The magnitude of gravitational field intensities at distance `r_(1)` and `r_(2)` from the centre of a uniform solid sphere of radius R and mass M are `I_(1)` and `I_(2)` respectively. Find the ratio of `I_(1)//I_(2)` if `r_(1) gt R` and `r_(2) lt R`

Options

A. `(R^(2))/(r_(1)r_(2))`
B. `(R^(3))/(r_(1)r_(2)^(2))`
C. `(R^(3))/(r_(1)^(2)r_(2))`
D. `(R^(4))/(r_(1)^(2)r_(2)r^(2))`

Answer

C

Explanation

We consider the following two cases
(i) when `r_(1) gt R`
Gravitatinal field `l_(1)=(GM)/(r_(1)^(2))` but `M=4/3 pi R^(3) p`
`l_(1)=(g4/3pi r^(3)p)/(r_(1)^(2))`
(ii) when `r_(2) lt R` gravitational field `l_(2)=(GM_(1))/(r_(2)^(2))`
where `M_(1)` is the mass of the body within the sphere of radius `r_(2)`
But `M_(1)=4/3 pi pr_(3)^(2) rarr l_(2)=(G4/3pi pr_(2)^(3))/(r_(2)^(2))`
on dividing eq i by eq ii we get `(l_(1))/(l_(2))=(R^(3))/(r_(1)^(2)r_(2))`