Waveform of Classic Electromagnetic Induction

[b.shahvir]

Hi guys,

Can someone please provide graphical representation (waveform) of emf induced in coil due to a bar magnet spinning perpendicular to axis of coil.

Thanks,
SB

 

[Baluncore]

A sinewave.

 

[b.shahvir]

A sinewave.

Yes, but in my opinion the waveform progression would be an irregular sinewave as there would be 2 positive peaks followed by 2 negative peaks (eg. M -- W) due to alternate rotation of N & S poles of the spinning bar magnet. I just wanted to visualize the graphical representation of my assumption.

 

[Baluncore]

The voltage is proportional to the rate of change of the magnetic flux passing through the coil.
For a spinning bar magnet that will be a sinewave.

 

[Paul Colby]

If the bar axis is perpendicular to the coil axis, isn’t the amplitude ideally 0? The total flux through the coil area is constant?

 

[b.shahvir]

If the bar axis is perpendicular to the coil axis, isn’t the amplitude ideally 0? The total flux through the coil area is constant?

The magnet spins perpendicular to axis of coil. So a sinewave will be induced in coil but it's pattern would alternate between double positive and negative peaks in progression as the N & S poles alternately sweep across the plane of the coil.

For eg., negative voltage peak as N pole approaches the plane of coil, then 0V at centre of coil when flux is maximum and again positive voltage peak as N pole leaves the plane of coil. Same pattern will repeat but with inverted voltage peaks when S pole sweeps along. I just wanted to confirm whether my assumption was correct.

 

[Paul Colby]

The magnet spins perpendicular to axis of coil.

This doesn’t define the geometry. The magnet’s axis of rotation is defined as perpendicular to the coil axis. What is the angle between the magnets rotation axis and the magnets magnetic moment? If this unspecified angle is zero, the amplitude will likewise be zero.

 

[b.shahvir]

The bar magnet rotates about it's axis which lies in a plane perpendicular to axis of the coil such that an emf will be induced in the coil. As i mentioned above the N & S poles will be sweep across the face of the coil.

 

[Delta2]

A picture worth a thousands words and I think in this case even more. If you can post a picture where you show the coil and the bar magnet and how exactly the bar magnet rotates.

 

[b.shahvir]

A picture worth a thousands words and I think in this case even more. If you can post a picture where you show the coil and the bar magnet and how exactly the bar magnet rotates.

The magnet is rotated about it's axis as illustrated in figure above. The axis of the magnet is perpendicular to the axis of the coil.

 

[Charles Link]

There is no simple expression for the magnetic flux that emerges from a bar magnet. The lines of flux exit the north pole and enter the south pole. I don't think there is a simple answer to what you are asking.

 

[b.shahvir]

It's classic case of electromagnetic induction but never found any graphical representation of same, hence the query.

 

[Charles Link]

The problem is too complicated for it to be one that is a standard textbook example. It may look simple enough, but another one in that category is the force between two bar magnets.

 

[b.shahvir]

The problem is too complicated for it to be one that is a standard textbook example. It may look simple enough, but another one in that category is the force between two bar magnets.

Actually this is common depiction in textbook physics but with output to galvanometer and not oscilloscope. I doubt if such arrangement is actually feasible in practical generators to generate proper sinewaves.

 

[Charles Link]

Actually this is common depiction in textbook physics but with output to galvanometer and not oscilloscope.

Faraday's law applies, and a signal is observed. To quantify it accurately would take some effort.

 

[Baluncore]

So a sinewave will be induced in coil but it's pattern would alternate between double positive and negative peaks in progression as the N & S poles alternately sweep across the plane of the coil.

The voltage is not generated when the flux cuts one side and then the other side of the cylindrical coil. The voltage is generated by the changing flux through the area surrounded by the circular coil.

 

[Delta2]

There is no simple expression for the magnetic flux that emerges from a bar magnet. The lines of flux exit the north pole and enter the south pole. I don't think there is a simple answer to what you are asking.

