Question

A uniform solid sphere of mass M and radius a is surrounded symmetrically by a uniform thin spherical shell of equal mass and radius 2 a. the ratio of gravitational field at a distance $\frac{3}{2}$ a froth the centre to $\frac{5}{2}$ a from the centre is..

• A. $\frac{25}{18}$
• B. $\frac{16}{19}$
• C. $\frac{25}{9}$
• D. 1

Answer 1

The correct option is A

$\frac{25}{18}$

Figure shows the situation. The point $P_1$ is at a distance $\frac{3}{2}$ a from the centre and $P_2$ is at a distance$\frac{5}{2}$ from the centre. As $P_1$ is inside the cavity of the thin spherical shell, the field here due to the shell is zero. The field due to the solid sphere is

E= $\frac{GM}{{\left(\frac{3}{2}a\right)}^2}$ = $\frac{4GM}{9a^2}$ .................(1)

This is also the resultant field. The direction is towards the centre. The point $P_2$ is outside the

sphere as well as the shell. Both may be replaced by single particles of the same mass at

the centre. The field due to each of them is

E= $\frac{GM}{{\left(\frac{5}{2}a\right)}^2}$ = $\frac{4GM}{25a^2}$

The resultant field is E=2E'=$\frac{8GM}{25a^2}$..........(2) towards the centre.