Waveform of Classic Electromagnetic Induction

[b.shahvir]

In this type of arrangement is induced emf influenced by flux cutting also or only rate of change of flux linkage?

 

[Baluncore]

In this type of arrangement is induced emf influenced by flux cutting also or only rate of change of flux linkage?

It is rate of change of flux linkage.
Draw the magnetic field of your magnet.
Draw a circle about the mid-point of the magnet with a radius equel to the distance to the coil.
Plot the rate of change of line density crossing that circle, around the circle, to get the induced voltage.

 

[b.shahvir]

It is rate of change of flux linkage.
Draw the magnetic field of your magnet.
Draw a circle about the mid-point of the magnet with a radius equel to the distance to the coil.
Plot the rate of change of line density crossing that circle, around the circle, to get the induced voltage.

I understand but aren't the side conductors of the coil also getting cut by the non uniform flux density sweeping across it and as a result also contributing to induced emf quantities? Just a thought.

 

[Baluncore]

I understand but aren't the side conductors of the coil also getting cut by the non uniform flux density sweeping across it and as a result also contributing to induced emf quantities?

It is the rate of change of the number of lines passing through the area of the coil. It has little to do with bits of wire, it is all to do with an area.

 

[b.shahvir]

It is the rate of change of the number of lines passing through the area of the coil. It has little to do with bits of wire, it is all to do with an area.

So if I may presume, as a hypothetical case for that matter, the side conductors of the coil are of significant length. Will the flux cutting have any considerable influence on the induced emf in this case?

 

[Baluncore]

So if I may presume, as a hypothetical case for that matter, the side conductors of the coil are of significant length. Will the flux cutting have any considerable influence on the induced emf in this case?

The area of the coil is defined by the wire. If a line of flux enters that area, at the same time as another line of flux exits the area, then no voltage will be generated.
You must forget about the local wire and see only the area of the coil defined by the windings.

 

[b.shahvir]

If a line of flux enters that area, at the same time as another line of flux exits the area, then no voltage will be generated.

The above statement is valid for uniform flux densities but in this case flux density is non uniform and obeys Inverse Square law. Hence the the no. of flux lines entering and leaving the coil sides may not not be equal at that instant. A resultant emf would be induced in this case due to flux cutting and may contribute to the emf induced due to rate of change of flux linkage.

 

[Delta2]

If the coil is made of thick wire (i.e. it has radius comparable to the radius of the coil area , and also radius big enough that we can't consider the variation of the magnetic field inside it negligible) then the calculations for the total EMF can become a lot more complex as it will be a weighted average of EMFs that are always given by Faraday's law.

 

[Baluncore]

A resultant emf would be induced in this case due to flux cutting and ...

When it comes to a coil, how do you define "flux cutting" ?

It is not the flux lines crossing the wire that is important, it is the derivative of the total number of lines that pass through the closed area of the coil.

 

[b.shahvir]

When it comes to a coil, how do you define "flux cutting" ?

A non uniform magnetic field when sweeping across the coil will also tend to cut the side conductors of the coil (assuming side conductors are of significant length).
A resultant emf will be induced in the side conductors of the coil due to this phenomenon (speed emf, dynamo effect). This emf would in turn contribute to the emf already being induced due to rate of change of flux linkage. This is my understanding.

 

[Paul Colby]

A non uniform magnetic field when sweeping across the coil will also tend to cut the side conductors of the coil (assuming side conductors are of significant length).

I think the point lost is that a time change of the field magnitude alone will cause emf to be generated, no cutting of field lines involved. EMF generated is related to change of flux which may or may not involve lines being cut.

 

[Charles Link]

For instructional purposes, it can be useful to work with a simpler scenario, e.g. a uniform magnetic field with a rotating coil, which is equivalent to a uniform but rotating magnetic field with a stationary coil. The magnetic field from a uniformly magnetized cylindrical magnet adds considerable mathematical complexity to the problem.
Determining the magnetic field from a cylindrical permanent (uniformly magnetized) magnet is a necessary part of this more complex calculation, while the determination of the EMF in the coil is an additional calculation that is most readily done numerically, (computer methods). I would have to believe that in general the waveform will be one with numerous significant Fourier components, besides the fundamental sinusoid.

See https://www.physicsforums.com/threads/a-magnetostatics-problem-of-interest-2.971045/ for computing the magnetic field of a cylindrical magnet.

 

[Paul Colby]

determining the magnetic field from a cylindrical permanent (uniformly magnetized) magnet is a necessary part of this more complex calculation, while the determination of the EMF in the coil is an additional calculation that is most readily done numerically, (computer methods).

Actually, the reciprocity integral relation I gave in #49 can be applied to this problem. One needs to use $J_2=\nabla\times M$ for the current. $E_2$ is then related to the generated voltage at the terminals to the coil. Integration by parts yields the integral of $B_1\cdot M$ over the volume of the magnet. $B_1$ is the field generated by supplying a current $J_1$ to the coil terminals. The only time (or angle) dependence in this integral comes from the angle $M$ makes with $B_1$ provided the both fields are uniform.

 

[b.shahvir]

This isn't helping in resolving my query .

 

[Paul Colby]

This isn't helping in resolving my query .

What would? The calculation I outlined in #73 can be done for uniform fields in closed form. All the principles remain valid without the uniform assumption. Non uniform fields lead to the waveform distortions as many have pointed out. What remains that is unclear?

 

[b.shahvir]

What remains that is unclear?

Contribution of induced emf in coil due to flux cutting.

