Waveform of Classic Electromagnetic Induction

[Paul Colby]

So for this case what are the appropriate linear constituative relations for $J_2$ . Assume it to be a run of the mill NdFeB cylindrical bar magnet magnetized axially.
I feel that you may not follow my argument.

$\mu=\mu_o$ is an adequate approximation for a fully saturated magnet. We’re talking about $\mu$ at 10 to 100 hertz, not the static component. All fields and currents have an implicit $e^{i\omega t}$ dependence in the calculations I’ve made.

 

[Paul Colby]

Meanwhile, if it surrounded by a long cylindrical solenoidal type coil, (or alternatively placed inside a Helmholtz coil), the EMF will be minimal. The magnet needs to be external to the solenoidal coil.

I can speak to actual experience. The Helmholtz coil was wound on 5” pvc pipe. 30 or so turns on each end. The magnets were rectangular about 1”x1/4”x1/8” or so. They were magnetized through the narrow dimension. Spinning at 10 or so hertz midway between the windings developed about 1/2 a volt of signal. This was 30 years ago so I’m going by recollections.

I have provided a fairly complete analysis of a configuration not being measured here. Applying the very same analysis with $B_1$ being very non uniform over the magnet will produce exactly the wave forms shown elsewhere.

 

[b.shahvir]

Yes, I am considering the equivalent principle regarding another configuration.

Referring to the magnet and coil arrangement of the OP, if we assume a stationary magnet and a rotating coil, do you think the induced EMF of the coil will be different?
I'm still not sure what the answer is, will it be asymmetric, so is the answer different?

This is another demonstration of the OP's magnet and coil arrangement.

Thanks very much for uploading this video. As a matter of fact this video is the central premise of this thread. Also thankful to Tom for his demo.

 

[hutchphd]

μ=μo is an adequate approximation for a fully saturated magnet. We’re talking about μ at 10 to 100 hertz, not the static component. All fields and currents have an implicit eiωt dependence in the calculations I’ve made.

My apologies I did not understand your argument. I agree with your analysis, and the results you quote are exactly what I would expect. The magnet is saturated and stays saturated. The AC response is $\mu_0$. Thanks

 

[b.shahvir]

The discussions on this thread have been intellectually stimulating and fruitful and I am thankful to all the contributors for making this possible. However, there is still a yearning for more...
It would be interesting to observe the experimental setup for the following 2 cases;
1) Double peaked output waveform with 0 state in between the two peaks
2) A perfectly sinusoidal output waveform

What extent of modifications will be required in the rotating magnet arrangement in order to obtain the above output waveforms?

 

[b.shahvir]

This is an interesting setup. Someone connect it to an oscilloscope.

 

[Charles Link]

I can speak to actual experience. The Helmholtz coil was wound on 5” pvc pipe. 30 or so turns on each end. The magnets were rectangular about 1”x1/4”x1/8” or so. They were magnetized through the narrow dimension. Spinning at 10 or so hertz midway between the windings developed about 1/2 a volt of signal. This was 30 years ago so I’m going by recollections.

I have provided a fairly complete analysis of a configuration not being measured here. Applying the very same analysis with $B_1$ being very non uniform over the magnet will produce exactly the wave forms shown elsewhere.

Thanks @Paul Colby . I have to question whether the two coils of the Helmholtz coil were both operational, and if so, that perhaps the polarity was reversed on the connection between the coils, or if the two were simply wound in opposite directions to get the EMF's to add rather than subtract.

 

[Paul Colby]

I have to question whether the two coils of the Helmholtz coil were both operational,

The coils were wired in series to give a constant field at the midpoint. The idea is to make as big and as uniform a magnetic field as possible for a given current. If they were wound to oppose then reciprocity would say no (or very little) EMF would be generated.

I know of no application of Helmholtz coils that would require opposite EMFs in the coils. That doesn't mean there isn't one.

 

[Charles Link]

Thanks @Paul Colby . I think I see my mistake. I was looking at a single pole (say the positive one), and thinking that when it starts to add more flux to the one coil that it is approaching, it will cause the flux to decrease in the other one. This is an incorrect and incomplete mathematical analysis. Instead, when the magnet is in the zero flux state (perpendicular), the change in flux is maximized or nearly maximized as it will increase in the same direction for both coils of the Helmholtz coil as the magnet is rotated from this position. Both poles are involved in the change that occurs, and both coils will see a change in the same direction. My apologies. :)
Edit: I see something else as well: Even for a single pole from the magnet, when the flux increases in one coil, the vector $\Delta B$ is in the same direction (say +z ) for both coils. (Note: $\Delta \Phi=\int \Delta B \cdot \hat{n} dA$ ).

