Question

# Suppose that a positive test charge q0 moves through a displacement d s→from one equipotential surface...

Suppose that a positive test charge q0 moves through a displacement d s→from one equipotential surface to the adjacent surface. The work the electric field does on the test charge during the move is –q0dV. The work done by the electric field may also be written as the scalar product (q0E→)·d s→, or q0E(cos θ) ds.

It's a general question what if the test charge is negative? what happens to the displacement and work?

If the test charge is negative, it will move exactly in the opposite direction that of the positive charge and hence the displacement will be negative. Instead of moving from higher to lower potential region(as the positive charge does), it will move towards higher potential region from lower. So the work done formula (-qdV) becomes like, q is negative and dV is also negative as change in potential is negative now( moving lower to higher). So the work done remains as it is.

For the electric field equation of the work done, it will remain same also. Because the sign of q is reversed and the direction of ds is reversed also. So the work done remains as it is.

So the displacement changes it sign but work done doesn't. There value remains same though.