Three concentric spherical metallic spheres `A,B` and `C` of radii `a , b` and `c(a lt b lt c)` have surface charge densities `sigma , -sigma` and `sigma` respectively.

Options

A. `V_(A) = (a+b+c)(sigma)/(epsilon_(0))`
B. `V_(B) = ((a^(2))/(b) - b+c)(sigma)/(epsilon_(0))`
C. `V_(C ) = ((a^(2)+b^(2))/(c) +c)(sigma)/(epsilon_(0))`
D. `V_(A) = V_(B) = V_(C ) = (a+b+c)(sigma)/(epsilon_(0))`

Answer

B

Explanation

Potential due to charged shell at an outside point `= (1)/(4pi epsilon_(0))(q)/(r )`
where r is the distance of point from the centre of the shell while that inside the shell `= (1)/(4pi epsilon_(0))(q)/(R )`
where R is the radius of the shell
Charges on the three shells are given by
`q_(A) = 4pi a^(2) sigma, q_(B) = -4pi b^(2) sigma, q_(C ) = 4pi c^(2) sigma`
`:. V_(A) = (1)/(4pi epsilon_(0))[(q_(A))/(a) + (q_(B))/(b) +(q_(C ))/(c )]`
`= (1)/(4pi epsilon_(0)) [(4pi a^(2) sigma)/(a) - (4pi b^(2)a)/(b) + (4pi c^(2) sigma)/(c )] = (sigma)/(epsilon_(0))(a-b+c)`
`V_(B) = (1)/(4pi epsilon_(0))[(q_(A))/(b) + (q_(B))/(b) + (q_(C ))/(c )]`
`= (1)/(4pi epsilon_(0))[(4pia^(2) sigma)/(b) - (4pi a^(2) sigma)/(b) + (4pi c^(2) sigma)/(c )]`
`= (4pi sigma)/(4pi epsilon_(0))[(a^(2))/(b) - b+c] = (sigma)/(epsilon_(0))((a^(2))/(b) - b + c)`
`V_(C ) = (1)/(4pi epsilon_(0))[(q_(A))/(c ) + (q_(B))/(c ) + (q_(C ))/(c )]`
`= (1)/(4pi epsilon_(0))[(4pi a^(2) sigma)/(c ) - (4pi b^(2) sigma)/(c ) + (4pi c^(2) sigma)/(c )] = (sigma)/(epsilon_(0))((a^(2) - b^(2))/(c ) + c)`