 

[Tom.G]

Post the movie elsewhere and give a link in your PF post. YouTube works.

Oops! YouTube requires a text message to a cell phone, I don't have a cell phone, only a landline.

Next?

 

[Tom.G]

First, many thanks to @Delta2 for posting the video to YouTube!

Here is a slow-motion video, 10% speed, showing the voltage waveform along with the magnet orientation relative to the coil.

The magnet is marked "N" at it's North pole. A bit easier to see is the end of the support string pokes out over the North pole of the magnet.

The waveform dip between the double peaks occurs when the magnet is perpendicular to, approximately on, the coil axis.

The waveform amplitude variations are from the variation of magnet speed at beginnning and end, and the slight off-axis drift of the magnet. These are most noticable near the start and end of the clip where the scope didn't trigger.

--(insert movie MVI_03-A.wmv)

--

SCOPE Tektronix 465

• Vert DC 10mV/div (+-40mV full screen)
• Horiz 50mS/div (500ms/sweep)
• Trigger AC+, slightly below 0

PROBE:

• TEK P6001 1X, one-times probe
• measured at probe tip as connected to scope
  • C=65.8pF, R=999K Ohm

INDUCTOR:

• L= 531uH
• R= 4.33 Ohm
• Winding
  • O.D. 1.2in.
  • Length 0.48in.
  • Wire 30AWG, Dia. over insulation: 0.018in.
  • Spool
    • Flange thickness: 0.050in.
    • Center bore: 0.375in.
  • Winding direction: CW as viewed
    • Connection:
    • Start (inner) to Scope
    • End (outer) to Gnd.

LCR Meter:

• BK PRECISION Model 875A LCR METER

CAMERA:

• CANON S3IS
  • Focal Length 6mm (wide angle, 35mm film equivalent 36mm)
  • Focus and Exposure: Auto

As a 'Bonus?', here is shot of the Work Station; the kitchen table. The camera was on the empty tripod on the lower right. The magnet was hung from the upper tripod with cotton twine, which was manually twisted up to provide magnet rotation.

--(insert still Work_Station.JPG)

--

Cheers,
Tom

 

[b.shahvir]

Great job! Keep up the good work.

 

[vanhees71]

I'm a bit puzzled. Couldn't one simply solve Maxwell's equations (retarded potential) for a harmonically oscillating magnetic dipole and then superimpose two such perpendicular dipoles with a $\pi/2$ phase shift, which is the same as a single rotating dipole, and then calculate the magnetic flux through the coil?

If I find the time, I can try to do that. It should be in close analogy to the corresponding textbook case of a harmonically oscillating dipole (Hertzian dipole). I'm sure, one can find also the magnetic-dipole case in some electroamgnetics textbook (Jackson?).

 

[alan123hk]

This is a cool scientific experiment.

Everything I tried to describe before was an imagination, and I did not perform any calculations to prove that what I said was correct. Now finally there is an excellent experiment to determine all the facts, for example, when the axis of the magnet is perpendicular to the axis of the coil, the local maximum and minimum between the two double peaks do appear.

 

[sophiecentaur]

I'm a bit puzzled. Couldn't one simply solve Maxwell's equations (retarded potential) for a harmonically oscillating magnetic dipole and then superimpose two such perpendicular dipoles with a $\pi/2$ phase shift, which is the same as a single rotating dipole, and then calculate the magnetic flux through the coil?

If I find the time, I can try to do that. It should be in close analogy to the corresponding textbook case of a harmonically oscillating dipole (Hertzian dipole). I'm sure, one can find also the magnetic-dipole case in some electroamgnetics textbook (Jackson?).

The physical version involves magnets of very finite sizes with small separations. That would affect the shape of the field that cuts the (also of finite size) coil and the individual emfs induced in the individual turns of the coil. The result wouldn't be expected to be that of a simple harmonically oscillating dipole.

