The electrostatic potential energy of a system of point charges is defined as: \( U = \sum_{i > j} \frac{k q_i q_j}{r_{ij}} \). Here, \( k \) represents the Coulomb constant, \( q_i \) and \( q_j \) denote the charges, and \( r_{ij} \) is the distance separating the charges.
Step 1: Identify charge pairs.
The system comprises three charges, each with magnitude \( +q \), positioned at the vertices of an equilateral triangle with side length \( a \). The potential energy arises from the mutual interactions between these charges. The identified charge pairs are:
The interaction between charges at vertices 1 and 2.
The interaction between charges at vertices 2 and 3.
The interaction between charges at vertices 3 and 1.
Step 2: Calculate the potential energy for a single pair.
The potential energy for any given pair of charges is calculated as: \( U_{\text{pair}} = \frac{k q^2}{a}. \)
Step 3: Sum the potential energy across all pairs.
Given that an equilateral triangle has three pairs of charges:
\( U_{\text{total}} = 3 \cdot U_{\text{pair}} = 3 \cdot \frac{k q^2}{a}. \)
Therefore, the total potential energy is: \( U_{\text{total}} = \frac{3kq^2}{a}. \)