JEE Main 2025 Jan 24 Shift 2, Physics Q7: Electric Potential

For 4 charges, there are $\left(\genfrac{}{}{0ex}{}{4}{2}\right)=6$ unique pairs. In each configuration count how many pairs are at distance "side" and how many at distance "diagonal".

Configuration (1) - charges at corners of a square of side $a$:

• 4 pairs along sides, separation $a$ - contribute $4\cdot \frac{K{q}_0^2}{a}$
• 2 pairs along diagonals, separation $a\sqrt{2}$ - contribute $2\cdot \frac{K{q}_0^2}{a\sqrt{2}}$

$U_1=\frac{K{q}_0^2}{a}\left(4+\frac{2}{\sqrt{2}}\right)=\frac{K{q}_0^2}{a}(4+\sqrt{2})$

Configuration (2) - charges at midpoints of the square sides. These four midpoints themselves form a smaller square whose side is $\frac{a}{\sqrt{2}}$ (midpoint-to-adjacent-midpoint along a 45° line) and whose diagonal is $a$ (e.g. G to H, opposite midpoints).

$U_2=\frac{K{q}_0^2}{(a/\sqrt{2})}\cdot 4+\frac{K{q}_0^2}{a}\cdot 2=\frac{K{q}_0^2}{a}(4\sqrt{2}+2)$

Difference:

$U_2-U_1=\frac{K{q}_0^2}{a}[(4\sqrt{2}+2)-(4+\sqrt{2})]=\frac{K{q}_0^2}{a}(3\sqrt{2}-2)$

Correct option: (4)