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?

Three infinitely long wires with linear charge density $$\lambda$$ are placed along the x-axis, y-axis, and z-axis respectively. We need to find the equation of an equipotential surface.

The electric potential at a perpendicular distance $$r$$ from an infinitely long line charge with linear charge density $$\lambda$$ is:

$$V = -\frac{\lambda}{2\pi\epsilon_0} \ln r + C$$

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

- Distance from the x-axis: $$r_x = \sqrt{y^2 + z^2}$$

- Distance from the y-axis: $$r_y = \sqrt{x^2 + z^2}$$

- Distance from the z-axis: $$r_z = \sqrt{x^2 + y^2}$$

$$V_{total} = -\frac{\lambda}{2\pi\epsilon_0}[\ln\sqrt{y^2+z^2} + \ln\sqrt{x^2+z^2} + \ln\sqrt{x^2+y^2}] + C'$$

$$= -\frac{\lambda}{4\pi\epsilon_0}\ln[(y^2+z^2)(x^2+z^2)(x^2+y^2)] + C'$$

For an equipotential surface, $$V_{total} = \text{constant}$$, which requires:

$$(y^2 + z^2)(x^2 + z^2)(x^2 + y^2) = \text{constant}$$

This can be rewritten as:

$$(x^2 + y^2)(y^2 + z^2)(z^2 + x^2) = \text{constant}$$

The correct answer is Option 3: $$(x^2 + y^2)(y^2 + z^2)(z^2 + x^2) = \text{constant}$$.