Theta-pinch, plasma physics

[indecicia]

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.

 

[Jhero]

Thank you for your question. It seems like you have made a good start on the problem by using Maxwell's equations to replace the current with the curl of the magnetic field. However, there are a few things that need to be clarified in order to fully answer the question.

Firstly, the differential relation that you are trying to find is the equilibrium condition for the plasma column. In other words, it is the condition that ensures that the plasma column is in a state of balance, with all forces and currents cancelling each other out. This is important because it allows us to make certain assumptions and simplifications in our calculations.

Secondly, in order to find the equilibrium condition, we need to consider all the forces and currents acting on the plasma column. This includes the external magnetic field, the induced diamagnetic current, and the gas pressure. In your attempt at a solution, you have only considered the force due to the induced diamagnetic current, but you have not taken into account the other forces and currents.

To find the equilibrium condition, we need to equate all the forces and currents acting on the plasma column. This will give us a differential equation in terms of the magnetic field and the gas pressure. We can then solve this equation to find an expression for the externally applied magnetic field as a function of the magnetic field and the gas pressure.

I hope this helps to guide you in the right direction. If you have any further questions or need clarification, please let me know.