1. Thermodynamic relations in a magnetic field
LL8 Section 31

2. Can B-fields do work on charges?

3. If B-fields do no work on charges, then why do we care about thermodynamics of B-fields?

4. Magnetic fields do no work on charges.
But B-fields that change induce E-fields, which can do work.
Power density =

5. Work done on currents j by the field during time dt =
Work done on the field by the EMF that drives the currents is

8. Work done by an infinitesimal change in the
B-field
Analogy: (10.2)
Magnetic induction
Electric induction
Imperfect analogy

9. Exploit analogy to obtain thermodynamic relations.
Small changes in
Total free energy
Total internal energy
If these total energies are normalized to unit volume, what additional term must appear?

10. Corresponding energy functions per unit volume
Chemical potential per unit mass
This term is absent in the differentials of the total energies, because the number of particles in the total volume doesn’t change, while the number of particles per unit volume can change.
B is the independent variable. How do we change it to H?

11. Legendre transformation changes independent variable from B to H.

12. Linear media: B = m H

13. Now the independent variable is the vector potential!

14. Or, we can make current density the independent variableHW

15. Currents are the sources of the H-field.
Charges are the sources of the electric field.
Would you expect the currents to be the analogs of charges in the energy equations?
Would you expect the vector potential to be the analog of the scalar potential?

16. Analogous formulas for E-fields
Currents j are analogs of potentials f, not charges r
Source of H
Source of E
Vector potentials A are analogs to charges r, not scalar potentials f