 

[b.shahvir]

Everything I tried to describe before was an imagination, and I did not perform any calculations to prove that what I said was correct.

Actually it's my proposal and the premise of my thread, please see post#3. But it's ok, I appreciate your efforts

 

[vanhees71]

The physical version involves magnets of very finite sizes with small separations. That would affect the shape of the field that cuts the (also of finite size) coil and the individual emfs induced in the individual turns of the coil. The result wouldn't be expected to be that of a simple harmonically oscillating dipole.

Then it becomes of course much more complicated ;-)).

 

[sophiecentaur]

Then it becomes of course much more complicated ;-)).

Natch. And possibly the basic approach is just not enough.
Sensitivity could be a problem but a much smaller diameter, short coil would be nearer to having all its parts in the same part of the magnet's field - more of a Probe, in fact.

 

[alan123hk]

Then it becomes of course much more complicated ;-)).

I believe that in the very uneven 3D magnetic field produced by a rotating magnetic dipole, calculating the rate of change of the magnetic flux passing through the 3D coil is a very difficult task. If it refers to the use of manual calculations, whether it is analytical calculations or numerical calculations, it is even more difficult to imagine and almost impossible to complete.

 

[b.shahvir]

One can redo this experiment with very short or significantly long permanent magnets. The waveforms will be pretty interesting to analyse.

 

[b.shahvir]

short coil would be nearer to having all its parts in the same part of the magnet's field - more of a Probe, in fact.

A search coil I presume. I think this arrangement might generate a perfect sinewave.

 

[Charles Link]

A search coil I presume. I think this arrangement might generate a perfect sinewave.

If you look at any images of the flux lines from a bar magnet, it is clear that the dependence is not $B=B_o \cos(\theta)$. The flux is concentrated on-axis. The signal then falls off much faster off-axis than simply $\cos(\theta)$. The result is that you will not get a perfect sinusoid.

If you rotate the small coil, instead of rotating the magnet, you do get a nearly perfect sinusoid.

 

[Charles Link]

This is an interesting setup. Someone connect it to an oscilloscope.

This one is interesting, but it seems to waste the flux from the magnets, using a coil with a long and thin iron core that is wrapped with many turns with multiple layers of turns. I think @Tom.G 's apparatus is much more efficient. :)

 

[hutchphd]

I'm a bit puzzled. Couldn't one simply solve Maxwell's equations (retarded potential) for a harmonically oscillating magnetic dipole and then superimpose two such perpendicular dipoles with a π/2 phase shift, which is the same as a single rotating dipole, and then calculate the magnetic flux through the coil?

This is inherently a near-field problem and the solution for a finite rotating dumbbell of charge (be it electric charge or "magnetic poles') needs to include the unipolar parts I think. I would be pleased if you see an easy way to do this but if the coil is not near the effect is lost

 

[Charles Link]

It is perhaps worth mentioning that @Tom.G is operating his apparatus at a very low frequency, (about 1 or 2 Hz) and that the signal levels ($\mathcal{E}=-\frac{d \Phi}{dt}$ ) would be considerably larger if the frequency were increased, e.g. to 60 Hz. It could be interesting to see some experimental results for higher frequencies, even in the range of 10 Hz.

 

[Tom.G]

It is perhaps worth mentioning that @Tom.G is operating his apparatus at a very low frequency, (about 1 or 2 Hz) and that the signal levels ($\mathcal{E}=-\frac{d \Phi}{dt}$ ) would be considerably larger if the frequency were increased, e.g. to 60 Hz. It could be interesting to see some experimental results for higher frequencies, even in the range of 10 Hz.

See the video referenced in post 173,
https://www.physicsforums.com/posts/6503116

Tom

 

[alan123hk]

I believe that in the very uneven 3D magnetic field produced by a rotating magnetic dipole, calculating the rate of change of the magnetic flux passing through the 3D coil is a very difficult task. If it refers to the use of manual calculations, whether it is analytical calculations or numerical calculations, it is even more difficult to imagine and almost impossible to complete.

Maybe I was too exaggerated before. In simple cases, this calculation may not be very difficult. For example, consider only the near field produced by an infinitesimal magnetic dipole and apply a one-turn coil (axial thickness is zero). But unfortunately I found that there is no double humps waveform in this case, so I want to raise a question below.

