Course:PHYS350/Tutorial 4 2007W

9. Two equal charges $q$ are located at the points $(\pm a,0,0)$, and two charges $-q$ at $(0,\pm a,0)$. Find the leading term in the potential at large distances.

24. Assume that the pressure $p$ in a star of radius $a$ and with spherical symmetry is related to the density $\rho$ by the equation of state $p={\frac {1}{2}}k\rho ^{2}$; this equation of state actually quite well approximates the pressure of nucleons in a neutron star. Use the fluid equilibrium equation

$\nabla p=\rho {\vec {g}}$

to find a relation between $\rho$ and $\Phi$, the gravitational potential. Hence show that Poisson's equation yields

${\frac {d^{2}\left[r\rho (r)\right]}{dr^{2}}}=-{\frac {4\pi G}{k}}r\rho (r).$

Solve this equation with the boundary conditions that $\rho$ is finite at $r=0$ and vanishes at the surface of the star. Show that the radius of the star is determined solely by $k$ and is independent of its mass $M$. Show also that $M=(4/\pi )a^{3}\rho (0)$. Evaluate the mass and radius of the star if $\rho (0)=2\times 10^{15}$g/cm³ (about ten times nuclear density) and $P(0)=0.1\rho (0)c^{2}$ (the maximum pressure for a given density is $P=\rho c^{2}$ - so this is pretty close).

You will find the expression for the Laplacian in spherical coordinates helpful:

$\nabla ^{2}\Phi ={\frac {1}{r}}{\frac {\partial ^{2}}{\partial r^{2}}}\left(r\Phi \right)+{\frac {1}{r^{2}\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial \Phi }{\partial \theta }}\right)+{\frac {1}{r^{2}\sin ^{2}\theta }}{\frac {\partial ^{2}\Phi }{\partial \phi ^{2}}}$