Question

In the first configuration (1) as shown in the figure, four identical charges \( q_0 \) are kept at the corners A, B, C and D of square of side length \( a \). In the second configuration (2), the same charges are shifted to mid points C, E, H, and F of the square. If \( K = \frac{1}{4\pi \epsilon_0} \), the difference between the potential energies of configuration (2) and (1) is given by:

• A. \( \frac{Kq_0^2}{a} (4\sqrt{2} - 2) \)
• B. \( \frac{Kq_0^2}{a} (4 - \sqrt{2}) \)
• C. \( \frac{Kq_0^2}{a} (3\sqrt{2} - 2) \)
• D. \( \frac{Kq_0^2}{a} (3 - \sqrt{2}) \)

Hint:

To solve problems involving potential energy, use the formula for the potential energy between two charges and sum over all pairs of charges. Pay attention to the distances between the charges in different configurations.

Answer 1

The Correct Option is A

The potential energy for a configuration of point charges is given by:

U = ∑_i<j \(\frac{K q_i q_j}{r_{ij}}\),

The distance between charges \( i \) and \( j \) is \( r_{ij} \).

Configuration (1): The charges are placed at the corners of the square. The distance between each pair of adjacent charges is \( a \), and the distance between diagonally opposite charges is \( \sqrt{2}a \).

Configuration (2): The charges are placed at the midpoints of the sides, so the distance between adjacent charges is \( \frac{a}{\sqrt{2}} \), and the distance between diagonally opposite charges is \( a \).

The potential energy of configuration (1) is:

\( U_1 = 4 \times \frac{K q_0^2}{a} + 2 \times \frac{K q_0^2}{\sqrt{2}a} \).

The potential energy of configuration (2) is:

\( U_2 = 4 \times \frac{K q_0^2}{\frac{a}{\sqrt{2}}} + 2 \times \frac{K q_0^2}{a} \).

The difference between the potential energies of configuration (2) and (1) gives the desired result:

\( \Delta U = U_2 - U_1 = \frac{K q_0^2}{a} (4\sqrt{2} - 2) \).

Final Answer: \( \frac{K q_0^2}{a} (4\sqrt{2} - 2) \).