Charge per Grain

The charging of the dust in the magnetic balloon will be due to the contribution of three effects: the collection of ions and electrons from the expanded plasma and photoionization due to UV radiation [Goertz, 1989]. The ion and electron contributions are given by their flux to the surface of the dust particles. Under the assumption of a Maxwellian plasma, these fluxes will be given by:

$$I_e = 4\pi a^2 \left( \frac{en_e}{4}\right) \left(\frac{8kT_e}{\pi m_e}\right)^{1/2} \exp{\left(\frac{eU}{kT_e}\right)}$$ (1)

$$I_i = 4 \pi a^2 \left(\frac{en_i}{4}\right) \left(\frac{8kT_i}{\pi m_i}\right)^{1/2} \left(1 - \frac{eU}{kT_i}\right)$$ (2)

where: $e$ is the electron charge, $n_i$ and $n_e$ are the ion and electron densities, $T_i$ and $T_e$ are the ion and electron temperatures, $m_i$ and $m_e$ are the ion and electron masses, and $U$ is the potential difference between the dust grain and the surrounding plasma. The photoelectric current is given by:

$$I_\nu =\pi a^3 Ke ^{-eU/kT_p}$$ (3)

where: $K$ is the photoelectron flux, $K = \eta(2.5\times10^{14})D^{-2}$, $\eta$ is the photoemission efficiency ($\eta \sim 1$ metals and $\eta \sim 0.1$ dielectrics), and $D$ is the distance from the sun in astronomical units (AU). $T_p$ is the temperature of the photoelectron and is assumed $T_p \sim 1$ eV.