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Find the electric field at (A) (a) due t...

Find the electric field at (A) (a) due to q, (b) due to charges induced on the inner surface of the shell and (c ) due to charges induced on the outer surface of the shell (Fig. 22-41).

Text Solution

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KEY IDEA
First we note that the charge q is not symmetrically located in the shell. Nevertheless, the electric field due to a point charge is easy to calculate.
`E_(q)=(q)/(4pi epsilon_(0)(5R//2)^(2))=(q)/(25pi epsilon_(0)R^(2))`.
For finding the electric field due to the inner shell, we note that since the charge q is not symmetrically placed, the charge distribution on inner surface will not be symmetric. At the points close to q, charge density will be higher and lower at points farther away from the shell. If the charge was distributed uniformly, we could possibly use the formula for the electric fiedl due to a uniformly charged shell at an external point. But this is not the case here. Not only that, we are not even aware of the pattern in which the induced charges will be distributed.
Calculations: This dilemma can be easily resolved by concept of electrostatic shielding. We know that the effect of the charge q will be cancelled by the charges induced on the inner surface of the cavity:
`vecE_(q)+vecE_("induced")=vec0`.
`E_("in")=(q)/(25pi epsilon_(0)R^(2))`.
To find the electric field due to the charge on the outer surface, we again do not know the pattern of induced charges on that surface. Without that, we cannot find the electric field at A. Let us use a neat trick. Suppose the charge q was at the center of the shell. What would the charge distribution on the outer surface look like? Yes, it would be uniform. From this position, when we move q, the charge on outer surface will not come to know about it because it has shielded from cavity. So the charge on the outer surface will still be uniform. Therefore, we can use the formula of the electric field due to a uniform shell:
`vecE_("outer")=(kq)/(R^(2))`.
So, the net electric field will be `vecE_("net")=(kq)/(R^(2))`.
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