At what distance 'd' from the surface of a solid sphere of radius 'R',

(a) potential is same as at a distance$\frac{R}{2}$ from the centre?

(b) field strength is same as at a distance $\frac{R}{4}$ from centre.

(a) Given, $V_{\text{outside }}=V_{\text{inside }}$

No such point will exist. Because potential at centre is $\frac{-15GM}{R}$. Potential at surface is $-\frac{GM}{R}$ and potential at infinity is zero. From centre to surface potential varies between $-\frac{1.5GM}{R}$ and $\frac{-GM}{R}$. From surface to infinity potential varies between $\frac{-GM}{R}$ and zero.

(b) Given, $E_{\text{inside }}=E_{\text{outside }}$

$\frac{GM}{R^3}\left(r_1\right)=\frac{GM}{r_2^2}$

Here, $r_1$and$r_2$ are the distances from centre.

$\frac{GM}{R^3}\left(\frac{R}{4}\right)=\frac{GM}{(R+d{)}^2}$

Solving this equation, we get

$d=R$