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Bennett Relation

\[\frac{\mu_0 I^2}{8\pi} = N k_B(T_e + T_i)\]

Variables

Symbol Meaning Units
\(I\) total axial current A
\(N\) ions per unit length (line density) m⁻¹
\(T_e,\,T_i\) electron and ion temperatures K
\(\mu_0\) permeability of free space H/m

Assumptions

  • Static, cylindrically symmetric Z-pinch equilibrium
  • Radial force balance \(dp/dr = -J_zB_\theta\) with \(B_\theta = \mu_0I(r)/2\pi r\)
  • Quasi-neutral, single ion species, isotropic pressure \(p = n k_B(T_e+T_i)\)
  • No axial field \(B_z\)

Derivation sketch

Multiply the force balance by \(r^2\) and integrate from 0 to \(a\). Integrating the pressure term by parts converts \(\int r^2(dp/dr)\,dr\) into \(-2\int rp\,dr\), i.e. the total pressure over the cross-section; the magnetic term likewise reduces to the total current. All profile information cancels, leaving a relation between integrated quantities only.

Notes

The profile-independence is the striking feature: whatever the density and temperature shapes, a given current confines a given line density at a given temperature. The relation is exact for the stated assumptions — and says nothing about stability, which is what actually killed the Z-pinch. See kink instability.