Field Strength of Permanent Magnets -- Inverse square or inverse cube?

[pete94857]

TL;DR Summary

Should we use the inverse square law or the inverse cube law or a combination to calculate feild strength.

I've just seen a video made by Dr. Usha Singh associate physics professor.

It showed a practical experiment giving results that the changing field is directly proportional to the inverse square law.



It involves two bar magnets.

I do however have some conflicting information. When analysing the feild between two magnets the feild if measured in the space between the magnets will be double because there are two magnets but if the force on one of the magnets is measured then depending on the distance the combination of both fields acting on that magnet varies with distance at best very close to each other a maximum of 90 % ranging to as low as 10 % at further distance.

 

[anuttarasammyak]

Say two bar magnets of length l are in a line with samedirection and the distance of the nearest poles is r.
Attraction between different sign poles
$$-Q_M^2(\frac{1}{r^2}+ \frac{1}{(r+2l)^2})$$
Repulsive force between same sign poles
$$Q_M^2\frac{2}{(r+l)^2}$$
where $\pm Q_M$ is magnetic charge at poles. We can investigate in total force is attractive/repulsive and how it depends on r.

 

[kuruman]

The analysis shown in the video is flawed primarily because it fails to convince the reader that the dependence of the repulsive force on pole separation follows an inverse square law. The presenter's conclusion that it does is based on the observation that when the force is plotted against $1/(r_{\text{eff.}})^2$ one gets a straight line. It ain't necessarily so.

I copied the data from the video and made two plots: (a) Force vs $1/(r_{\text{eff.}})^2$ (same as the video), shown below left and (b) Force vs $1/(r_{\text{eff}})^3$, shown below right. Variable $r_{\text{eff.}}$ is the distance provided in the video. I fitted straight lines to both. The fitting parameters and R-squared (goodness of fit) are included. It is clear that the straight line fits are equally good and that one cannot tell from these straight lines which power law is the case.

It is also disturbing that the slope and intercept of the straight line fit are ignored. Usually, when one produces a linearized fit the slope and intercept carry useful information. So how can we figure out the power law?

Note that if we assume a power law of the form $F=Cr^n$ and take the logarithm on both sides, we get $$\ln(F)=\ln C+n\ln(r)$$ Then we make a log-log plot which should be a straight line in which the intercept is the multiplicative constant C and the slope is the power to which $r$ is to be raised. This plot is shown on the right.

Now we can write the dependence of the force on pole separation as $$F=\frac{e^{14.85}}{r^{2.833}}$$ and plot the data on the same graph as the calculated values from the above equation (see right). The same result can be obtained by fitting a power law on a spreadsheet.


The video also disingenuously

• Does not show the measured pointer-to-pointer distances AB but the massaged data values of $r_{\text{eff.}}$ I reluctantly had to reconstruct the measured values AB from the massaged data.
• Does not justify the offset value of the poles from the ends of the magnets. Where did the particular factor of 0.85 come from?
• Uses pressure units (dynes/cm²) for force to label the ordinate of the plot.

Bottom line: The power law for the repulsive force between the magnets is closer to an inverse cube law than to an inverse square law. How close is "closer" depends on error analysis but only the person who performed the experiment knows what the measurement uncertainties are.

I said more than I intended, but it was worth it because I had fun processing these data.

(Edited to fix typos.)

 

[pete94857]

The analysis shown in the video is flawed primarily because it fails to convince the reader that the dependence of the repulsive force on pole separation follows an inverse square law. The presenter's conclusion that it does is based on the observation that when the force is plotted against $1/(r_{\text{eff.}})^2$ one gets a straight line. It ain't necessarily so.

I copied the data from the video and made two plots: (a) Force vs $1/(r_{\text{eff.}})^2$ (same as the video), shown below left and (b) Force vs $1/(r_{\text{eff}})^3$, shown below right. Variable $r_{\text{eff.}}$ is the distance provided in the video. I fitted straight lines to both. The fitting parameters and R-squared (goodness of fit) are included. It is clear that the straight line fits are equally good and that one cannot tell from these straight lines which power law is the case.