If we do the simplifying assumption that the magnetic field near the bar magnet is very similar to a homogeneous magnetic field then we can say that the resulting voltage will be a sine wave. But I don't know this assumption how close is to reality.

 

[b.shahvir]

If we do the simplifying assumption that the magnetic field near the bar magnet is very similar to a homogeneous magnetic field then we can say that the resulting voltage will be a sine wave. But I don't know this assumption how close is to reality.

I never mentioned flux cutting. But this arrangement will obey faraday's laws of EM such that voltage peak polarities will change each time the flux linking the coil changes due to instantaneous position of N and S poles with time.
I was anticipating waveform patterns to support by assumptions.

 

[b.shahvir]

If we do the simplifying assumption that the magnetic field near the bar magnet is very similar to a homogeneous magnetic field then we can say that the resulting voltage will be a sine wave. But I don't know this assumption how close is to reality.

It would be a sine wave but with repeated double positive and negative peaks for N and S poles linkage respectively. I did not find any waveforms depicting this phenomenon exactly.

 

[Charles Link]

I don't follow the double positive and double negative features. The pole when it is farther away from the coil will have a much lesser effect. To quantify this is somewhat difficult, but it is not a sine wave at twice the frequency.

 

[b.shahvir]

I don't follow the double positive and double negative features. The pole when it is farther away from the coil will have a much lesser effect. To quantify this is somewhat difficult, but it is not a sine wave at twice the frequency.

Frequency depends on speed of spinning magnet so it would remain constant. But i presume the nature of sine wave patterns will differ in time.

 

[Baluncore]

If the bar magnet dipole was short relative to it's distance from the coil, then the voltage generated would closely approximate a sinewave. If the bar magnet was to remain fixed some distance away, while the coil was rotated in the field, then the coil voltage would be a sinewave.

I do not understand the alleged mechanism that might generate a polarity symmetric double hump, which in essence would constitute only odd harmonic distortion.
The sum of two sinewaves, (one generated by each end of the dipole), would also be a single hump sinewave.

 

[b.shahvir]

The sum of two sinewaves, (one generated by each end of the dipole), would also be a single hump sinewave.

It cannot be a single hump sinewave due to 0 flux positions (between alternate N and S pole linkages) at regular intervals during rotation of magnet.

 

[Baluncore]

The zero flux occurs when the two poles effectively cancel. The zero flux positions are usually the points at which the flux is changing at the greatest rate. They are therefore the voltage peaks.

 

[b.shahvir]

The zero flux occurs when the two poles effectively cancel. The zero flux positions are usually the points at which the flux is changing at the greatest rate. They are therefore the voltage peaks.

As per the swinging magnet experiment in the attached pdf, the voltage peak occurs when the magnet pole is just near the midpoint of the coil (for both cases of pole approaching and leaving the coil) as the rate of change of flux is maximum at that point. In my opinion this may be closest simulation of my rotating magnet case.

 

[Baluncore]

The field rotates with the rotating magnet. With the swinging magnet the field translates and there is no reversal of the field direction.

 

[b.shahvir]

The field rotates with the rotating magnet. With the swinging magnet the field translates and there is no reversal of the field direction.

Exactly. That's why the double hump sinewave phenomenon as the peaks will repeat twice and invert as each pole sweeps across the coil in alternation.

 

[Baluncore]

You must define some parameters.
1. The length of the coil.
2. The diameter of the coil.
3. The length of the bar magnet.
4. The distance from the centre of the coil to the centre of the bar magnet.
This discussion is pointless without those parameters.

 

[b.shahvir]

You must define some parameters.
1. The length of the coil.
2. The diameter of the coil.
3. The length of the bar magnet.
4. The distance from the centre of the coil to the centre of the bar magnet.
This discussion is pointless without those parameters.

1. 0.5cm
2. 1cm
3. 1cm
4. 1cm

 

[hutchphd]

The zero flux occurs when the two poles effectively cancel. The zero flux positions are usually the points at which the flux is changing at the greatest rate. They are therefore the voltage peaks.

