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Find the potential at point A due to (a)...

Find the potential at point A due to (a) charge Q, ( b) charge on the outer surface of the conducting shell, and (c) total potential at A (Fig. 24-45).

Text Solution

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It is easy to find the potential due to a point charge. But the charge on the surface of the conducting shell is nonuniform and its potential can be tricky. However, we can use the fact that the entire conductor is an equipotential surface.
Calculation: The first part, potential due to Q is easy. It can be seen that
`V_(Q ) = (q)/( 6 pi epsilon_(0) R)`
There are induced charges on outer surface of the shell As discussed before, they are nonuniformly distributed. So, we cannot calculate the potential due to a nonuniform shell at A. But let us try to calculate the potential due to this shell at C, the center of the shell. We cannot use the formula for the uniformly charged shell, but the center is equidistant from all the induced charges at the surface. Since potential is a scalar, Vat the center due to the induced charges will be
`V_( C "in" ) = (sum q_("in"))/( 4 pi epsilon_(0) R)`.
Since the shell is initially uncharged, the total charge on it will be zero. o, the potential at C due lo induced charges will be zero. Therefore, the total potential at C will be
`V_("cq") + V_("c in") = (Q) /( 8 pi epsilon_(0) R)+0.`
As we discussed before, the potential at all the points within the cavity should be the same. So, `V_(A) = V_( C)`
From this, we can say that
`V_(A) = (Q) / (Q pi epsilon_(0) R)`
`V_A = V_("AQ") + V_(" A in") = (Q)/( 8 pi epsilon_(0) R)`
`V_(" A in ") = (-Q)/( 24 pi epsilon_(0) R).`
Learn: We can see why the potential at A is negative. The induced charges near Q will be negative, and equal and opposite charges will be induced on the other side of the conductor. They will be farther off, producing a net negative potential at A.
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Knowledge Check

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