Three infinitely long wires with linear charge density \(\lambda\) are placed along the \(x-a x i s, y\)-axis and \(z-\) axis respectively. Which of the following denotes an equipotential surface?

Potential due to an infinite wire is \(V=2 k \lambda \ln r\), where \(r\) is the distance from the wire.
Taking the point in space \(P(x, y, z)\)
Distance from wire along \(x\)-axis is \(r_x=\sqrt{y^2+z^2}\)
Distance from wire along \(y\)-axis is \(r_y=\sqrt{x^2+z^2}\)
Distance from wire along \(z\)-axis is \(r_z=\sqrt{x^2+y^2}\)
\(\Rightarrow\) Potential at \(P\) due to wire along \(x\)-axis is
\(V_x=2 k \lambda \ln r_x\)
Potential at \(P\) due to wire along \(y\)-axis is
\(V_y=2 k \lambda \ln r_y\)
Potential at \(P\) due to wire along \(z\)-axis is
\(V_z=2 k \lambda \ln r_z\)
\(\Rightarrow\) Not potential at \(P=V=V_x+V_y+V_z\)
or \(\quad V=2 k \lambda \ln r_x+2 k \lambda \ln r_y+2 k \lambda \ln r_z\)
i.e. \(V=2 k \lambda \ln \left(r_x r_y r_z\right)\)
or
\(\begin{aligned}
& V=2 k \lambda \ln \left(\sqrt{y^2+z^2} \sqrt{z^2+x^2} \sqrt{x^2+y^2}\right) \\
& =k \lambda \ln \left(y^2+z^2\right)\left(z^2+x^2\right)\left(x^2+y^2\right)
\end{aligned}\)
\(\Rightarrow\) For equipotential surface
\(\left(x^2+y^2\right)\left(y^2+z^2\right)\left(z^2+x^2\right)=\text { constant }\)