In this geometry (small coil, long thin magnet):
The zero flux positions occur when the magnet axis is 90 deg from the coil axis (this happens twice per rotation).
The position of maximum flux change occurs approximately when the magnet pole engages the coil edge (this happens twice for each pole face = four times per rotation)
The position of extremal flux is when the axes align and this gives a zero crossing for the voltage (two crossings per rotation)
So there can be double humps between zero crossings as the OP indicates. The rest is as you indicate detail.

 

[Delta2]

The position of maximum flux change occurs approximately when the magnet pole engages the coil edge (this happens twice for each pole face = four times per rotation)

Hmm , I semiagree to this, I can't understand why it happens twice per pole per rotation, do you mean once as it engages it from the bottom edge and one from the top edge?

 

[hutchphd]

Yes: on the way in (toward closest approach) and on the way out. In between these approach events the flux is small and changes sign but the derivative preserves sign.

 

[vanhees71]

I don't know, whether one can find this in the literature. Maybe it's worth trying to calculate the em. (wave) field for a rotating magnetic point dipole. The magnetization should be something like
$$\vec{M}(t,\vec{x})=\begin{pmatrix} \mu \cos(\omega t) \\ \mu \sin(\omega t) \\0 \end{pmatrix} \delta^{(3)}(\vec{x}).$$
The Maxwell equations read
$$\vec{\nabla} \times \vec{B} + \frac{1}{c} \partial_t \vec{E} = \vec{\nabla} \times \vec{M}, \quad \vec{\nabla} \cdot \vec{B}=\vec{\nabla} \cdot \vec{E}=0, \quad \vec{\nabla} \times \vec{E} + \frac{1}{c} \partial_t \vec{B}.$$
Perhaps one can even solve this analytically by calculating the retarded potentials.

Then you can put your current loop/coil and simply calculate the magnetic flux going through and the EMF by taking its time derivative.

 

[hutchphd]

Except this is not a point dipole. And that is the crux of the confusion. Its really two monopoles.

 

[b.shahvir]

Hmm , I semiagree to this, I can't understand why it happens twice per pole per rotation, do you mean once as it engages it from the bottom edge and one from the top edge?

The waveform from the swinging magnet experiment in attached pdf may help to understand this phenomenon of double hump sine wave.

 

[vanhees71]

Except this is not a point dipole. And that is the crux of the confusion. Its really two monopoles.

Why that? I'd say it's a very much idealized model for a rotating permanent magnet by just approximating the magnetization of the extended true magnet by a point dipole.

How would you write down a true point dipole? How is it different from two close (fictitious) magnetic monopoles of opposite magnetic charge (in analogy to the electric case)?

 

[hutchphd]

How would you write down a true point dipole?

I don't want to here.
It is not infinitesimal and this is very much near field. Perhaps I misunderstand your intent but the poles must far apart compared to the distance to and size of the coil for the OP effect to show up.

 

[b.shahvir]

Just imagine it to be a primitive hand driven permanent magnet generator.

 

[hutchphd]

Your result will be somewhere in between the very double-humped output of your video and the textbook example of a centered short dipole in a flat loop (which has a simple smooth sinusoid).. Do you need to know the exact solution ? The physics is pretty clear from those two limits for me.

 

[alan123hk]

The waveform from the swinging magnet experiment in attached pdf may help to understand this phenomenon of double hump sine wave.

I believe this phenomenon can be briefly described as follows.

When the north pole of the magnet is approaching to the coil, the induced voltage is +A, and when the north pole of the magnet is leaving the coil, the induced voltage is -A.

When the south pole of the magnet is approaching to the coil, the induced voltage is -A, and when the south pole of the magnet is leaving the coil, the induced voltage is +A.

The above process occurs in one cycle, so when the magnet is rotating, the timing of coil induced voltage becomes $+A~ →~ - A ~→~0~→~ -A~→+A$

Am I right?

 

[hutchphd]

Am I right?

Except it doesn't quite go to zero except at the crossings.

 

[alan123hk]

Except it doesn't quite go to zero except at the crossings.

I think you are right. In theory, only zero crossings can occur. Although the slope may be small, the coil induced voltage will not remain zero even for a short period of time.

