Expert: Philip A. Stahl - 2/17/2008

Question
Help! Can you give advice on solving this?

Show that: m(dv_z/ dt) = -mu (@B/@z)

where @ are partials, and mu is a constant of the motion
Let there be a slowly-convergiing magnetic field: B(r, z, phi)= B_rr^ + B_zz^

where the field is axi-symmetric and assume: B_r^2 << B_z^2

and B_r = r/2[ dB_z/ dz]

Answer
Hello,

You certainly have a lot of plasma physics problems these days, don't you?

The key point to note in the solution is that the field is axisymmetric so that

B_phi = 0 and @/ @phi = 0

(Note – the field designation is also more accurately B(r, phi, z))

To proceed with your working you will need the force-motion eqn.

F = q(v X B)

And be aware that you have to resolve v, B into components as per the vector cross product function.

This means, you need to find: F_r,   F_phi, and F_z

To remind you, the basics of the cross-product in matrix form are given here:

http://www.physics.uoguelph.ca/tutorials/torque/Q.torque.cross.html


Now, you should be able to obtain for your case:

F_r = q(v_phiB_z – v_zB_phi)

F_phi = q(-v_rB_z + v_zB_r)

F_z = q( v_rB_phi – v_phi B_r)

Since we know B_phi = 0, the above reduces to:


F_r = q(v_phiB_z)

F_phi = q(-v_rB_z + v_zB_r)

F_z = -q( v_phi B_r)

Now, for the motion with which you’re concerned – the terms for the usual Larmor gyration are not needed, so the terms in B_z are omitted. Same with the “v_zB_r” term since it vanishes on the z-axis.

Thus, you are left with:

F_z = -q v_phi B_r

From one of your assumptions you have:  B_r = r/2[dB_z/ dz]

So:

F_z = - q v_phi (r/2)[dB_z/ dz] =  -(q v_phi r)/ 2  [dB_z/ dz]

From here you should be able to complete the solution fairly quickly.

Tips:

Take an average over one gyration period. Let the guiding center lie on the z-axis, so then:

v_phi = const.

(during a gyration)

If then, r = r_c (guiding center radius) the average force can be found

Further hint: Go back to the previous problem you asked about, concerning gyration periods, times, gyro-frequency etc!

With the correct substitutions you should have your result!