Clarification on magnetization and temperature

[IBphysics]

Summary:: Hello, I am doing a report for my IB course that requires some mathematics to be done. I have found this thread incredibly useful and I would like to reach out to ask for some clarification.

My experiment recorded the magnetic flux density directly above neodymium disc magnet, at different temperatures by submerging the magnet in water, the range of temperature is about 0-100 C. I plotted some diagrams in order to do analysis and here is where I ran into troubles, I am looking a theory or formula compare my results to, but most information I can find online are too complex. Here are the information I have collected and would like verification on.

Formula: M/M0 = ((Tc-T)/Tc)^γ

M = B(u0)

B = magnetic flux density (tesla/T)

M = magnetization (Am)

T = temperature (kelvin/K)

u0 = permeability of free space = 4π x 10-7

Remanence of neodymium ~ 1-1.3T

M0 = 1.3(u)

Tc = Curie Temperature = 320C = 593K
Please tell me if any of these definitions, formulas and constants are correct, I thank you in advance for your help.

 

[Charles Link]

Hello @IBphysics

See https://www.physicsforums.com/threa...tionship-in-ferromagnets.923380/#post-6599179 post 27.

Looks like you got some very good data. :)

Edit: additional comment: It would be nice if the data went higher in temperature, but within the constraints of using water for the temperature bath, the data is very good.

and with a closer look at the data, it appears your second set might have the decimal in the wrong place. $mT$ is $10^{-3} T$, and magnetization in A/m is $B (in \, Tesla)/(4 \pi E-7)$.

and I see you also don't have 1.0-1.3 T for the magnetic field strength at low temperature. How was the magnetic field measured? Was it with a sensor that was at some distance from what presumably is a cylindrical magnet? These extra details are needed to properly analyze the data.

 

[kuruman]

You are on the right track. I agree with @Charles Link that your data look good considering that you used water to make your measurements. Here are my suggestions if you wish to do more with this.
1. Experimental
I understand that you had to stop at 100 °C because you used water up to its boiling point. The Curie temperature for a neodymium magnet is about 320 °C. Have you considered going higher than that with something other than water, e.g. cooking oil? If you do that, please be very careful and take good precautions because hot oil can give real nasty burns.

2. Data analysis
Forget the linear and polynomial fits in Excel. They only put solid lines near the data points but the fitted parameters are not very informative. You have a theory that says $$\frac{M}{M_0}=\left(\frac{T_C-T}{T_C}\right)^{\gamma}$$ Unless you have measurements all the up to $T_C$, you can accept that T_C = 320 °C and see if you can pluck the critical exponent $\gamma$ from the data.

Use Excel to make a plot of $\ln\!\left(\frac{M}{M_0}\right)$ vs. $\ln\!\left(\frac{T_C-T}{T_C}\right)$. That should be a straight line with slope $\gamma$ which makes the graph much more informative.

 

[IBphysics]

Hello @IBphysics

See https://www.physicsforums.com/threa...tionship-in-ferromagnets.923380/#post-6599179 post 27.

Looks like you got some very good data. :)

Edit: additional comment: It would be nice if the data went higher in temperature, but within the constraints of using water for the temperature bath, the data is very good.

and with a closer look at the data, it appears your second set might have the decimal in the wrong place. $mT$ is $10^{-3} T$, and magnetization in A/m is $B (in \, Tesla)/(4 \pi E-7)$.

and I see you also don't have 1.0-1.3 T for the magnetic field strength at low temperature. How was the magnetic field measured? Was it with a sensor that was at some distance from what presumably is a cylindrical magnet? These extra details are needed to properly analyze the data.

Thank you for the quick reply! I measured the magnetic field strength using a sensor from Pasco which was connected to a laptop for data collection:

The magnets I used were circular, flat disc magnets:

I pointed the top of the sensor to the flat side of the magnets during measurements, the tip of the sensor was touching the magnet when measurements were taken.

This is a good point that you have raised, since I was unsure if I can simply cite the remenance of neodymium magnets, given that I couldn't in an experimental setting reach absolute zero. Would it be wise for me to reference an online source in order to find M0 for my formula?

 

[IBphysics]

You are on the right track. I agree with @Charles Link that your data look good considering that you used water to make your measurements. Here are my suggestions if you wish to do more with this.
1. Experimental
I understand that you had to stop at 100 °C because you used water up to its boiling point. The Curie temperature for a neodymium magnet is about 320 °C. Have you considered going higher than that with something other than water, e.g. cooking oil? If you do that, please be very careful and take good precautions because hot oil can give real nasty burns.

2. Data analysis
Forget the linear and polynomial fits in Excel. They only put solid lines near the data points but the fitted parameters are not very informative. You have a theory that says $$\frac{M}{M_0}=\left(\frac{T_C-T}{T_C}\right)^{\gamma}$$ Unless you have measurements all the up to $T_C$, you can accept that T_C = 320 °C and see if you can pluck the critical exponent $\gamma$ from the data.

Use Excel to make a plot of $\ln\!\left(\frac{M}{M_0}\right)$ vs. $\ln\!\left(\frac{T_C-T}{T_C}\right)$. That should be a straight line with slope $\gamma$ which makes the graph much more informative.

