Gravitational field of a hollow sphere

Question:
A hollow sphere has region $a<r<b$ filled with mass of uniform density $\rho$. Find the magnitude of the gravitational field between $a$ and $b$.

Answer:
Utilize Gauss's  law for gravitational fields.
\[
\int_{S} g d\alpha = -4\pi GM
\]
As we know, if there is no mass in a sphere, no gravitational field is detected. Thus, when $r<a$, $g=0$. We can also find the field when $r>b$.
\[
\int_{S} g d\alpha = -4\pi GM  \\
\rightarrow 4 \pi r^2 g = -4 \pi GM  \\
\rightarrow g = \frac{GM}{r^2}
\]
The integral of left hand side gives surface area of a sphere. For $a<r<b$, the mass, $M$, depends on the volume.
\[
M_{a-b} = \int \rho dV = \rho \int^{r}_{a}r^2 \int^{\pi}_{0}\sin \theta d\theta \int^{2\pi}_{0}d\phi \\
=\rho\frac{4\pi}{3}(r^3-a^3)
\]
From Gauss's law,
\[
-4\pi r^2 g = -4\pi G \rho\frac{4\pi}{3}(r^3-a^3)
\]
Therefore, we have the gravitational field in $a<r<b$.
\[
g = \frac{4\pi}{3}G\rho\left(r-\frac{a^3}{r^2}\right)
\]

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