Module 2 : Electrostatics
Lecture 8 : Properties Of Conductors
Take a gaussian surface entirely within the conducting shell, completely enclosing the cavity. Everywhere on the gaussian surface $\vec E = 0$. The flux and therefore, the charge enclosed is zero within the gaussian surface. As the cavity contains a charge $Q$, the inner surface of the cavity must have charge $-Q$. As the conductor has distributed charge $2Q$, the charge on the outside surface is $3Q$.
  The electric field outside the shell is same as that due to a point charge located at the centre of the sphere. The reason is that the charge in the cavity is compensated exactly by the induced charges in the surface of the cavity. Thus the effect of the charge $Q$is not felt at points outside the sphere. The uniforml distributed charge of $3Q$ over the surface of the sphere gives the same field as that of a point charge located at the centre.

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