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(B) Consider a solid sphere of uniform density $\rho$ and radius $R$. The gravitational field at any point inside a solid sphere is given by $\vec{g}

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(B) Consider a solid sphere of uniform density $\rho$ and radius $R$. The gravitational field at any point inside a solid sphere is given by $\vec{g} = -\frac{4}{3} \pi G \rho \vec{r}$,where $\vec{r}$ is the position vector from the center of the sphere.When a spherical cavity is created,the gravitational field inside the cavity is the vector sum of the field due to the original solid sphere and the field due to a smaller sphere of negative mass (representing the cavity) placed at the same location.Let $\vec{r_1}$ be the position vector from the center of the large sphere and $\vec{r_2}$ be the position vector from the center of the cavity.The total field $\vec{g}_{net} = \vec{g}_{large} + \vec{g}_{cavity} = -\frac{4}{3} \pi G \rho \vec{r_1} - (-\frac{4}{3} \pi G \rho \vec{r_2})$.Since $\vec{r_1} - \vec{r_2} = \vec{d}$,where $\vec{d}$ is the constant vector connecting the center of the sphere to the center of the cavity,we get $\vec{g}_{net} = -\frac{4}{3} \pi G \rho \vec{d}$.Since $\rho$,$G$,and $\vec{d}$ are constants,the gravitational field inside the cavity is uniform.

If a cavity is made inside a solid sphere,then the gravitational field inside the cavity is

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