Gravitation Inside Earth — Question 3

Question 3

A thick spherical shell has three distinct field regions

A spherical body has uniform density $\rho$ only for $a<r<b$, with an empty concentric cavity at $r<a$. Use the shell theorem.

1. Find the inward field magnitude in all three radial regions.

2. Check continuity at $a$ and $b$, and determine whether the field increases through the material.

Question 3 - Solution

A thick spherical shell has three distinct field regions

(a) Derivation and calculation

Only enclosed mass contributes. With $C=4\pi G\rho/3$, $$M(r)=\frac{4\pi\rho}{3} \begin{cases}0,&r<a,\\r^3-a^3,&a<r<b,\\b^3-a^3,&r>b.\end{cases}$$ Thus the inward magnitude is $$\boxed{g(r)=C\begin{cases}0,&r<a,\\r-a^3/r^2,&a<r<b,\\(b^3-a^3)/r^2,&r>b.\end{cases}}$$

(b) Result and interpretation

At the boundaries, $$g(a^-)=g(a^+)=0,\qquad g(b^-)=g(b^+)=C\frac{b^3-a^3}{b^2}.$$ Within the material, $$g'(r)=C(1+2a^3/r^3)>0,$$ while outside $g'=-2g/r<0$. Hence the global maximum is at $r=b$.

Physical check

Finite volume density changes field slope at a boundary without creating a jump in the field.