MHD Equilibrium

In magnetohydrodynamics (MHD), a plasma is in equilibrium when the pressure gradient force is balanced by the magnetic force:

$\nabla p = \mathbf{J} \times \mathbf{B}$

This fundamental relation constrains the geometry of magnetically confined plasmas.

Z-Pinch: Bennett Relation

In a z-pinch (current flows along the z-axis), the magnetic field wraps around the plasma column. The Bennett relation gives the current required to confine a plasma with line density (N) (particles per meter) at temperature (T):

$I^2 = \frac{8\pi}{\mu_0} N k_B T$

where (\mu_0 = 4\pi \times 10^{-7}) H/m. This shows that higher current can confine denser or hotter plasmas.

Theta-Pinch: Pressure Balance

In a theta-pinch (current flows azimuthally), the axial magnetic field provides confinement. Outside the plasma, the magnetic pressure must balance the plasma kinetic pressure:

$\frac{B^2}{2\mu_0} = n k_B T$

Solving for the required field: (B = \sqrt{2\mu_0 n k_B T}).

Tokamak Safety Factor

In a tokamak, field lines wind helically around the torus. The safety factor (q) measures how many toroidal turns a field line makes per poloidal turn:

$q = \frac{r B_T}{R B_P}$

where (r) is the minor radius, (R) is the major radius, (B_T) is the toroidal field, and (B_P) is the poloidal field. Stability requires (q > 1) (Kruskal-Shafranov condition).