View attachment 353594
It is also disturbing that the slope and intercept of the straight line fit are ignored. Usually, when one produces a linearized fit the slope and intercept carry useful information. So how can we figure out the power law?

View attachment 353597Note that if we assume a power law of the form $F=Cr^n$ and take the logarithm on both sides, we get $$\ln(F)=\ln C+n\ln(r)$$ Then we make a log-log plot which should be a straight line in which the intercept is the multiplicative constant C and the slope is the power to which $r$ is to be raised. This plot is shown on the right.



View attachment 353598Now we can write the dependence of the force on pole separation as $$F=\frac{e^{14.85}}{r^{2.833}}$$ and plot the data on the same graph as the calculated values from the above equation (see right). The same result can be obtained by fitting a power law on a spreadsheet.


The video also disingenuously

• Does not show the measured pointer-to-pointer distances AB but the massaged data values of $r_{\text{eff.}}$ I reluctantly had to reconstruct the measured values AB from the massaged data.
• Does not justify the offset value of the poles from the ends of the magnets. Where did the particular factor of 0.85 come from?
• Uses pressure units (dynes/cm²) for force to label the ordinate of the plot.

Bottom line: The power law for the repulsive force between the magnets is closer to an inverse cube law than to an inverse square law. How close is "closer" depends on error analysis but only the person who performed the experiment knows what the measurement uncertainties are.

I said more than I intended, but it was worth it because I had fun processing these data.

(Edited to fix

After further analysis of the video I completely agree that the video is flawed. And although your analysis of the data produced is correct the data produced is also completely flawed based on the equipment used to perform the experiment. If a straight bar is suspended by a bearing at its top (like a pendulum) with a weight at its base (the magnet) it will obviously use quite a substantial amount of the energy that has been converted from the magnetic force. Therefore the readings should be higher than they are recorded. The weight of the pan only accounts for a particular position. Any other positions the rod takes and the countering effect caused by the pan are no longer relevant. So any data produced by the readings from the video are also unfortunately obsolete. Therefore the energy to lift the assumed steel pole and counter bearing resistance is not accounted for.

 

[kuruman]

I think you totally misunderstood the experiment. The readings are a direct measurement of the force between the two magnets. As Assoc. Prof. Usha Singh mentioned this is a null measurement. Here is how it works.

1. In the initial position the suspended magnet on the left is hanging in the vertical position, there is separation $r$ between the magnets and weight $mg$ is pulling on the vertical rod. The system is in equilibrium which means that the tension in the string to the right is the same as magnetic force to the left. The distance AB between the midpoints of the two magnets and the hanging weight are recorded.
2. The screw mechanism is turned to push the magnet on the right to the left by some distance. This moves the rod away from the vertical position. Enough weight is added to the pan to bring the rod back to the vertical position. The new distance AB between the midpoints of the two magnets and new hanging weight are recorded.
3. Repeat.

One ends up with two columns of data, the repulsive force between magnets and the effective separation between the poles of the magnets obtained from the AB distance measurements. I see no flaw in this procedure and I trust the numbers that were obtained. If I didn't, I wouldn't spend my time analyzing them.

The data are what they are and you have no basis to make the claim

If a straight bar is suspended by a bearing at its top (like a pendulum) with a weight at its base (the magnet) it will obviously use quite a substantial amount of the energy that has been converted from the magnetic force. Therefore the readings should be higher than they are recorded.

If you want to bring in energy considerations to substantiate your claim, you need to generate a theoretical background and then show that the numbers don't fit in. It is not as obvious as you make it seem.

 

[pete94857]

I think you totally misunderstood the experiment. The readings are a direct measurement of the force between the two magnets. As Assoc. Prof. Usha Singh mentioned this is a null measurement. Here is how it works.

