Consider a solid sphere of density `rho` and radius `4R`. Centre of the sphere is at origin. Two spherical cavities centred at `(2R,0)` and `(-2R,0)` are created in sphere. Radii of both cavities is `R`. In left cavity material of density `2rho` is filled while second cavity is kept empty. What is gravitational field at origin.

Options

A. `(GrhopiR)/(3)`
B. `(2GppiR)/(3)`
C. `(4GrhopiR)/(3)`
D. `(3GrhopiR)/(2)`

Explanation

Above distribution can be represented as shown in figure. Gravitational field due to sphere of radius R at a distance 3R
`E_(g)=(Grho(4)/(3)piR^(3))/(4R^(2))=(GrhopiR)/(3)`
so net field at center will be `2F_(g)=(2GrhopiR)/(3)`