= Axial potential of a uniform hemispherical shell
{title2=$\phi(h)=-\frac{2\pi G\rho}{3h}\left[b^3-a^3+\frac32h(b^2-a^2)-(b^2+h^2)^{3/2}+(a^2+h^2)^{3/2}\right]$}
For constant <mass density> $\rho$ on the lower half of the <spherical shell> $a\leq|\mathbf r'|\leq b$, with $0<a<b$, the displayed <Newtonian gravitational potential> applies at $(0,0,h)$ for every $h>0$, with zero at infinity. Use source <spherical coordinates> with polar angle $\alpha\in[\pi/2,\pi]$. For a fixed source radius $s$,
$$
\int_{\pi/2}^{\pi}\frac{\sin\alpha\,d\alpha}{\sqrt{s^2+h^2-2sh\cos\alpha}}
=\frac{s+h-\sqrt{s^2+h^2}}{sh}.
$$
Multiplication by $-2\pi G\rho s^2$ and <integration> from $a$ to $b$ leaves the <antiderivative> $s^3/3+hs^2/2-(s^2+h^2)^{3/2}/3$, proving the formula. The entire positive axis is outside the matter, even for $a<h<b$, so there are no interior/exterior radial cases. The <limit> at the origin is $-\pi G\rho(b^2-a^2)$; far away the leading term is $-GM/h$, with total <mass> $M=2\pi\rho(b^3-a^3)/3$.
Back to article page
