Plasma Beta

Plasma beta is a ratio of two pressures: the pressure of the gas in a plasma over the pressure of the magnetic field threading it. The numerator $n\,k_B\,T$ is the gas pressure, the number of protons per volume times Boltzmann's constant times their temperature. The denominator $B^{2}/2\mu_0$ is the pressure of the field. The two forms in the formula are the same ratio, with the $2\mu_0$ moved up. Because a pressure is divided by a pressure, $\beta$ carries no unit.

Beta tells you which of the two is in charge. Below one, the field is in charge: it guides the gas and the gas cannot bend it. Above one, the gas is in charge and drags the field along with it. The solar wind sits near one, a fusion device is built to run at a few per cent, and the solar corona and the Earth's plasmasphere are well below one. The gas pressure counted here is the protons' alone; the electrons would add about as much again, so the number is a proton beta.

The page's example takes 5 protons per cm³ at 100 000 K in a field of 5 nT and returns $\beta$ = 0.6946, a little under one, so the gas is close to balancing the field. The result was checked against the practical form in the NRL Plasma Formulary (Huba, 2019), $\beta = 4.03\times10^{-11}\,n\,T/B^{2}$, with $n$ in cm⁻³, $T$ in eV and $B$ in gauss. Converted, 100 000 K is 8.6173 eV and 5 nT is $5\times10^{-5}$ G, and the two routes agree to within 0.08 %, the formulary's coefficient being rounded to three figures.

Setting $\beta$ = 1 asks what balance looks like. For 5 nT, $B^{2}/2\mu_0$ is $9.9472\times10^{-12}$ Pa, and at 5 per cm³ the temperature that matches it is 144 094 K, or 143 821 °C. Since $n$ and $T$ sit in the numerator and $B$ is squared in the denominator, hotter or denser gas needs a stronger field to hold the same beta, and the page can solve for any one of $n$, $T$ or $B$ from the other three.