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

$\mathrm{\nabla }p=\rho \overrightarrow{g}$

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

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

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:

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