A spherical hole is made in a solid sphere of radius R. The mass of the sphere before hollowing was M. The gravitational field at the centre of the hole due to the remaining mass is

The correct option is GM/2R².

To find the gravitational field at the centre of the spherical hole in the hollowed solid sphere, we can use the principle of superposition. The gravitational field due to the remaining mass is equal to the gravitational field of the entire solid sphere (before hollowing) minus the gravitational field of the sphere that was removed to create the hole.

Let the solid sphere have radius $R$ and mass $M$. The volume of this sphere is:
$V_0=\frac{4}{3}\pi R^3$
The uniform density of the sphere is:
$\rho =\frac{M}{\frac{4}{3}\pi R^3}=\frac{3M}{4\pi R^3}$

A spherical hole is created. Since the hole is within the sphere, let's consider the standard geometry where a spherical cavity of radius $r=\frac{R}{2}$ is made such that its center is at a distance $d=\frac{R}{2}$ from the center of the original solid sphere, touching the outer surface. The center of this hole lies at a distance of $\frac{R}{2}$ from the center of the original sphere.
Let $C$ be the center of the original sphere and $C^{\prime }$ be the center of the hole. Thus, the distance $C{C}^{\prime }=\frac{R}{2}$, and the radius of the hole is $r=\frac{R}{2}$.

The mass of the sphere that was removed to create the hole, $M^{\prime }$, is:
$M^{\prime }=\rho \cdot V_{\text{hole}}=\rho \cdot \frac{4}{3}\pi r^3=\rho \cdot \frac{4}{3}\pi {\left(\frac{R}{2}\right)}^3=\frac{M}{8}$

Now, we calculate the gravitational field at the center of the hole, $C^{\prime }$:
1. The gravitational field due to the entire solid sphere of mass $M$ at a point inside it (at distance $r=\frac{R}{2}$ from the center $C$) is given by the formula:
$E_{\text{original}}=\frac{GMx}{R^3}$
where $x=\frac{R}{2}$ is the distance from the center. Substituting $x$:
$E_{\text{original}}=\frac{GM\left(\frac{R}{2}\right)}{R^3}=\frac{GM}{2R^2}$
This field is directed towards the center of the original sphere, $C$.

2. The gravitational field at the center of the hole due to the removed sphere of mass $M^{\prime }$ is:
$E_{\text{removed}}=0$
since the point is at the center of this removed sphere.

By the principle of superposition, the gravitational field due to the remaining mass at the center of the hole is:
$E_{\text{remaining}}=E_{\text{original}}-E_{\text{removed}}$
$E_{\text{remaining}}=\frac{GM}{2R^2}-0=\frac{GM}{2R^2}$

Therefore, the magnitude of the gravitational field at the centre of the hole due to the remaining mass is $\frac{GM}{2R^2}$.