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Three concentric spherical shells have r...

Three concentric spherical shells have radii `a, b` and `c(a lt b lt c)` and have surface charge densities `sigma, -sigma` and `sigma` respectively. If `V_(A), V_(B)` and `V_(C)` denote the potentials of the three shells, then for `c = q + b`, we have

A

`V_(C)=V_(B)=V_(A)`

B

`V_(C)=V_(A) ne V_(B)`

C

`V_(C) =V_(B) ne V_(A)`

D

`V_(C) ne CV_(B) ne V_(A)`

Text Solution

Verified by Experts

The correct Answer is:
B
`q_(A)=4 pi a^(2) sigma` ,
`q_(B)=-4 pi b^(2) sigma ` ,
`q_(c)=4pic^(2) sigma , c=a+b`
`V_(A)=(1)/(4 pi in_(0))((q_(A))/(a)+(q_(B))/(b)+(q_(C))/(c))`
`=(2 sigma a )/(in_(0)) `
`v_(B)=(1)/(4pi in_(0))((q_(A))/(a)+(q_(B))/(b)+(q_(C))/(c))`
`=(sigma)/(in_(0))((a^(2))/(b)-b+c)`
`=(sigma)/(in_(0))(a+(a^(2))/(b))`
`V_(C)=(1)/(4pi in_(0))((q_(A))/(a)+(q_(B))/(b)+(q_(C))/(c))`
`=(sigma)/(in_(0))((a^(2)-b)/(c)+c)=(2sigmaa)/(in_(0))`
So , `V_(C)=V_(A) ne V_(B)`
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