Potential and Field for a sphere with a central core of differing density [closed]

Question

A spherically symmetric planet of radius $a$ consists of a central core of radius $b(<a)$ of uniform density $\rho_1$ surrounded by an outer region of uniform density $\rho_2$. Obtain an expression for the gravitational potential and field at all points in space.

To find the potential and field inside, I used Gauss' theorem and applied that to a sphere of radius $b$ and a shell of radius $a$ and thickness $a-b$. I obtained the expressions $$F = -\frac{4G \pi \rho_1}{3b^3}r^4 \ \text{and} \ \phi = \frac{2G\pi r^3\rho_1(r^2-3b^2)}{3b} - 4\pi a G \rho_2(a-b).$$

I did a similar thing for the region outside of both spheres, obtaining $$F = -\frac{G}{r} \left( \frac{4}{3} \pi r^3 \rho_1 \right)-\frac{4G\pi a^2\rho_2(a-b)}{r^2} \ \text{and} \ \phi = - \frac{4\pi b^3 \rho_1 G}{3} - \frac{4\pi Ga^2\rho_2(a-b)}{r}.$$

Assuming this is correct, I am struggling to determine the field and potential for the region where $a < r < b$.

Answer 1

Applying Gauss at a distance $r$ from the centre of the sphere you get
$g(r)\;4\pi r^2 = - 4 \pi G M_{\text{enclosed}} \Rightarrow g = - \dfrac{GM_{\text{enclosed}}}{r^2}$ where $g$ is the gravitational field.
So all you need to do is to find $M_{\text{enclosed}}$ for each of the regions.

What this equation tells you is that you can treat the enclosed mass $M_{\text{enclosed}}$ as a point mass at the centre of the sphere.
That being so the potential $\phi = - \dfrac{GM_{\text{enclosed}}}{r}$

For $r \le b$, $M_{\text{enclosed}}$ is the mass of a sphere of density $\rho_1 $ and radius $r$.

For $b \le r \le a$, $M_{\text{enclosed}}$ is the mass of a sphere of density $\rho_1 $ and radius $b$ and a thick shell of density $\rho_2$ with inner radius $b$ and outer radius $r$.

For $r \ge a$, $M_{\text{enclosed}}$ is the mass of a sphere of density $\rho_1 $ and radius $b$ and a thick shell of density $\rho_2$ with inner radius $b$ and outer radius $a$.

You can check your solutions by making sure that the values for $r\le b$ and $b \le r \le a$ that you have found are the same at $r=b$ and do a similar thing at $r=a$.