Controlled Thermonuclear Fusion — Question 6

Question 6

Plasma beta compares particle pressure with magnetic energy density

A quasineutral D-T plasma has ion and electron densities each $10^{20}\,\mathrm{m^{-3}}$, with kT=10.0 keV for both. Let B=5.00 T, $\mu_0=4\pi\times10^{-7}$ SI and $1\,\mathrm{keV}=1.602\times10^{-16}$ J. Define $\beta=p/[B^2/(2\mu_0)]$.

1. Calculate total particle pressure and beta.

2. At fixed density and temperature, find the change in beta if B increases by 20 percent.

Question 6 - Solution

Plasma beta compares particle pressure with magnetic energy density

(a) Method and calculation

$$\boxed{p=(n_i+n_e)kT=2(10^{20})(10)(1.602\times10^{-16})=3.204\times10^5\,\mathrm{Pa}}.$$$$\boxed{\beta=\frac{2\mu_0p}{B^2}=.0322101=3.22101\%}.$$

(b) Interpretation and further calculation

$$\boxed{\beta\prime/\beta=(B/B\prime)^2=1/(1.20)^2=.694444}.$$$$\boxed{\beta\prime=.0223681=2.23681\%}.$$

Physical check

Beta is a pressure ratio, not by itself a stability criterion. Geometry and pressure profiles matter in determining allowable equilibria.