1. Linear conductors with currents Section 33

2. Assume B =  H everywhere. Then Ignore this term, which is unrelated to the currents Or better Since j = 0 outside of the wires

3. Since Maxwell’s equations are linear in the fields, the superposition of field solutions is also a solution = sum over fields from each current alone. Free self energy Energy of a th conductor in its own field Interaction energy Energy of a th conductor in the field of the b th conductor

4. Since

5. Also Field at dV a due to j a Field at dV b due to j a Field at dV a due to j b a th conductor b th conductor a th conductor

6. For a given current-density distribution, depends only on the total current a Cross section

7. Both j a and the field A a produced by j a are proportional to total current J a Self energy Proportionality constant L aa = “self-inductance” Depends only on geometry

8. Similarly and So interaction energy is A different proportionality constant. L ab = “mutual inductance” Also depends only on geometry

9. Magnitude of free energy depends on relative sense of currents Total free energy of system No restriction a>b now, due to factor ½ applied to interaction term

10. >0 (Positive definite) Puts conditions on the coefficients L aa > 0 for all a (e.g. consider a single conductor) L aa L bb > L ab 2 Mutual inductance is always less than or equal to the product of self inductances of two circuits. L ab = k Sqrt[L aa L bb ] Coupling coefficient |k|

11. Calculation of for arbitrary 3D currents is hard. If  = 1 in both conductors and surrounding media, the problem simplifies. Then  (T) is not a factor in, which then is independent of the thermodynamic state of the materials. Then = U, i.e. “free energy” -> “energy”.

12. Non-magnetic  = 1 case (30.12), same as in vacuum Then the self energy is a th conductor a th conductor

13. Similarly, the interaction energy is Linear circuits Depends only on shape, size, relative position, and relative direction of currents Does not require  = 1 for the linear conductors, since their magnetic energy is tiny. If surrounding medium has  > 1, L ab  L ab.

14. The total energy is The vector potential of the total field from all conductors at dl a of the a th conductor. Linear circuits =  = Magnetic flux through a th circuit

15. ThenFlux from all circuits through the a th circuit For a linear current J in an external B-field No self-energy for the sources of B. No factor of 2 to correct for double counting of interaction pairs.

16. For uniform external B-field and non-permeable medium surrounding conductor External field without linear circuitMagnetic moment of linear circuit

17. How to find forces on linear current-carrying conductors. If free energy is known as a function of shape, size, and relative positions Then forces on conductors are found by differentiation with respect to the proper coordinates. Which thermodynamic potential should we use? It is easy to hold currents constant during the differentiation. It is difficult to hold fluxes constant when the wires move. The free energy with respect to currents is

18. Generalized force F q in the direction of the generalized coordinate q. Omitting terms that are independent of currents,See (31.7)

19. About the sign of the force vector… L ab can be positive or negative, depending on the relative direction of the currents Both ccw: repel One ccw and one cw: attract

20. Forces exerted on a conductor by its own magnetic field For given J and T, spontaneous irreversible changes occur that reduce until it reaches a minimum.

21. Forces on a conductor, which tend to minimize, will tend to maximize L L is always positive [L] = length (HW) L~ dimension of conductor A magnetic field from J in a conductor tends to increase its size.