From a solid sphere of mass M and radius R, a spherical portion of radiu `(R )/(2)` is removed as shown in the figure. Taking gravitational potential V = 0 at `r = oo`, the potential at the centre of the cavity thus formed is (G = gravitational constant)

Options

A. `(-GM)/(R )`
B. `(-2GM)/(3R)`
C. `(-2GM)/(R )`
D. `(-GM)/(2R)`

Answer

A

Explanation

Mass of radius R is M
Mass of radius `(R )/(2)` is `(M)/(8)`
Potential at O. due to mass M (Due to entire solid sphere)
`V_(M) = (-GM)/(2R^(3))[3R^(2) - (R^(2))/(4)]`

Potential at centre O. for mass M/8
`V_(M//8) = (G_(M//8))/(2((R )/(2))^(3))[3((R )/(2))^(2) - 0]`
`= (-3GM)/(8R)`
Potential due to remaining mass of sphere at point O..
`V = V_(M) - V_(M//8)`
`= (-GM)/(2R^(3))[3R^(2) - (R^(2))/(4)] + (3GM)/(8R)`
`= (-3GM)/(2R) + (GM)/(8R) + (3GM)/(8R)`
`= (-11 GM)/(2R) + (3GM)/(8R) = (-GM)/(R )`