A conducting sphere S1 of radius r is attached to an insulating handle. Another conducting sphere S2 of radius R(r) is mounted on an insulating stand, S2 is initially uncharged. S1 is given a charge Q. Brought into contact with S2 and removed. S1 is recharged such that the charge on it is again Q and it is again brought into contact with S2 and removed. This procedure is repeated n times.
(a) Find the electrostatic energy of S2 after n such contacts with S1.
(b) What is the limiting value of this energy as n→∞ ?

Question

A conducting sphere S1​ of radius r is attached to an insulating handle. Another conducting sphere S2​ of radius R(>r) is mounted on an insulating stand, S2​ is initially uncharged. S1​ is given a charge Q. Brought into contact with S2​ and removed. S1​ is recharged such that the charge on it is again Q and it is again brought into contact with S2​ and removed. This procedure is repeated n times.
(a) Find the electrostatic energy of S2​ after n such contacts with S1​.
(b) What is the limiting value of this energy as n→∞ ?

Filo · Accepted Answer

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}^{'}

.

∴ \frac{q _{1}}{q _{1}^{'}} = \frac{q _{1}}{Q - q _{1}} = \frac{C _{2}}{C _{1}} = \frac{R}{r}

It implies that

Q

charge is to be distributed in

S_{2}

and

S_{1}

in the ratio of

R / r

.

∴ q_{1} = Q (\frac{R}{R + r})

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 (1 + \frac{R}{R + r})

This charge is again distributed in the same ratio. Therefore, charge on

S_{2}

in second contact,

q_{2} = Q (1 + \frac{R}{R + r}) (\frac{R}{R + r})

= Q [\frac{R}{R + r} + (\frac{R}{R + r})^{2}]

Similarly,

q_{3} = Q [\frac{R}{R + r} + (\frac{R}{R + r})^{2} + (\frac{R}{R + r})^{3}]

and

q_{n} = Q [\frac{R}{R + r} + (\frac{R}{R + r})^{2} + \dots + (\frac{R}{R + r})^{n}]

or

q_{n} = Q \frac{R}{r} [1 - (\frac{R}{R + r})^{n}] \dots

(ii)

[S_{n} = \frac{a ( 1 - r ^{n} )}{( 1 - r )}]

Therefore, electrostatic energy of

S_{2}

after

n

such contacts

U_{n} = \frac{a _{n}^{2}}{2 C} = \frac{q _{n}^{2}}{2 ( 4 π ε _{0} R )} or U_{n} = \frac{q _{n}^{2}}{8 π ε _{0} R}

where

U_{n} = \frac{a _{n}^{2}}{2 C} = \frac{q _{n}^{2}}{2 ( 4 π ε _{0} R )} or U_{n} = \frac{q _{n}^{2}}{8 π ε _{0} R}

can be written from Eq. (ii).

(b)

q_{n}

q_{n} = \frac{QR}{R + r} [1 + \frac{R}{R + r} + \dots + \dots + (\frac{R}{R + r})^{n - 1}]

as n \to \infty

q_{\infty} = \frac{QR}{R + r} (\frac{1}{1 - \frac{R}{R + r}})

= \frac{QR}{R + r} (\frac{R + r}{r}) = Q \frac{R}{r} [S_{\infty} = \frac{a}{1 - r}]

∴ U_{\infty} = \frac{q _{\infty}^{2}}{2 C} = \frac{Q ^{2} R ^{2} / r ^{2}}{8 π ε _{0} R}

Question Text	A conducting sphere $S_{1}$ of radius $r$ is attached to an insulating handle. Another conducting sphere $S_{2}$ of radius $R (> r)$ 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. (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 $n \to \infty$ ?
Topic	Electric Charges and Fields
Subject	Physics
Class	Class 12
Answer Type	Text solution:1
Upvotes	66

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