inside a sphere there is a cavity of an arbitrary shape . a chage q is kept inside the cavity . there will be a uniform distribution of charge on the outer surface of the sphere . how can i prove that ?
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If you are actually going to use Gauss's law to prove this you'd better state the problem a lot more precisely. Is the sphere conducting? Does the sphere have any net charge on it? Otherwise, it's not even true.
well it can not be proved using gauss law .... the cavity is of any arbitrary shape .... All i can say is that the potential of the metallic sphere will be constant . after that which gaussian surface do i take ? i definitely have a non uniform charge distribuition on that irregular shaped cavity .... so how do i procceed now ?
Take a gaussian surface inside the conductor. Tells you total charge enclosed is zero - so there's a surface charge on the cavity surface cancelling the enclosed charge. Now take a gaussian outside the conductor. The field must correspond to the charge in the cavity since the conductor is net neutral. So there's a surface charge on the outside equal to the charge in the cavity. You can argue that the charge distribution on the outside of the sphere is uniform since there is no field coming through the conductor to the outer surface to disturb it.