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Question

Two thin conducting shells of radii R and 3R are shown in the figure. The outer shell carries a charge +Q and the inner shell is neutral. The inner shell is earthed with the help of a switch S.


A
With the switch S open, the potential of the inner sphere is equal to that of the outer
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B
When the switch S is closed, the potential of the inner sphere becomes zero
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C
With the switch S closed, the charge attained by the inner sphere is 1/3 times that of charge present on the outer sphere.
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D
By closing the switch the capacitance of the system increases
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Solution

The correct option is D By closing the switch the capacitance of the system increases
When the switch S is open, no charge is present on inner shell.

Vinner shell=Vouter shell=kQ3R

So, Option (a) is correct

Let the charge on the inner shell after closing the switch be Q


Since the inner shell is earthed, the net potential must be zero for it.
Vinner=0

The net potential at the inner shell is the sum of potential due to the inner shell itself and due to the outer shell.

kQR+kQ3R=0Q=Q3

So, Option (b) and (c) are correct.

Now from the definition of capacitance,
Ci=4πε0(3R)

The charge on inner shell is zero initially when switch is open, hence only capacitance of outer sphere is taken into account.

& Cf=4πε0(3R)+4πϵ0(1R13R)

After the switch is closed the capacitance of both spheres add up, since they get connected in parallel.

Cf>Ci

Hence, options (a) , (b) , (c) and (d) are the correct answers.

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