In Chen problem 10.1 it's stated, "A z-pinch of radius a has a uniform current (J_z) zbar and a plasma pressure p(r) which is balanced by the J × B force. Derive the parabolic form of p(r)."

I'd like to clarify if my attempt is correct since there isn't a solution in Chen.

In a cylindrical column of plasma, starting with B = (B_θ) θbar and the equilibrium condition grad(p) = J × B we get

dp/dr = (B_θ)(J_z)

Using B_θ = (μ_ο)I/2πr from Ampere's law and J_z = I/πa² we get

dp/dr = (μ_ο)I²/2π²a²r

Integrating with a dummy variable ∫ dr' from r to a gives p(r) that scales as

p(r) ~ ln(a/r)

Is that right? I think it makes sense that for r < a there's p > 0 and for r = a there's p = 0, because the plasma is fully contained inside the cylinder, there's no plasma at the boundary of the cylinder to exert pressure. But it isn't in a parabolic form that Chen suggests.