Gravitation Inside Earth — Question 7

Question 7

An off-center spherical cavity contains a uniform nonzero field

A sphere of uniform density $\rho$ has a spherical cavity entirely inside it. The cavity center is displaced by vector $\vec d$ from the original sphere center. Use the uniform-sphere interior field $\vec g=-C\vec r$, where $C=4\pi G\rho/3$.

1. Use subtraction of the missing sphere to find the field at any point inside the cavity.

2. State its direction and explain why the empty cavity does not generally have zero field.

Question 7 - Solution

An off-center spherical cavity contains a uniform nonzero field

(a) Derivation and calculation

Let $C=4\pi G\rho/3$. A cavity point has original-center position $\vec r$ and cavity-center position $\vec r-\vec d$: $$\vec g_{\mathrm{full}}=-C\vec r,\qquad \vec g_{\mathrm{removed}}=-C(\vec r-\vec d).$$ Subtracting the missing material, $$\boxed{\vec g=\vec g_{\mathrm{full}}-\vec g_{\mathrm{removed}}=-C\vec d}.$$

(b) Result and interpretation

All dependence on the observation point cancels: $$\frac{\partial g_i}{\partial r_j}=0,\qquad |\vec g|=\frac{4\pi G\rho}{3}|\vec d|.$$ The uniform field points opposite $\vec d$, toward the side containing the original center. Only the concentric case gives $$\vec d=\vec0\quad\Longrightarrow\quad\vec g=\vec0.$$

Physical check

Superposition applies because every cavity point lies inside both the original uniform sphere and the sphere removed from it.