Find the mutual inductance between the coil and the wire in two ways

[sapz]

Homework Statement

We have a toroidal coil, and an infinite wire that runs through it on its symmetry axis. (See picture)

We need to find the mutual inductance between the coil and the wire in two ways:

1. By running a current through the wire, and finding the induced voltage in the toroidal coil.
2. By running a current through the toroidal coil, and finding the induced voltage in the wire

Homework Equations

The Attempt at a Solution

So, I manage when trying to solve (1): If there's a current in the wire, there's a magnetic field in the Theta direction, through the coil, and we can find L.

However, if I try to solve it through (2): If I run a current through the coil, then the magnetic field would exist only inside the coil! So how can I find the flux on the wire, if it's outside the coil?... It would just be zero all the time, wouldn't it?

Any help would be appreciated!

 

[zephyr5050]

I'm not sure if this is part of the problem, but it appears to me that the toroidal solenoid is not complete - that is, there's a gap in the wiring. If this is truly part of the problem, then there would be a field which "comes out" of the gap and thus a field on the wire.

If the picture is just misleading and the toroid is supposed to be complete, then my only offering is that saying there is no magnetic field outside the solenoid is not technically correct. That is only the case in an ideal system (in the same world where all cows are spherical). In the real world there will be a field outside of the solenoid, albeit it will be very small and decay rapidly.

I don't have an answer to your problem but hopefully those points might be insightful enough for you.

 

[sapz]

Ive got it! You take the infinite wire, and close it in infinity.
That way you have a closed loop that the flux through it is the flux through the toroid, and therefore there's a current and we can get L.

 

[Basic_Physics]

The magnetic field that the toroidal coil creates is identical to the magnetic field that one finds around a wire with current in it. Can one exists without the other?

 

[Nickie]

I would suggest using the concept of Faraday's Law to find the mutual inductance between the coil and the wire. Faraday's Law states that the induced electromotive force (EMF) in a closed loop is equal to the negative rate of change of magnetic flux through the loop.

In the first method, when a current is run through the wire, it creates a changing magnetic field that passes through the toroidal coil. This changing magnetic field induces an EMF in the coil, which can be measured using a voltmeter. By using Faraday's Law, we can relate the induced EMF to the rate of change of magnetic flux and thus determine the mutual inductance between the coil and the wire.

In the second method, when a current is run through the toroidal coil, it creates a changing magnetic field that passes through the wire. Again, using Faraday's Law, we can relate the induced EMF in the wire to the rate of change of magnetic flux through the wire. This can then be used to determine the mutual inductance between the coil and the wire.

It is important to note that in both cases, the magnetic flux passing through the loop is the same, as the wire and the coil are in close proximity. Therefore, the mutual inductance between the coil and the wire is the same in both cases.