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 ?

To determine the equipotential surfaces for three infinitely long wires with linear charge density $\lambda$ placed along the x-axis, y-axis, and z-axis, we need to analyze the electric potential distribution.

The electric field due to an infinitely long charged wire with linear charge density $\lambda$ is given by:

• $$E = \frac{\lambda}{2\pi\epsilon_0 r}$$

where $r$ is the perpendicular distance from the wire.

The electric potential $V$ at a point distance $r$ from a wire is obtained by integrating the electric field:

• $$V = - \int E \cdot dr = \frac{\lambda}{2\pi\epsilon_0} \ln r + C$$

Since potential depends on $\ln r$, it is constant for a given radial distance from the wire. Hence, equipotential surfaces are cylindrical around each wire.

Equipotential Surfaces for Three Wires

Wire along x-axis ($\hat{i}$):

• The potential is constant on cylinders centered along the x-axis, i.e., surfaces of the form: $$y^2 + z^2 = \text{constant}$$

Wire along y-axis ($\hat{j}$):

• The potential is constant on cylinders centered along the y-axis, i.e., surfaces of the form: $$x^2 + z^2 = \text{constant}$$

Wire along z-axis ($\hat{k}$):

• The potential is constant on cylinders centered along the z-axis, i.e., surfaces of the form: $$x^2 + y^2 = \text{constant}$$