Question

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?

• A. \( (x + y)(y + z)(z + x) = \text{constant} \)
• B. \(xyz = \text{constant}\)
• C. \( xy + yz + zx = \text{constant} \)
• D. \( (x^2 + y^2)(y^2 + z^2)(z^2 + x^2) = \text{constant} \)

Hint:

In electrostatics, equipotential surfaces are where the electric potential is constant, and they often involve simple relationships between the distances from the source charges.

Answer 1

The Correct Option is C

The potential due to a long charged wire is proportional to the logarithm of the distance from the wire. To find the equipotential surface, we sum the potentials from the three wires.

Step 1: For each wire, the potential depends on the perpendicular distance from the wire.

Step 2: The equipotential surface is where the total potential from the three wires is constant.

Step 3: After analyzing the expressions, we conclude that the correct relation is \( xy + yz + zx = \text{constant} \), which satisfies the condition for an equipotential surface.

Final Conclusion: The equation \( xy + yz + zx = \text{constant} \) represents the equipotential surface, which corresponds to Option (3).