Question

Three charges \( +q \) are placed at the corners of an equilateral triangle of side \( a \). What would be the total electrostatic potential energy (in terms of \( k \))?

• A. \( kq^2 / a \)
• B. \( 2kq^2 / a \)
• C. \( 3kq^2 / a \)
• D. \( 4kq^2 / a \)

Hint:

For symmetric charge distributions, calculate the potential energy for one pair and multiply by the number of pairs. In an equilateral triangle, there are 3 charge pairs.

Answer 1

The Correct Option is C

The electrostatic potential energy of a system of point charges is given by:

\( U = \sum_{i > j} \frac{k q_i q_j}{r_{ij}} \)

where: \( k \) is the Coulomb constant, \( q_i \) and \( q_j \) are the charges, \( r_{ij} \) is the distance between the charges.

Step 1: Identify charge pairs.

In the given system, there are three charges \( +q \) at the vertices of an equilateral triangle of side \( a \). The potential energy is due to the pairwise interactions between the charges. The charge pairs are:

Between charges at vertices 1 and 2,

Between charges at vertices 2 and 3,

Between charges at vertices 3 and 1.

Step 2: Calculate the potential energy for one pair.

The potential energy for one pair of charges is:

\( U_{\text{pair}} = \frac{k q^2}{a}. \)

Step 3: Sum the potential energy for all pairs.

Since there are three pairs of charges in an equilateral triangle:

\( U_{\text{total}} = 3 \cdot U_{\text{pair}} = 3 \cdot \frac{k q^2}{a}. \)

Thus:

\( U_{\text{total}} = \frac{3kq^2}{a}. \)