Three infinitely long wires with linear charge density $\lambda$ are placed along the $x$-axis, $y$-axis and $z$-axis respectively. Which of the following denotes an equipotential surface?
Answer: (C) $(x^2+y^2)(y^2+z^2)(z^2+x^2) = \text{constant}$
The potential of an infinite line charge at perpendicular distance $r$ is $V = -\dfrac{\lambda}{2\pi\varepsilon_0}\ln r + C$.
Distances of a point $(x, y, z)$ from the three axes:
$$r_x = \sqrt{y^2+z^2}, \quad r_y = \sqrt{z^2+x^2}, \quad r_z = \sqrt{x^2+y^2}$$
$$V = -\frac{\lambda}{2\pi\varepsilon_0}\ln(r_xr_yr_z) + C' = -\frac{\lambda}{4\pi\varepsilon_0}\ln\left[(y^2+z^2)(z^2+x^2)(x^2+y^2)\right] + C'$$
$V$ is constant exactly when $(x^2+y^2)(y^2+z^2)(z^2+x^2)$ is constant.
Solution by Sreeraj P, M.Sc Physics