Consider two concentric conducting spheres. The outer sphere is hollow and initially has a charge `-7Q` on it. The inner sphere is solid and has a charge `+2Q` on it.
a. How much charge is on the outer surface and inner surface of the outer sphere.
b. If a wire is connected between the inner and outer sphere, after electrostatic equilibrium is established how much total charge is on the outer sphere? How much charge is on the outer surface and inner surface of the outer sphere? Does the electric field at the surface of the inside sphere change when the wire is connected?
c. We return to original condition in (a). We now connect the outer sphere to ground with a wire and then disconnect it. How much total charge will be on the outer sphere? How much charge will be on the inner surface and outer surface of the outer sphere?

Explanation

a. The charge on the inner sphere induces eqyal magnitude of charge, but opposite in sign, on the inner surface of the outer sphere. Sum of all the induced charges is always zero. Therefore, an equal amount of charge must appear on the surface. The outer and inner surface of the outer sphere have charges `-5Q and -2Q`, respectively.

b. When outer and inner spheres are connected by a wire, the entire charge is transferred to the outer surface from the inner sphere. In electrostatic equilibrium, charge does not reside inside a conductor. Total charge on the outer surface of the outer sphere is `-5Q`.
Total charge on the inner surface is 0.
The electric field at the surface of the inside sphere goes to zero after connection. Consider a Gaussian surface just on the surface of the inner sphere.
`phi_E = E(4pir^2) = Q_(enclosed)/epsilon_0 = 0, as Q_(enclosed)=0`
Thus, we have E = 0
c. When the outer sphere is grounded the charge on the outer surface is transferred to ground, thus charge is reduced to zero on the outer surface. The final charge distribution is shown in figure.
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