Consider an electric dipole as in the figure. There is a vertical equipotential line/surface. If I understand correctly, electric potential is the amount of work done per unit charge in bringing a charge from infinity to a distance r from a charge.
But consider moving along the equipotential line from infinity to a point along the equipotential line. Surely the work done to move the charge must be zero, since we are moving along an equipotential line. However, the electrostatic field is conservative, so work done in moving from infinity to a given point should be the same along any path. Therefore, work done in moving a charge from infinity to a point close to a dipole is zero.
This is clearly not the case, where have I gone wrong?? Thank you!