Electrostatics 7 Question 30
32. A conducting sphere $S _1$ of radius $r$ is attached to an insulating handle. Another conducting sphere $S _2$ of radius $R$ is mounted on an insulating stand. $S _2$ is initially uncharged.
$S _1$ is given a charge $Q$, brought into contact with $S _2$ and removed. $S _1$ is recharged such that the charge on it is again $Q$ and it is again brought into contact with $S _2$ and removed. This procedure is repeated $n$ times.
(1998, 8M)
(a) Find the electrostatic energy of $S _2$ after $n$ such contacts with $S _1$.
(b) What is the limiting value of this energy as $n \rightarrow \infty$ ?
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Solution:
- Capacities of conducting spheres are in the ratio of their radii. Let $C _1$ and $C _2$ be the capacities of $S _1$ and $S _2$, then
$$ \frac{C _2}{C _1}=\frac{R}{r} $$
(a) Charges are distributed in the ratio of their capacities. Let in the first contact, charge acquired by $S _2$ is $q _1$. Therefore, charge on $S _1$ will be $Q-q _1$. Say it is $q _1^{\prime}$
$$ \begin{aligned} \therefore \quad \frac{q _1}{q _1{ }^{\prime}} & =\frac{q _1}{Q-q _1} \\ & =\frac{C _2}{C _1}=\frac{R}{r} \\ \therefore \quad q _1 & =Q \frac{R}{R+r} \end{aligned} $$
In the second contact, $S _1$ again acquires the same charge $Q$.
Therefore, total charge in $S _1$ and $S _2$ will be
$$ Q+q _1=Q \quad 1+\frac{R}{R+r} $$
This charge is again distributed in the same ratio. Therefore, charge on $S _2$ in second contact,
$$ \begin{aligned} q _2 & =Q 1+\frac{R}{R+r} \frac{R}{R+r} \\ & =Q \frac{R}{R+r}+\frac{R}{R+r}^{2} \end{aligned} $$
Similarly, $q _3=Q \frac{R}{R+r}+\frac{R}{R+r}^{2}+\frac{R}{R+r}^{3}$
and $\quad q _n=Q \frac{R}{R+r}+\frac{R}{R+r}^{2}+\ldots+\frac{R}{R+r}^{n}$ or
$$ q _n=Q \frac{R}{r} 1-\frac{R}{R+r}^{n} \quad S _n=\frac{a\left(1-r^{n}\right)}{(1-r)} $$
Therefore, electrostatic energy of $S _2$ after $n$ such contacts
$$ =\frac{q _n^{2}}{2\left(4 \pi \varepsilon _0 R\right)} \text { or } U _n=\frac{q _n^{2}}{8 \pi \varepsilon _0 R} $$
where, $q _n$ can be written from Eq. (ii).
(b) As $n \rightarrow \infty \quad q _{\infty}=Q \frac{R}{r}$
$$ \therefore U _{\infty}=\frac{q _{\infty}^{2}}{2 C}=\frac{Q^{2} R^{2} / r^{2}}{8 \pi \varepsilon _0 R} $$
or $\quad U _{\infty}=\frac{Q^{2} R}{8 \pi \varepsilon _0 r^{2}}$
