# B Field Confinement for Toroidal Coils Driven by AC Current

1. Aug 14, 2012

### -dove

It is said that toroidal inductors/coils (see http://fa.tu-sofia.bg/te/Brandisky/images/toroidal_coil.jpg [Broken]) that are perfectly axially symmetric will completely confine B fields. That is, the B field inside will be nonzero and will circle the toroid, but the B field outside the toroid will be zero. This makes sense for a DC current. The magnetic field inside will be constant and there will be a nonzero, constant, electric field wrapping around the oustide of the toroid that has no curl.

Now, from what I've read, this natural confinement is supposedly taken advantage of by toroidal transformers. However, transformers are driven by AC currents, and I can't figure out how a toroidal inductor driven by an AC current can possibly have zero external B field. The external E field would have to oscillate in order to obey Faraday's law (since the flux inside is oscillating), but then the external E field would be time varying. So, since there is no external current, Ampere's law would then imply that the external B field must have a non-zero curl (at least at some times) and thus at certain times the external B must be nonzero. Thus, I don't see how toroidal inductors could possibly confine B fields when driven by AC currents. Now, transformers are different from inductors; they have a secondary winding in addition to the first one. But the whole situation has left me suspect to whether or not anyone is even claiming that toroidal coils and transformers confine B fields when driven by AC.

Do 'ideal' toroidal transformers perfectly confine B fields (when driven by AC)? And if so is it just a result of the secondary winding cancelling out the external field? Or what?

Last edited by a moderator: May 6, 2017
2. Aug 15, 2012

### -dove

EDIT: The end of that first paragraph should read
"This makes sense for a DC current or a steadily increasing one. In the former case, the magnetic field will be constant inside and zero outside; the electric field outside will be zero as well. In the latter case, the magnetic field inside will steadily increase, the magnetic field outside will remain zero, and the electric field outside will be nonzero, constant, have zero curl, and wrap around the toroid."

The question regarding AC (or any case where the internal B has a nonzero second derivative w.r.t. time) remains the same.