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 \( xy + yz + zx = constant} \), which satisfies the condition for an equipotential surface.