5.8.9: Solid Sphere

$\text{FIGURE V.24A}$

The potential outside a solid sphere is just the same as if all the mass were concentrated at a point in the centre. This is so, even if the density is not uniform, and long as it is spherically distributed. We are going to find the potential at a point $\text{P}$ inside a uniform sphere of radius $a$, mass $M$, density $ρ$, at a distance $r$ from the centre ($r < a$). We can do this in two parts. First, there is the potential from that part of the sphere “below” $\text{P}$. This is $-GM_r/r$, where $M_r = \frac{r^3M}{a^3}$ is the mass within radius $r$. Now we need to deal with the material “above” $\text{P}$. Consider a spherical shell of radii $x$, $x + δx$. Its mass is $δM = \frac{4 \pi x^2 δx}{\frac{4}{3} \pi a^3} \cdot M = \frac{3Mx^2 δ x}{a^3}$. The potential from this shell is $-GδM/x = -\frac{3GMxδx}{a^3}$. This is to be integrated from $x = 0$ to $a$, and we must then add the contribution from the material “below” $\text{P}$. The final result is

$$ψ = -\frac{GM}{2a^3} \left(3a^2 - r^2 \right) . \label{5.8.23} \tag{5.8.23}$$

Figure $\text{V.25}$ shows the potential both inside and outside a uniform solid sphere. The potential is in units of $−GM/r$, and distance is in units of $a$, the radius of the sphere.

$\text{FIGURE V.25}$