Question

The density inside a solid sphere of radius a is given by $\rho$ =${\rho }_0$a/r where ${\rho }_0$ is the density at the surface and r denotes the distance from the centre. The gravitational field due to this sphere at a distance 2a form its centre is..

• A. $\frac{3}{2}\pi G{\rho }_{0}a$
• B. $\frac{1}{4}\pi G{\rho }_{0}a$
• C. $\pi G{\rho }_{0}a$
• D. $\frac{1}{2}\pi G{\rho }_{0}a$

Answer 1

The correct option is D

$\frac{1}{2}\pi G{\rho }_{0}a$

The field is required at a point outside the sphere. Dividing the sphere in concentric shells, each shell can be replaced by a point particle at its centre having mass equal to the mass of the shell. Thus, the whole sphere can be replaced by a point particle at its centre having mass equal to the mass of the given sphere. If the mass of the sphere is M, the gravitational field at the given point is

$E=\frac{GM}{(2a{)}^2}=\frac{GM}{4a^2}--------\left(i\right)$

The mass M may be calculated as follows. consider a concentric shell of radius r and thickness dr.

its volume is dV=$\left(4\pi r^2\right)$dr and its mass is dM=$\rho$dV=$\left({\rho }_0\frac{a}{r}\right)\left(4\pi r^{2}dr\right)$

$=4\pi {\rho }_{0}ardr.$

The mass of the whose sphere is

$M=\underset{0}{\overset{a}{\int }}4\pi {\rho }_{0}ardr$

$=2\pi {\rho }_0{a}^3$.

Thus, by (i) the gravitational field is

$E=\frac{2\pi G{\rho }_0{a}^3}{4a^2}=\frac{1}{2}\pi G{\rho }_{0}a.$

mass of the sphere is M, the gravitational field at the given point is