Plasma Beta

Plasma beta ($\beta$) is one of the most important dimensionless parameters in plasma physics. It is the ratio of the thermal pressure to the magnetic pressure:

$\beta = \frac{p_{thermal}}{p_{magnetic}} = \frac{n k_B T}{B^2 / (2\mu_0)}$

Where:

• $n$ = number density (m⁻³)
• $k_B = 1.381 \times 10^{-23}$ J/K
• $T$ = temperature (K)
• $B$ = magnetic field strength (T)
• $\mu_0 = 4\pi \times 10^{-7}$ H/m (permeability of free space)

Pressure Components

Thermal pressure (kinetic pressure of the plasma): $p_{thermal} = n k_B T$

Magnetic pressure (energy density of the magnetic field): $p_{magnetic} = \frac{B^2}{2\mu_0}$

Physical Significance

$\beta$ tells us which pressure dominates the plasma dynamics:

• Low-$\beta$ plasmas: magnetic field controls particle motion; good confinement
• High-$\beta$ plasmas: particles can distort or escape the field

In tokamak fusion reactors, achieving higher $\beta$ means more plasma pressure (and thus more fusion power) for a given magnetic field — a key engineering goal.

Implement the three pressure functions below. Keep all constants ($k_B$, $\mu_0$) inside the function bodies.