Question 4.1.2: A Z-pinch device contains a cylindrical plasma discharge in ......
A Z-pinch device contains a cylindrical plasma discharge in which the axial electric current, j_{z}(r), is concentrated at the radial center and falls to zero at the external boundary, r = R. The plasma thermal pressure, p(r), also peaks at r = 0 and gradually decreases to zero at r = R. Show, that for magnetostatic equilibrium, when the radial plasma thermal pressure gradient is balanced by the magnetic force, the net discharge current, given by I = \int j_{z}dS, and the plasma diamagnetic response, P = \int pdS, are related to each other as P = I^{2}/2c^{2} (the so-called Bennett relation).
The radial component of the magnetostatic equilibrium condition reads
− \frac{dp}{dr} + \frac{1}{c} (\vec{j} × \vec{B})_{r} = −\frac{dp}{dr} −\frac{1}{c} j_{z}B_{θ} = 0 (4.5)
Since \vec{j} = \frac{c}{4π} (\vec{∇} × \vec{B}), the axial current density
j_{z} = \frac{c}{4πr} \frac{d(rB_{θ})}{dr} , (4.6)
and, thus, equation (4.5) can be re-written as
\frac{dp}{dr} = − \frac{B_{θ}}{4πr} \frac{d(rB_{θ})}{dr} = − \frac{1}{8πr^{2}} \frac{d(r^{2}B^{2}_{θ})}{dr}
Itegration of this equation yields
\frac{R^{2}B^{2}_{θ}(R)}{8} = −π \int_{0}^{R} r^{2} \frac{dp}{dr} dr = −(πr^{2}P)\mid ^{R}_{0} + \int_{0}^{R} 2πrp(r)dr = P (4.7)
On the other hand, it follows from equation (4.6) that
RB_{θ}(R) = \frac{2}{c} \int_{0}^{R} j_{z}2πrdr = \frac{2I}{c} ,
which, together with equation (4.7), results in the Bennett relation.