 

[Paul Colby]

Lacking a definition of flux cutting no one can say. The calculation outlined yields the complete answer. It’s the best I can do.

 

[Delta2]

Contribution of induced emf in coil due to flux cutting.

Can you give us a reference paper where they calculate emf due to flux cutting by a conductor?
If you find anything that has integrals that contain $\mathbf{B} \times \mathbf{v}$ then don't bother, it is equivalent to faraday's law of induction.

 

[b.shahvir]

The equations for faraday's laws of EM are common to both methods of induced emf (rate of change of flux linkage as well as flux cutting). I was wondering if there was a way to distinguish between the two in the rotating magnet case.

 

[Delta2]

There is really only one method of calculating EMF and that's the rate of change of flux. What you refer as flux cutting I interpreted it as calculating integrals that contain $\mathbf{B}\times \mathbf{v}$ and is suitable for systems that contain moving wires with velocity $\mathbf{v}$. But this method is equivalent to the rate of change of flux method.

Here in this example in the rest frame of the rotating magnet the wire of the coil is moving, so you can use the second method but its going to be a hell of more complicated.

 

[b.shahvir]

Here in this example in the rest frame of the rotating magnet the wire of the coil is moving, so you can use the second method

Provided the results are same with both methods.

 

[alan123hk]

I believe that Faraday's law applies to all situations, for example, the magnet is stationary and the coil is moving, or the coil is stationary and the magnet is moving, or both the coil and the magnet are moving, or the mutual inductance of the two coils that move relative to each other, or others.

 

[Delta2]

Provided the results are same with both methods.

They are guaranteed to be the same, check https://en.wikipedia.org/wiki/Faraday's_law_of_induction sections 3 (Proof) and 4 (EMF for non thin wires)

 

[b.shahvir]

I believe that Faraday's law applies to all situations, for example, the magnet is stationary and the coil is moving, or the coil is stationary and the magnet is moving, or both the coil and the magnet are moving, or the mutual inductance of the two coils that move relative to each other, or others.

Absolutely. But in this particular case the contention was which method is best applicable (or both) to the rotating magnet case.
From the arrangement it implies that two situations are applicable to this case;
1) 'Statically' induced emf due to rate of change of flux linkage with coil (transformer simulation)
2) 'Dynamically' induced emf due to cutting of magnetic flux lines by side conductors of the coil as a non uniform magnetic field sweeps across the coil (dynamo simulation)
I believe the above 2 types of EM induction would give rise to individual emfs which may contribute as a whole to the total induced emf value in the coil at a particular instant.

 

[alan123hk]

Absolutely. But in this particular case the contention was which method is best applicable (or both) to the rotating magnet case.
From the arrangement it implies that two situations are applicable to this case;
1) 'Statically' induced emf due to rate of change of flux linkage with coil (transformer simulation)
2) 'Dynamically' induced emf due to cutting of magnetic flux lines by side conductors of the coil as a non uniform magnetic field sweeps across the coil (dynamo simulation)
I believe the above 2 types of EM induction would give rise to individual emfs which may contribute as a whole to the total induced emf value in the coil at a particular instant.

If this is a permanent magnet, so the strength of the magnetic field source itself does not change, then this should be a motion-induced electromotive force. You can use the Lorentz force (the cutting of the magnetic field) or the change in the magnetic flux through the coil to calculate the induced voltage. Both methods should get the same answer, but I don't think the two methods should be used at the same time. As for the method to be used, it depends on the actual situation and each person's choice.

For the example you put forward in #1, because the magnetic field generated by the magnet is very uneven in space, when the magnet rotates, the relative angle of the coil and the magnet is constantly changing, so no matter which method is used, I believe that manually calculating the actual induced voltage is very difficult.

 

[hutchphd]

I believe the above 2 types of EM induction would give rise to individual emfs which may contribute as a whole to the total induced emf value in the coil at a particular instant.

All changes in flux through the coil will result in "cutting" of flux lines. Either the lines move apart or the coil moves to a region of different line density. The results comport with Faraday's Law in either case.
But you cannot calculate a result by endlessly using semantics. Set up a model and calculate.
Edit: I recommend a program called Vizimag which I have used and offers a free 30 day trial I believe. This problem is inherently a 2D problem. Print out some fields (you can choose the line value) and look at how the fields and line cuttings behave. I think your intuition has been pretty good here.
You need to think about line cuttings and line "uncuttings" as well. It is the net change that matters for a loop.

 

[Charles Link]

Suggest you try the experiment if you have access to an oscilloscope. You could spin a cylindrical magnet that is about 10 cm long with an area of 1.0 cm^2. Just some ballpark numbers: The magnetic field at the endface $B \approx 1.0$ W/m^2, so that the flux $\Phi \approx 1.0$E-4 . It should be easy to achieve a $\Delta t=.1$ seconds for a half cycle. With $N=10$ turns, and assuming the entire flux goes through the coils, you get an EMF $\mathcal{E}=10$ mV. I think you are correct when you stated previously that the sinusoid will be distorted with a double "up" hump, and a double "down" hump.

Notice also, to get a stronger signal, you can spin the magnet faster.

 

[Baluncore]

Post #29 specified the dimensions, but not the number of turns on the coil.

 

[b.shahvir]

Suggest you try the experiment if you have access to an oscilloscope.

No I don't, that's why put query on forum.

 

[Charles Link]

No I don't, that's why put query on forum.

I presently don't have access to an oscilloscope either. It would be interesting to see some experimental results.