______ is an expression of capacitance of a spherical capacitor.

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None of these

answer is A.

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Detailed Solution

Concept:

Consider a spherical capacitor having +Q charge on the inner surface
–Q on the outer surface.
Let R and r be the radii of the outer surface and inner surface respectively.
These two charged surfaces are separated by a radial distance of (R-r).
Let the potential at the inner spherical surface be V1
and potential at the outer spherical surface be V2
Let C be the required capacitance of the assumed spherical capacitor.
As we know that the electric field due to a charged sphere having charge on the surface Q and radius r is given by

where

denotes the permittivity in vacuum (

Farad/metre)
Using above equation, electric field due to inner spherical surface having charge +Q and radius r is given by

-------------1
We knew that relation between electric field and potential difference

----------------2
By comparing equations (1) and (2) ,
we can see that the LHS of both these equations are same
hence, the RHS of both these equations will be equal

Integrating on both sides,

can be taken outside the integration as it is constant. Question Image

Where ΔV=V2−V1 represents the increase in potential produced when moving from the inner spherical surface to the outer spherical surface
capacitance of any capacitor=

By substituting equation (3) in the above equation
Question Image

The above equation gives the expression for the capacitance of the spherical capacitor with inner surface radius as r and outer surface radius as R. Hence, option 1 is correct

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