the system of charges given in the figure.

Hint:

The total electric potential energy of a system of point charges is calculated by summing the potential energies of all pairs of charges, where each pair's potential energy is given by \( \frac{k_e q_i q_j}{r_{ij}} \). For symmetric charge configurations, the distances can be easily calculated using geometry.

Answer 1

The electric potential energy of a system of point charges is the work required to assemble the charges from infinity to their given configuration. It is mathematically expressed as \[ U = \sum_{i<j} \frac{k_e q_i q_j}{r_{ij}}, \] where \(q_i\) and \(q_j\) are the charges and \(r_{ij}\) is the distance between them. In the given figure, four charges are placed at the corners of a square of side \(a\), with \(+q\) at the top left, \(-q\) at the top right, \(+q\) at the bottom right, and \(-q\) at the bottom left. For the interactions between adjacent charges separated by a distance \(a\), each pair consists of unlike charges, so the potential energy contribution from each side is \(-\frac{k_e q^2}{a}\). Since there are four sides, the total contribution from adjacent charges is \(-\frac{4 k_e q^2}{a}\). Next, consider the interactions along the diagonals, where the distance between charges is \(\sqrt{2}a\). Along one diagonal, we have two like charges, giving a positive contribution of \(\frac{k_e q^2}{\sqrt{2}a}\) each, while along the other diagonal, the charges are unlike, giving a negative contribution of \(-\frac{k_e q^2}{\sqrt{2}a}\) each. When summed, these diagonal contributions cancel out to zero. Therefore, the total electric potential energy of the system is obtained by adding the adjacent and diagonal contributions: \[ U_{\text{total}} = -\frac{4 k_e q^2}{a}. \] Hence, the total potential energy of the system of charges arranged at the corners of the square is negative, equal to \(-\frac{4 k_e q^2}{a}\).