A dipole and a charge

A dipole is placed on the x-axis with its center on the origin. A positive point charge will be moved from very far away on the y-axis to the origin. In Case 1 it will be moved straight down the y-axis to the origin. In Case 2 it will follow a complicated path but its starting and ending points will be the same as in case 1.

Which case takes more work?

  1. Case 1
  2. Case 2
  3. The work done is the same in both cases

Like gravity, the electrostatic force is conservative. When the only forces acting are conservative it doesn't matter how an object gets from A to B, the work done is always the same.

How much work is required to bring the charge from very far away to the center of the dipole?

  1. Zero
  2. The work done is positive
  3. The work done is negative

The potential at the two end-points is the same, zero. The change in potential energy is:

DU = q DV = 0

The work done by the field is -DU. We would have to do an amount of work +DU to bring in the charge against the field, but since DU = 0 no work is done.