An idealized theta-pinch geometry is an infinitely long, cylindrically symmetric (d-by-d theta = 0), translationally-invariant (d-by-d z = 0) plasma column with an externally applied axial magnetic field B_z_ext. This induced a large diamagnetic azimuthal current which produces its own magnetic field which opposes the external magnetic field. Assume the plasma column is in MHD equilibrium with velocity v = 0 and with mag field B_z(r) and gas pressure p(r).
I'm stuck on the first part of the question which is find the differential relationwhich the field B_z and the gas pressure must satisfy.
gradp = j cross B
maxwell's equations to replace the current with the curl of B
The Attempt at a Solution
grad p = (del cross B_z z-hat) cross B_z z-hat
del cros B_x z-hat = - d-bydr B_z theta-hat
grad p = d-by-dr p
(1/mu_0)(- d-by-dr B_z)theta-hat cross B_z z-hat = (1/mu_0)(- d-by-dr B_z) = d-by-dr p
Is that the correct answer? Do I need to go further?
Next I need to integrate to find an expression for the externally applied mag field as a function of B_z and p and I just don't see how i can integrate the expression i found above.