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## Homework Statement

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.

## Homework Equations

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.