(a) Calculate the escape energy for a particle of mass $m$ and charge $q$ situated at the surface of a star of mass $M$, charge $Q$, and radius $R$.
(b) Calculate the equilibrium potential $V$ of the star. Assume a sphere of fully ionized atomic hydrogen, with the electrons and protons at the same temperature and zero net current. The fraction of the electrons, or protons, that possess enough energy to escape is
$$
\exp \left(-\frac{\text { escape kinetic energy }}{k T}\right)
$$
where $k$ is Boltzmann's constant, $1.37 \times 10^{-2 x}$ joule per degree. At equilibrium, the electron and proton currents are equal. It is this phenomenon that causes the solar wind.
(c) Show that $V \approx 10^{3}$ volts for the sun.
This phenomenon does not appear to have any appreciable astrophysical significance. Even giant galaxies have center-to-surface potential differences that are only of the order of 1000 volts, like the sun. Show that, for any $r, I=2 \pi r \rho \mathcal{M E}$, where $\rho$ is the space charge density $\epsilon_{0} E / r .$
(b) The drift velocity of the dust particles is given by Stokes's law: it is the force $E Q$ divided by $6 \pi \eta a$, where $\eta$ is the viscosity of the gas.
Show that their drift velocity $v$ is $2 \epsilon_{0} E^{2} a / \eta$.
(c) Calculate $I, \rho, v$, and the time required for a dust particle to drift from the cathode to the anode when $\mathscr{M}=2 \times 10^{-4}$ meter $^{2} /$ (volt-second), $a=5$ micrometers, and $\eta=2 \times 10^{-5}$ kilogram/(meter-second).
This simplified theory neglects turbulence, which is important in practice.