A point charge +Q is placed at the centre of an uncharged spherical conducting shell of inner radius a and outer radius b.
i. Find the electric field for `rlta`.
ii. What is the magnitude and sigh of the induced charge q' on the inner shell surface?
iii. What is the electric field at points `rgtb`?
iv. What id the surface charge on the outer surface of the conductor?

Explanation

i. Consider a Gaussian surface `S_1` of radius `rltR` inside
the cavity , centered on charge Q. From Gauss's law,
`phi_E = oint E* dA = E(4pir^2) = (+Q)/epsilon_0`
From this we find the electric field to be
`E = 1/(4piepsilon_0) Q/r^2`
ii. Consider a Gaussian surface `S_2` inside the conducting material. We do not know if there is a charge on the inside surface of the conductor or not. We assume that the charge is q', if q' is zero, the result of Gauss's law will show it. Because the Gaussian surface is inside the conductor, the electric field is zero. From Gauss's law,
`oint vecE * dvec(A) = E (4pir^2) = Q_(enclosed)/epsilon_0= 0`

or `Q_(enclosed)/epsilon_0 = (Q+q')/epsilon_0 = 0 or q' = -Q`
There is a charge on the inside surface of the conductor. The total charge induced on the inside surface of the cavity is the charge placed at its center.
iii. For `E, (rgtb)`, consider a Gaussian surface `S_3`. From Gauss's law,
`phi_E = ointE*dA = Q_(enclosed)/epsilon_0`
or `E(4pir^2) = (+Q)/epsilon_0`
or `E = Q/(4piepsilon_0r^2)`
It was stated in the problem that the conducting sphere has no net charge. Consequently, the total charge inside our Gaussian surface `S_3` is the sum of charge `+Q` and induced charges `-Q` on the inner surface of the conductor and `+Q` on the surface. Once more we can see that the field outside the sphere is the same as for a point charge. The conducting sphere has no shielding effect at all.
However, such a conducting shield prevents
electrostatic fields from charges outside the shell from
entering it.
iv. The conducting shell has no net charge, yet there is a surface charge -Q on its surface. Because the net charge on the shell is zero and no charge can reside inside a conductor, there must be +Q on the outer surface of the conductor