A metal sphere has radius R and mass M. A spherical hollow of diameter R is made in this sphere such that it
surface passes through the centre of the metal sphere and touches the outside surface of the metal sphere. A
unit mass is placed at a distance a from the centre of metal sphere. The gravitation field at that point is

Options

A. `("GM")/(R^(2))(1-(1)/(8(1-(2R)/(a))^(2)))`
B. `("GM")/(a^(2))(1-(1)/(8(1-(2R)/(a))^(2)))`
C. `("GM")/((R+a)^(2))(1-(1)/(8(1-(R)/(2a))^(2)))`
D. `("GM")/((a-R)^(2))(1-(1)/(8(1-(2a)/(R))^(2)))`

Answer

B

Explanation

Net field , I = `("GM")/(a^(2))-("GM")/((a-(R)/(2))^(2))`
Where M. is the mass of the sphere taken out from the main sphere.
But M= `(4)/(3) pi(R/(2))^(3)rho`and `M=(4)/(3) piR^(3)rho` `therefore M=(M)/(8)`
Then I = `("GM")/(a^(2))-("GM")/(8(a-(R)/(2))^(2))=("GM")/(a^(2))[1-(1)/(8(1-(R)/(2a))^(2))]`