Thank you for your quick reply! I did include the use of oil and dry ice in my preliminary experiment design, unfortunately, my teacher rejected the use of hot oil and dry ice since the school did not have the appropriate apparatus to safely conduct this experiment / he didn't want me to make a mess and hurt myself. This partially explains why I had to experiment within the temperature range mentioned.

I understand what you are proposing and indeed this analysis would be greatly beneficial to my report, I have been wondering how to process the data and not appear descriptive, but also analytical. This way I can draw parallels between my experimental results, and published theories, which place γ between 0.32 and 0.39.

 

[kuruman]

Good luck with your report. When it's done and if you have the time, please summarize your findings here. I am curious to see how it all turned out. Thanks.

 

[Charles Link]

@IBphysics For a permanent magnet, the $M_o$ value depends on the shape of the magnet. The number 1.3 is for a long cylinder, (and for this case I believe the value of $B$ at the surface is .65 T). It is somewhat difficult to do any accurate calculation for a flat disc shape, but from first principles using pole theory and surface current concepts, the $M_o$ value will be much lower than 1.3, as you determined. [Edit: See post 13. The $M_o$ value is somewhat lower, but the magnetic field $B$ at the surface is much lower than even $\mu_o M/2$].

For the analysis, you might try a curve fit as @kuruman suggested, using the known $T_c$, along with your experimental $M_o$ of around $M_o=80$ E-3 T, and show that your data is consistent with this. You don't have data that takes you all the way to the Curie temperature, but that is ok. (Meanwhile, the data you have goes low enough in temperature, so that you can use an estimated value of 80 E-3 T for your curve fit. [Edit: See also post 8 below. You might find that your $M_o$ is closer to 100 E-3 T]). [Edit: See post 13 for a correction to this].

and please check your decimal point on the second set of data, as mentioned in post 2 above.

It might be of some interest, if you have some extra time, to get a set of data using either a longer cylinder or a number of discs stacked together.(You should then get a measured $B$ that is around .65 T at the surface). In any case, the data you already have is ok for a good analysis.

 

[Charles Link]

One additional comment in looking at the data, (I think you might find this to be the case if you try a curve fit with $T_c=593$), is that the data seems to trend downward somewhat sooner than T=320 C. If that is indeed the case, it could have something to do with the hysteresis curve, and the flat disc shape. In any case, you might find you get improved results if you do the experiment with a somewhat longer cylinder shape, rather than a flat disc. I look forward to seeing some results of the curve fitting.

See also https://www.physicsforums.com/threa...tionship-in-ferromagnets.923380/#post-6599179 post 19 for curve fits of some similar data.

 

[IBphysics]

@IBphysics For a permanent magnet, the $M_o$ value depends on the shape of the magnet. The number 1.3 is for a long cylinder, (and for this case I believe the value of $B$ at the surface is .65 T). It is somewhat difficult to do any accurate calculation for a flat disc shape, but from first principles using pole theory and surface current concepts, the $M_o$ value will be much lower than 1.3, as you determined.

For the analysis, you might try a curve fit as @kuruman suggested, using the known $T_c$, along with your experimental $M_o$ of around $M_o=80$ E-3 T, and show that your data is consistent with this. You don't have data that takes you all the way to the Curie temperature, but that is ok. (Meanwhile, the data you have goes low enough in temperature, so that you can use an estimated value of 80 E-3 T for your curve fit).

and please check your decimal point on the second set of data, as mentioned in post 2 above.

It might be of some interest, if you have some extra time, to get a set of data using either a longer cylinder or a number of discs stacked together.(You should then get a measured $B$ that is around .65 T at the surface). In any case, the data you already have is ok for a good analysis.

I have redid my math and I think I have the correct decimal places now.
Additionally, I would like to inquire if M0 is theoretically the measured B at absolute zero (0K / -273C)?
You previously mentioned using my experimental results of M0 = 80 E-3 T, wouldn't this be incorrect since my experimental data are in Celsius? Or am I confused about the definition of M0?
Additionally, is there a source for the 0.65T data? I previously planned to use M0 = (1.3)/(4*3.14*10^-7), it seems this is too optimistic of an estimation.
Attached is the new graph of Magnetization and Temperature in Celsius.

 

[Charles Link]

Yes, the $M_o$ is for very low temperatures. See the "link" in post 8 for how the shape of the graph will look as you go to lower temperatures.

For a flat disc $M_o$ will be much smaller than what you computed using the 1.3 T. (see post 7 above).

Your data looks good. You need to try a couple curve fits with your formula,($M/M_o=((T_C-T)/T_C)^{\gamma}$), where you go from T=0 K to T=T_c, and you can try a couple different $M_o$ values. With your latest data set, I think I would try an $M_o=90000$ A/m with a $T_C=593$ K and see how it intersects your data. $\gamma=.36$ should work ok.

Be sure and see post 27 of
https://www.physicsforums.com/threa...tionship-in-ferromagnets.923380/#post-6599179
for units of $M$, etc.