 

[Paul Colby]

I worked at a company in the late 90s that manufactured linear stepper motors as part of a wafer prober. Each motor had 4 rare Earth magnets. At one point they had manufacturing issues with these motors. One theory was perhaps the magnets varied in strength enough to cause the observed out of spec wobble. I wound myself a Helmholtz coil and extracted a motor out of a floppy disk drive. Spinning the magnets generated a sinusoidal voltage proportional to the magnetic moment of the magnet. The measurements were quite precise, very reproducible. I was able to rule out variations of magnet strength as the culprit. Those were fun times.

 

[b.shahvir]

The above process occurs in one cycle, so when the magnet is rotating, the timing of coil induced voltage becomes $+A~ →~ - A ~→~0~→~ -A~→+A$

Am I right?

Yes, but there will be 0 emf state whenever the poles align with the axis of coil ( as rate of change of flux linkage is 0) and also 0 emf state when the magnet poles are 90 deg wrt axis of the coil (again rate of change of flux linkage is 0). So the result should be
0 +A 0 -A 0 and 0 -A 0 +A 0 per cycle
I wonder what the waveform will look like.

 

[Baluncore]

and also 0 emf state when the magnet poles are 90 deg wrt axis of the coil (again rate of change of flux linkage is 0).

No, the flux is zero at that instant, but it is changing through zero.

 

[b.shahvir]

No, the flux is zero at that instant, but it is changing through zero.

Yes but this may be more suitable in case of speed emfs (dynamo) rather than statically induced emfs as flux density obeys inverse square law and the magnetic flux density linking the coil would be zero when magnet is vertical (90 deg wrt coil plane). So rate of change of flux when magnet is passing through 0 would be 0, hence V=0 at that instant.

 

[vanhees71]

I don't want to here.
It is not infinitesimal and this is very much near field. Perhaps I misunderstand your intent but the poles must far apart compared to the distance to and size of the coil for the OP effect to show up.

I still don't understand your objection.

I just want to have a single rotating magnetic point dipole. It's of course the superposition of two perpendicular dipoles oscillating with a phase shift of $\pi/2$ relative to each other. Thus it's the analogue of the analogous case of a harmonically oscillating point dipole (the Hertz dipole). I'm sure, you can solve this problem as easily as you solve the Hertzian dipole you find in any textbook. Then you have a model field valid everywhere. In the near zone you have the quasistatic approximation, but I guess it's not much simpler to get than the complete retarded solution. If I find the time over the weekend, I'll post my result.

 

[Baluncore]

So rate of change of flux when magnet is passing through 0 would be 0, hence V=0 at that instant.

As the polarity changes, the derivative cannot be zero because the value is changing.
The sine curve rises through the origin, at that instant the value; sin( 0 ) = 0; but the rate of change is; cos( 0 ) = 1;

 

[Paul Colby]

Then you can put your current loop/coil and simply calculate the magnetic flux going through and the EMF by taking its time derivative.

It’s actually simpler than that. If one looks just at the time harmonic fields, there are a set of reciprocity integral relations which relate fields between two physical solutions. For harmonic electric fields and currents, for example, one has,

$\iiint E_1\cdot J_2 dv =\iiint E_2\cdot J_1 dv$

Where fields 1 are due to antenna 1 only and 2 due to antenna 2 only. All other boundary conditions are held fixed. For the magnetic case the B field in the coil due to an applied voltage is constant over the volume of the magnet (assuming proper design). This allows one to express the generated voltage as an integral of the constant magnetization in the magnet. Sadly, I’ve lost all the details to antiquity.

 

[rude man]

The emf will be a very irregular - not at all a sine wave - due to the irregularity of magnetic fields surrounding a permanent magnet.

The one thing you can say is that the average emf will be $4fN\phi$ where $\phi$ is the maximum net flux in the coil (when the perm. magnet points along the coil axis), N = no. of turns, f = frequency of perm. magnet rotation in Hz.
EDIT: I meant the average emf over 1/4 rotation of the perm. magnet.
The avg. emf for any integer number of rotations is of course zero.