Increase in plasma energy from the increase in an ultraweak field

[Veles]

Homework Statement

Consider a time-dependent magnetic field B(t). Let the variation be slow so that the magnetic moment is constant. Let B be made to decrease by a factor of two - say, due to plasma motions. If the field is in a uniform Maxwellian plasma and they are both in a cylinder of radius r, then the energy of the plasma must decrease by a factor of two. This is true even if the energy of the field is infinitesimal compared to the energy of the plasma, provided B(t) decreases slowly compared to the cyclotron period . Where does the extra energy go?

Relevant Equations

None

As B increases, a circular E-field is setup by Faraday's Law, which accelerates the ions/electrons into a gyrating motion. The gyrating electrons/ions have an effective magnetic moment that opposes the applied B-field (Lenz's Law). When the B-field is decreased, a circular induced E-field decelerates the ions/electrons, reducing the magnetic moment and again opposing the change in the B-field.

The work done to increase the B-field (part of which went into acceleration the electrons/ions, and part into the B-field density) must be returned to the mechanism that generated it. This is analogous to the current returning to the circuit in a RLC circuit after the inductor discharges.

 

[NateD]

The returning current (in the form of a circular E-field) induces a voltage in the circuit. Thus, as B increases, a circular E-field is setup. As B decreases, a circular induced E-field is setup, and this opposes the change in the B-field.