Prove or disprove (with a counterexample) the following
Theorem: Suppose a conductor carrying a net charge Q, when placed in an external electric field E _{e}, experiences a force F; if the external field is now reversed \left( E _{e} \rightarrow- E _{e}\right), the force also reverses (F → –F)
What if we stipulate that the external field is uniform?
The theorem is false. For example, suppose the conductor is a neutral sphere and the external field is due to a nearby positive point charge q. A negative charge will be induced on the near side of the sphere (and a positive charge on the far side), so the force will be attractive (toward q). If we now reverse the sign of q, the induced charges will also reverse, but the force will still be attractive.
If the external field is uniform, then the net force on the induced charges is zero, and the total force on the conductor is Q E _{c}, which does switch signs if E _{e} is reversed. So the “theorem” is valid in this very special case .