The gravitational potential of two homogeneous spherical shells `A` and `B` of same surface density at their respective centres are in the ratio `3:4`. If the two shells collapse into a single one such that surface charge density remains the same, then the ratio of potential at an internal point of the new shell to shell `A` is equal to

Options

A. `3 : 2`
B. `4 : 3`
C. `5 : 3`
D. `5 : 6`

Answer

C

Explanation

`M_(A) = sigma 4 pi R_(A)^(2), M_(B) = sigma 4 pi R_(B)^(2)`,
where `sigma` is surface density
`V_(A) = (-GM_(A))/(R_(A)), V_(B) = (- GM_(B))/(R_(B))`
`(V_(A))/(V_(B)) = (M_(A))/(M_(B)) (R_(B))/(R_(A)) = (sigma 4 pi R_(A)^(2))/(sigma 4 pi R_(B)^(2)) (R_(B))/(R_(A)) = (R_(A))/(R_(B))`
Given `(V_(A))/(V_(B)) = (R_(A))/(R_(B)) = (3)/(4) "then " R_(B) = (4)/(3) R_(A)`
for new shell of mass M and radius R
`M = M_(A) + M_(B) = sigma 4 pi R_(A)^(2) + 4 sigma R_(B)^(2)`
`sigma 4 pi R^(2) = sigma 4 pi (R_(A)^(2) + R_(B)^(2))`
then `(V)/(V_(A)) = (M)/(R) (R_(A))/(R_(B)) = (sigma 4 pi (R_(A)^(2) + R_(B)^(2)))/((R_(A)^(2) + R_(B)^(2))^(1//2)) = (R_(A))/(sigma 4 pi R_(A)^(2))`
= `(sqrt(R_(A)^(2) + R_(B)^(2)))/(R_(A)) = (5)/(3)`