 

[IBphysics]

Yes, the $M_o$ is for very low temperatures. See the "link" in post 8 for how the shape of the graph will look as you go to lower temperatures.

For a flat disc $M_o$ will be much smaller than what you computed using the 1.3 T. (see post 7 above).

Your data looks good. You need to try a couple curve fits with your formula,($M/M_o=((T_C-T)/T_C)^{\gamma}$), where you go from T=0 K to T=T_c, and you can try a couple different $M_o$ values. With your latest data set, I think I would try an $M_o=90000$ with a $T_C=593$ K and see how it intersects your data.

I tried different values for M0 from 1T to 0.08T (Maximum B measured during experiment at 274 K), the γ value measured by the log vs log graph seems too high at 0.7, should I change the curie temperature for this analysis?
Also, should I use absolute values for the log graph?

 

[Charles Link]

Looks like my preliminary assessment in post 8 might be accurate=the data does seem to trend downward before the Curie temperature. The best recommendation I have is that you might try getting a data set with a longer cylinder magnet. (The flat disc is really a poor geometry for trying to match theory with experiment for a couple of reasons. The magnetic field it has is much weaker than that of the longer cylinder, and with the weaker magnetic field, it may lose its permanent magnet nature somewhat below the Curie temperature.)

For the log-log graph, multiplying the numbers, (e.g. the magnetization), by a constant will not change the slope of the graph. You could try adjusting the Curie temperature=it is possible, as mentioned above, that the shape of the magnet (being a flat disc) is resulting in the magnet losing its permanent magnetism sooner than the Curie temperature=I'm going to have to try to google that to see what comes up. The literature does seem to be somewhat lacking on this topic.

See https://www.azonano.com/article.aspx?ArticleID=5353

They do mention in this article, that longer magnets are less susceptible to demagnetization from heating than thin ones. That confirms what I thought was rather likely.

 

[Charles Link]

See https://www.physicsforums.com/threads/magnetization-of-ferromagnetic-material.1011486/#post-6590317
posts 5 and 8 for the results of a couple of calculations for the magnetization values for a (similar material) permanent magnet with different shapes=namely a toroid with a small gap, and a sphere. It uses the hysteresis curve that came supplied with the problem in the OP. For a flat disc, the magnetization could be expected to be even less than that for a sphere. A long cylinder should give a result that is similar to, (perhaps slightly less than), a toroid with a small gap.

I could add that the above problem, which just appeared on Physics Forums, is the first one of that type that I have seen. Perhaps there is starting to be an increased interest in problems of this type.

In any case, I would be very interested in seeing the results for a repeat of this experiment with a magnet that is a longer cylinder if @IBphysics is able to get some additional data.

One other item of interest: There are basically 2 different types of MKS units for the magnetization $M$ in common use: The older textbooks like to use the formula $B=\mu_o H+M$, while the newer textbooks seem to be switching to $B=\mu_o H+\mu_o M$. For the first formula, $M$ has units of $T$,(Tesla=Webers/meter^2), while the second one will have units of $A/m$, (amperes/meter).

I prefer using $T$, but some others prefer the $A/m$. (In much of this, I used the second convention for $M$, but when I mention $M$, I use it as $\mu_o M$ so it relates to units of $T$.)

One other item: For your data, the magnetic field $B$ is always proportional to the magnetization $M$. The magnetization $M$ for the flat disc is difficult to compute precisely from the magnetic field $B$ at the surface, but we can always write $B/B_o=M/M_o \approx ((T_C-T)/T_C)^{\gamma}$. For your graphs, a curve of $B/B_o$ vs. $T$ may be the best way to present it.
(For the long cylinder, the calculation is easier, with the result $B_{surface}=\mu_o M/2$. For the flat disc, $\mu_o M$ will be considerably greater than $2 B_{surface}$, but good estimates of exactly what it is are difficult for that shape.(Note that for the flat disc $B_{surface}$ will be considerably less than $\mu_o M/2$, and $\mu_o M$ is likely to be somewhat less than 1.3 T). Note: Because of the flat disc shape of the sample, we don't have a good number for $\mu_o M_o$. Because of the shape, calculations at the top of this post, (see the "link"), suggest it may be considerably less than 1.3 T, but it is likely to be considerably more than $2 B_o \approx 160$ E-3 T ). Note: Post 7, paragraph 2 is incorrect, and I will place an edit there, and also at paragraph 1. We have a $B_o$, but we don't have a good number for $M_o$ from the $B_o$. (additional note:: Given precise dimensions of the disc, it might be possible to do some approximate calculations to determine the proportionality constant relating $M$ to $B_{surface}$.
Edit: and one result is $B_{surface}=\mu_o Md/(2R)$, where $d$ is the thickness of the disc, and $R$ is the radius. The calculation should work so long as the disc isn't real thick). See also https://www.physicsforums.com/threa...tionship-in-ferromagnets.923380/#post-6599179 post 6 with $X=0$ and $d=L$. The result is $B_{surface}=\frac{\mu_o ML}{2 \sqrt{L^2+R^2}}$. This should work for any $d=L$.