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 the 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 \(a<r<b\).

\[  g = \frac{4\pi}{3}G\rho\left(r-\frac{a^3}{r^2}\right)  \]