Gravitational Acceleration Under a Well

Let $g(r)$ be the magnitude of the gravitational acceleration at a distance $r$ from Earth's center, let $F(r)=4\pi r^2g(r)$ be the flux of the gravitational field through a sphere of radius $r$ centered at the origin, and let $V(r)=\frac{4}{3}\pi r^3$ be the volume of a sphere of radius $r$ .

For a body of constant density, Gauss' Law tells us that $\frac{F(r)}{V(r)}=\frac{3g(r)}{r}$ is constant for $r\leq R$ (meaning that $g(r)$ is proportional to $r$ inside Earth). Applying this to the relevant radii, $r=R$ and $r=R-d$ , we find that $\frac{g(R)}{R}=\frac{g(R-d)}{R-d}$ and $\frac{g(R-d)}{g(R)}=\frac{R-d}{R}=\boxed{1-\frac{d}{R}}$ .

Gauss' Law also implies that the gravitational acceleration inside a spherical shell of constant area density is zero.