Is it possible use multiple magnetic dipoles arranged in a long strip to simulate a bar magnet, so to find a magnetic field equivalent to a bar magnet rotating in space ?

If this method works, place a one-turn coil in this magnetic field, and then according to the rate of change of the magnetic flux passing through this one-turn coil, the induced EMF be obtained. I hope this induced EMF waveform is consistent with the experiment, that is, the aforementioned local minimum/maximum will appear between the double peaks.

 

[b.shahvir]

It would be interesting to observe the experimental setup for the following 2 cases;
1) Double peaked output waveform with 0 state in between the two peaks
2) A perfectly sinusoidal output waveform

What extent of modifications will be required in the rotating magnet arrangement in order to obtain the above output waveforms?

I suppose the above would be more interesting to analyse

 

[b.shahvir]

Or this arrangement for that matter.

 

[Charles Link]

Is it possible use multiple magnetic dipoles arranged in a long strip to simulate a bar magnet, so to find a magnetic field equivalent to a bar magnet rotating in space ?

This is one way to interpret the pole method of magnetism, where all of the opposite matching poles in the material cancel, except at the end faces. In any case, the calculation is one of just two poles. See https://www.physicsforums.com/threads/a-magnetostatics-problem-of-interest-2.971045/ which I mentioned previously in post 72.

Post 1 of the "link" summarizes both the magnetic pole method and the magnetic surface current method of computing the result. Both methods get the identical result for the magnetic field. The pole model is the much simpler one in this case.

Someone who is handy with computer programming could probably somewhat routinely put together a program that will generate the waveform for a specified geometry of the rotating magnet with a single coil. $B(t)$ would need to be computed for the two poles for about 1000 points in the cycle. It would not even be necessary to integrate $\int B \cdot dA$ in a very simplified approach, but doing the integral numerically should not be difficult to program. In any case, $\mathcal{E}=- \frac{d \Phi}{dt}$ is readily computed for each of the 1000 points .

It may be worth mentioning, (I think I may be stating the obvious), that the computation of the magnetic field $B$ for the rotating bar magnet is a completely static type calculation. It is not necessary to account for any motion of the moving magnetic poles. The calculation of the magnetic field $B$ for the rotating bar magnet is rather straightforward.

 

[Charles Link]

I should mention as a follow-on to the above, that the magnetic pole of magnetic surface charge density $\sigma_m=\pm M$ and area $A$ can be assumed to be a point magnetic charge at the center of the end face. This gives point magnetic charges $q_m= \pm \sigma_m A=\pm M A$, on the end faces, and the $H$ field is simply an inverse square analogous to $E$, with $\mu_o$ instead of $\epsilon_o$. Finally $B=\mu_o H$. The north pole is the positive one, and the south pole is the negative one. Only the component of $B$ perpendicular to the plane of the coil is needed for computing the flux.

I presently don't have any computer computing capability on my Chromebook or I would program this up. It should be fairly straightforward to numerically generate the waveform for a bar magnet that is 5 cm long and is 10 cm from a small coil. I do expect the double hump feature will emerge.

 

[b.shahvir]

I think the best option would be a practical demonstration to help us better understand and analyse the results.

 

[hutchphd]

I have done an analytical model and obtained results consistent with my previous posts (and the lovely experiments).. The model consists if a N and a S magnetic monopole on the ends of a rotating stick of length d. A small sensing coil is placed outside the radius of rotation. By varying d (while keeping the dipole moment fixed) one easily reproduces the cos for small d and then flatter then double hump for d approaching the coil. There are no surprises in the model.
Sorry for the tease but I will slog through the LaTeX in the next few days. Too much sh*t on my fan right now!
EDIT: I meant that literally I need to fix my fridge!

 

[alan123hk]

I think the best option would be a practical demonstration to help us better understand and analyse the results.

Experiments and practical demonstrations are of course very important, but scientists and engineers do need to develop some theoretical-based calculation methods to simulate and predict the results of some physical processes, which may be necessary for the design of complex and sophisticated systems.

 

[alan123hk]

I tried to use the magnetic charge model to simulate a rotating bar magnet, and then found an equation for the change of magnetic flux through the stationary coil with time in the simplest case, then I differentiated this equation, and I finally found the equation for the induced EMF of the coil.

The calculated induced EMF is similar to what I had imagined before, but it does have a certain degree of difference.