The density inside a solid sphere of radius r varies as `rho (r ) = rho_(0) ((r )/(R ))^(beta), " where " rho_(0) and beta` are constants and r is the distance from the centre. Let `E_(1) and E_(2)` be gravitational fields due to sphere at distance `(R )/(2)` and 2R from the centre of sphere. If `(E_(2))/(E_(1)) =4`, teh value of `beta` is

Options

A. 2
B. 2.5
C. 3
D. 4

Answer

C

Explanation

Nass endclosed in a sphere of radius r is

`M_(r) = int rho dV = underset(0)overset(r ) int rho_(0) ((r )/(R ))^(beta) . 4pi r^(2) dr`
`=[(4pi rho_(0))/(R^(beta)).(r^(beta + 3))/(beta + 3)]_(0)^(r )`
So, mass enclosed in a sphere of radius `(R )/(2)` is
`M_(1) = (4pi rho_(0))/(R^(beta)) (((R )/(2))^(beta + 3))/((beta + 3)) = (4pi rho_(0)R^(3))/((beta + 3)) xx (1)/(2^(beta + 3))`
and mass enclosed in radius R is
`M_(2) = (4pi rho_(0))/(R^(beta)) .(R^(beta + 3))/(beta + 3) = (4pi rho_(0) R^(3))/(beta + 3)`
So, gravitational field intensities are
`E_(1) = (GM_(1))/(((R )/(2))^(2)) = (4G)/(R^(2)) xx (4pi rho_(0)R^(3))/((beta + 3)) xx (1)/(2^(beta + 3))`
and `E_(2) = (GM_(2))/((2R)^(2)) = (G)/(4R^(2)) xx (4pi rho_(0)R^(3))/(beta + 3)`
As `(E_(2))/(E_(1)) =4`, we get
`((G)/(4R^(2)) xx (4pi rho_(0)R^(3))/(beta + 3))/((4G)/(R^(2)) xx (4pi rho_(0)R^(3))/(beta + 3) xx (1)/(2^(beta+ 3)))=4`
`rArr (2^(beta + 3))/(16) = 4 rArr 2^(beta + 3) = 64`
`2^(beta).2^(3) = 2^(6) rArr 2^(beta) = 8 rArr 2^(beta) = 2^(3) rArr beta = 3`