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It is widely understood that $\nabla \times \vec A =\vec B = 0$ outside a toroidal solenoid with constant current, $I=k$ ($I = \vec J$ in the below illustration).

enter image description here

It is not so widely understood that this condition also holds with a constantly increasing current, $d I / d t =k$.

Faraday's Law says this linearly increasing current will cause a poloidal $\vec E$ field outside the solenoid, thence current in a conducting poloidal circuit outside the solenoid. This is exemplified in a toroidal current transformer driven in reverse of this diagram: enter image description here

However, since $$\nabla \times \vec A = \vec B = 0$$ $$\vec H = \vec B / \mu = 0$$ and $$\vec E \times \vec H = \vec S = 0$$ induction energy must be transferred via some mechanism other than the Poynting vector $\vec S$.

$\vec E = -\partial \vec A / \partial t$ but energy cannot be transferred via the changing vector potential $\vec A$ as this would reify $\vec A$ as a classical quantity, which is outside mainstream physics.

How is it transferred?

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If I understand you right, your question is simply this:

since there is external electric field (in your example, the induced field of the torus coil) that does positive work on the charges bound to the secondary circuit (since current in this secondary circuit increases), but the external magnetic field is zero near the secondary circuit, isn't the Poynting vector there zero? How can EM energy move into the wires of the secondary circuit in agreement with the energy interpretation of the Poynting flux from the Poynting theorem?

The answer is that the Poynting theorem does not relate flow of EM energy at some point of space to the external fields at that point (like the fields of the torus coil), but it relates it to the total electric and total magnetic field (sum of all contributions due to all bodies). In the vicinity of the secondary circuit, this total field includes not only contribution due to the torus coil, but also contribution due to the secondary circuit.

As soon as there is non-zero electric current in the secondary (which requires arbitrarily small energy), total magnetic field is not exactly zero in the spatial neighborhood of the secondary circuit wires, because the secondary produces its own magnetic field.

Thus except the time when current in secondary is zero, the Poynting vector near the secondary is not zero. Initially, the secondary current is small, so the Poynting vector is small, so the rate of energy transfer to the secondary is small. But as the secondary current builds up, rate of work of the induced field of the primary on the secondary increases and the Poynting vector flux into the secondary increases as well.

If you're thinking what happens when we suppress the secondary magnetic field too, by making the secondary to be another ideal torus coil with zero magnetic field outside it, then you are right that the transfer of energy will not go well. Of course, real torus coils made from real wire will not have exactly zero magnetic field outside, because there will be stray fields due to concentrations of current in the wires and lack of current in between them. So some transfer of energy will still happen, but the geometry of mutually tangled toruses seems to be inhibiting this transfer. I did not do the calculation, but most likely mutual inductance of such tangled torus coils is very small, because each turn of one circuit has almost zero flux due to the other circuit. The more tightly the coils are wound the lower the mutual inductance and the slower the energy transfer between the two coils.

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  • $\begingroup$ Your paraphrase is inaccurate. My question entails energy flow out of the solenoid and through the intervening space (presumably at $c$) as well as into receiving conductor. $\endgroup$ – James Bowery Jul 9 at 22:27
  • $\begingroup$ My answer is consistent with your description here - the energy flow in the space between the torus coil and the secondary circuit is non-zero, because the magnetic field in that space is actually non-zero. Magnetic field is non-zero because the secondary circuit creates its own magnetic field. $\endgroup$ – Ján Lalinský Jul 9 at 22:30
  • $\begingroup$ This implies a startup transient in which a reverse Poynting vector provides energy from the secondary to the primary before an additional dynamical process ramps up the forward Poynting vector to the energy flow predicted by Faraday's Law. If you can't describe this process in quantitative terms can you at least provide a literature cite to such? $\endgroup$ – James Bowery Jul 10 at 0:18
  • $\begingroup$ I don't see where this reverse energy flow idea comes from. In my answer, I say that the secondary circuit accepts positive EM energy and this is consistent with the Poynting theorem, because the Poynting vector is nonzero, and moreover, it points towards the wire of the secondary circuit. $\endgroup$ – Ján Lalinský Jul 10 at 1:33
  • $\begingroup$ You said: "As soon as there is a non-zero electric current in the secondary, total magnetic field is not exactly zero." You also said, "the magnetic field in that space is actually non-zero". The origin of the magnetic field is in the secondary and travels at $c$ to pervade the space. Until it reaches the primary, there is no path for energy flow from the primary. During this time, it comprises both $\vec E$ and $\vec H$ in propagation hence a reverse Poynting vector initiates. Correct? $\endgroup$ – James Bowery Jul 10 at 2:55

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