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 $a<r<b$.
$$g = \frac{4\pi}{3}G\rho\left(r-\frac{a^3}{r^2}\right)$$

Powered by Hirophysics.com