Q:

Force on charge inside cavity

Suppose we have a conductor with a cavity of arbitrary shape inside it.Now we introduce a point charge + q inside the cavity in an arbitrary position.Will the charge experience any force due to the charge induced by it on the cavity,or not?If not,then how can it be shown mathematically?also if yes,how to do that?
At first glance it appears that as the charge induced on the walls is entirely due to the point charge the overall field on it should be zero,but this being a very vague thought,it has to be treated mathematically.How to do this?

A:

Will the charge experience any force due to the charge induced by it
  on the cavity,or not?

(1) the force on the point charge (if any) must be from a non-zero electric field (within the cavity) due to the induced charge on the inner surface
(2) if there is no force on the point charge, there is zero electric field due to the induced charge
(3) if there is zero electric field due to the induced charge, the electric potential (within the cavity) due to the induced charge is constant
(4) if the electric potential due to the induced charge is constant, the potential at the inner surface, due to the induced charge, is constant
(5) the (total) potential at the inner surface, due to both the induced charge and the point charge, is constant (conductor is an equipotential)
(6) the potential at the inner surface due to the point charge is not, in general, constant
(7) thus, the potential at the inner surface due to the induced charge is not, in general, constant
(8) thus, the potential within the cavity due to the induced charge is not, in general, constant
(9) thus, the electric field within the cavity due to the induced charge is no, in general, zero
(10) thus, the force on the point charge in the cavity is not, in general, zero

