# Electric Field due to charges inside cavity

Imagine a conducting body as shown:

A positive charge +q is placed inside the cavity of the body. Now due to induction a negative induced charge -q has appeared on inside surface of cavity.

Is the electric field due to all the charges inside cavity(+q and induced -q) always 0 outside cavity(outside cavity and outside conducting body).

If it is, how can you prove it using properties of conductor and gauss law(I searched about this but all statements and proofs were using uniqueness theorem which i haven't studied till now).

Inside the cavity we have placed $$+q$$ charge. Due to the electric field of the $$+q$$ charge in the cavity (radiating outwards), the free electron gets drifted towards the inside surface of cavity (opposite to the radial outward direction of positive charge in the cavity). As a result, the inside surface of cavity gets negative charge an outer surface of sphere gets positive charge. So, inside the conductor an electric field is induced which is opposite to the direction as that of charge $$+q$$ inside the cavity. If there is some residual electric field inside the cavity (i.e.$$\mathbf{E}_{+q}>\mathbf{E}_{induced\;charge}$$), then more free electrons get drifted to the inside of the cavity surface and thus $$\mathbf{E}_{induced\;charge}$$ gets increased till the net electric field inside the conductor gets $$0$$ (as conductors have plenty of free electrons, any electric field inside the conductors causing them to drift in such a way that net eectric field inside them is 0).