The density inside a solid sphere of radius a is given by `rho=rho_0/r`, where `rho_0` is the density ast the surface and r denotes the distance from the centre. Find the graittional field due to this sphere at a distance 2a from its centre.

Explanation

The field is required at a point outside the sphere. Dividing the sphere in concentric shells, each shell can be replaced by a point pareticle at ilts cenre 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 equla to the mass of the given sphere. If the mass of the sphere is M the gravitatioN/Al field at the given point is
`E=(GM)/((2a)^2)=(GM)/(4a^2)`.............i
The mass M may be calculated as follows. Consider a concentreic shell of radius r nd thickness dr. Its volume is
`dV=4pi^2)dr`
and its mass is
`dM=rhodV=(rho_0a/r)(4pir^2dr)`
The mass of the whole sphere is
`M=int_0^a4pirho_0ardr`
Thus by i the gravitatioN/Al field is
`E=(2piGrho_0a^3)/94a^I2)=1/2piGrho_0a`.