1. In the initial position the suspended magnet on the left is hanging in the vertical position, there is separation $r$ between the magnets and weight $mg$ is pulling on the vertical rod. The system is in equilibrium which means that the tension in the string to the right is the same as magnetic force to the left. The distance AB between the midpoints of the two magnets and the hanging weight are recorded.
2. The screw mechanism is turned to push the magnet on the right to the left by some distance. This moves the rod away from the vertical position. Enough weight is added to the pan to bring the rod back to the vertical position. The new distance AB between the midpoints of the two magnets and new hanging weight are recorded.
3. Repeat.

One ends up with two columns of data, the repulsive force between magnets and the effective separation between the poles of the magnets obtained from the AB distance measurements. I see no flaw in this procedure and I trust the numbers that were obtained. If I didn't, I wouldn't spend my time analyzing them.

The data are what they are and you have no basis to make the claim

If you want to bring in energy considerations to substantiate your claim, you need to generate a theoretical background and then show that the numbers don't fit in. It is not as obvious as you make it seem.

You maybe correct but there are still bearing resistance x2 and the fact the the magnets are not aligned as one is fixed to the bottom of a pendulum the other is held straight. Saturation is definitely a problem, plus i doubt the accuracy is correct. Although over all it would give a very loose estimate. As for producing an energy chart there's not nearly enough information available, materials, resistances etc.As you have now said you trust the data in the video previously you basically said you do not trust the data hence the charts and explanations you produced with respect I'm confused by your answers.

 

[kuruman]

As you have now said you trust the data in the video previously you basically said you do not trust the data hence the charts and explanations you produced with respect I'm confused by your answers.

I never said I do not trust the data. In post #3 I said

The analysis shown in the video is flawed primarily because it fails to convince the reader that the dependence of the repulsive force on pole separation follows an inverse square law.

This means that I disagree with the interpretation and conclusions of the presenter not with the data. I then proceeded to provide evidence why the analysis is flawed and, finally, provided the correct analysis of these very same data for the benefit of anyone reading this thread in the future. I am sorry you found it confusing.

 

[pete94857]

I see, I considered as you were disagreeing with the interpretation you were in a way disagreeing with the actual purpose of her attemps at proving the inverse square law as a way to calculate the force between 2 opposing magnets therefore in essence not trusting the data. Your interpretation of her data is a more sound approach this I can see. I'm going to make my own experiment to determine a more accurate result with an open minded approach. If you like I'd be happy to send you the results. I'm going to firstly test 1 magnet with a ferrous material to gauge an attractive reaction then 2 magnets in attraction and then in opposition. I will be using an old micrometer as a distance gauge. The magnets will be mounted to plastic at such a distance from any materials such as metals etc not to effect the results. With everything moving parallel and minimal friction although I'll try to account for it. I would appreciate your analysis on it.

 

[tech99]

The field strength from an isolated pole follows the inverse square law (according to Gauss). However, real magnets have two poles and the two are opposing. So at a large distance so we see the field falling off more rapidly than inverse square, depending on distance and orientation, but essentially at a distance broadside to the magnet at a distance more than the length we start to see the inverse cube law. Seems unnecessarily complicated to use two magnets for a demonstration as there are many confounding factors

 

[kuruman]

I would be happy to look at your experiment, but it isn't necessary to send it to me personally. Just post it on this forum for everyone to see and comment if they like. Be sure to explain everything in detail, and ideally provide a photograph or two.

A very simple approach, if I may make a suggestion, is to place two cylindrical magnets in repulsion inside a transparent plastic tube of slightly larger diameter than the magnets (see figure on the right). Measure their initial separation then add non-magnetic weights such as copper pennies on the top magnet and measure the distance between the magnets after each addition. The force between the magnets is the weight of the top magnet plus the added weights. Easy to explain and analyze, only one moving part, no fuss - no muss.

Clearly, you need to pay some attention to the accuracy of measuring the magnet-to-magnet separation. For starters, I would tape a 12-inch flexible plastic ruler on the containment tube and refine from there.

Good luck.