JEE Main 2025 Jan 28 Shift 1, Physics Q4: Electric Potential

The potential at distance $r$ from an infinite line charge with density $\lambda$ is $V=-2k\lambda \ln r+C$, where $k=\frac{1}{4\pi {ϵ}_0}$.

For a point $(x,y,z)$:

Perpendicular distance from the x-axis $=\sqrt{y^2+z^2}$.
Perpendicular distance from the y-axis $=\sqrt{z^2+x^2}$.
Perpendicular distance from the z-axis $=\sqrt{x^2+y^2}$.

Total potential (superposition):

$V=-2k\lambda \left[\ln \sqrt{y^2+z^2}+\ln \sqrt{z^2+x^2}+\ln \sqrt{x^2+y^2}\right]+C^{\prime }$

$=-k\lambda \ln \left[(x^2+y^2)(y^2+z^2)(z^2+x^2)\right]+C^{\prime }$

$V$ is constant iff the product $(x^2+y^2)(y^2+z^2)(z^2+x^2)$ is constant.

$\therefore$ Option (3).