Three conducting spherical shells have radii (a, b, and c) such that `a lt b lt c` (Fig. 3.113). Initially, the inner shell is uncharged, the middle shell has a positive charge Q, and the outer shell has a negative charge `-Q`.
.
a. Find the electric potential of the three shells.
b. If the inner and outer shells are now connected by a wire that is insulated as it passes through the middle shell, what is the electric potential of each of the three shells ? Also, what is the final charge on each shell ?

Answer

a. `V_A - V_B = Q/(4 pi epsilon_0)[(c - b)/(b c)], V_C = 0`
b. Charge on `A - (Q a)/b [(c - b)/(c -a)]`
Charge on `B, Q_b = Q`
Charge on `C, (Q c)/b[(c -b)/(c - a)]`.

Explanation

a. Potential of shell A
`V_A = V_("due to charge on A") + V_("due to charge on B") + V_("due to charge on C")`
=`0 + Q/(4 pi epsilon_0 b) - Q/(4 pi epsilon_0 c)`
Potential of shell B
`V_B = V_("due to charge on A") + V_("due to charge on B") + V_("due to charge on C")`
=`0 + Q/(4 pi epsilon_0 b) - Q/(4 pi epsilon_0 c)`
Potential of shell C
`V_C = V_("due to charge on A") + V_("due to charge on B") + V_("due to charge on C")`
=`0 + Q/(4 pi epsilon_0 c) - Q/(4 pi epsilon_0 c) = 0`
B After connecting inner and outer shells, we get
.
`V_A = V_C`
or `q/a - q/b + (Q + q)/b - (Q + q)/( c) = (q - q + Q + q -(Q + q))/( c)`
or `q = (Q a)/b ((b - c))/((c - a))`
Final potential of A and C is zero.
`V_B = k [q/a - q/b + (Q + q)/b - ((Q + q))/( c)] = k (Q +q)[(c - b)/(b c)]`
=`k [Q + (Q a(b - c))/(b (c - a))][(c - b)/(b c)] = k Q[((b - a)(c - b))/(b^2(c - a))]`.