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I'm currently reading Bleaney & Bleaney 3rd Ed. On pg 136 they discuss the case of 2 coils. Their logic is as follows:

If a second coil is brought near to a coil carrying a current $I_1$, there will be a magnetic flux $\Phi_2$ through the second coil due to the current in the first coil. This is linearly proportional to $I_1$ so we can write $\Phi_2 = L_{21} I_{1}$ for a constant $L_{21}$. There will also be a flux $\Phi_1$ through the first circuit due to a current $I_2$ in the second circuit, given by $\Phi_1 = L_{12} I_{2}$ for a constant $L_{12}$.

The potential energy of the system can be found from the flux of either coil due to the field of another: $U = -\Phi_1 I_1 = -L_{21} I_2 I_1 = -\Phi_2 I_2 = -L_{12} I_1 I_2$.

Hence $L_{12} = L_{21}$.

However I am struggling to understand the setup. Is there a current $I_1$ and $I_2$ flowing in both coils simultaneously? If so, why is the potential energy not $U = -\Phi_{11} I_1 - \Phi_{21} I_1 -\Phi_{22} I_2 - \Phi_{21} I_2$ where $\Phi_{ij}$ is the flux produced by coil i that threads coil j.

(If there is only a current through one coil at a time, I still dont understand their logic, as why should the potential energy when coil 1 has a current $I_1$ be equal to the potential energy when coil 2 has current $I_2$, given we haven't assumed that the coils are identical)

So I have 2 questions.

  1. In this derivation for $L_{12} = L_{21}$ is there a current through both coils simultaneously?
  2. Why can "The potential energy of the system can be found from the flux of either coil due to the field of another". Why can the flux produced by e.g. coil 1 which threads coil 1 be ignored?
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  • $\begingroup$ @hyportnex thanks! would still be good to understand where my intuition is wrong in my above working though. $\endgroup$
    – psychgiraffe
    Commented Apr 5 at 23:07
  • $\begingroup$ Each loop carries a fixed current kept independently from the other irrespective of their relative positions. In fact, the EM work it takes to keep the currents constant as the loops are moved relative to each other is the magnetic energy stored in the system. To prove $L_{12}=L_{21}$ there is a simple "physical" proof and there is also a straightforward vector calculus based "analytical" proof, both are here: physics.stackexchange.com/questions/150068/… $\endgroup$
    – hyportnex
    Commented Apr 5 at 23:10
  • $\begingroup$ Have you already read p.147? $\endgroup$
    – Peltio
    Commented Apr 5 at 23:12
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    $\begingroup$ @psychgiraffe uhm... I think it's more what hyportnex said. Let's see if I can give you an alternate pov: keep your coils close, make a current I1 in the first coil and leave the second open. The flux from the first coil will be intercepted by the second one. Now, if you want to pass a current in the second coil to generate a given flux you need to first cancel the flux that was there without current and then build your own flux. So, in one computation this extra cancellation work is not included, while in the other it is. $\endgroup$
    – Peltio
    Commented Apr 6 at 0:33
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    $\begingroup$ If you wish to understand how the energy gets built to which @Peltio alluded to, please, read Jackson: Classical Electrodynamics, Section 5.16 and then if something is still not clear then ask in a separate new question specifically what steps need clarification. Mutual inductance is in the next Section 5.17 $\endgroup$
    – hyportnex
    Commented Apr 6 at 22:30

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