$$\require{cancel}$$

5.4.10: Bubble Inside a Uniform Solid Sphere

$$\text{FIGURE V.11}$$

$$\text{P}$$ is a point inside the bubble. The field at $$\text{P}$$ is equal to the field due to the entire sphere minus the field due to the missing mass of the bubble. That is, it is

$\textbf{g} = -\frac{4}{3} \pi G ρ \textbf{r}_1 - (-\frac{4}{3} \pi G ρ \textbf{r}_2) = -\frac{4}{3} \pi G ρ ( \textbf{r}_1 - \textbf{r}_2) = -\frac{4}{3} \pi G ρ \textbf{c}. \label{5.4.26} \tag{5.4.26}$

That is, the field at $$\text{P}$$ is uniform (i.e. is independent of the position of $$\text{P}$$) and is parallel to the line joining the centres of the two spheres.