Experiment measuring the distance dependence of a suspended metal bolt from a magnet

[Sidsid]

Homework Statement

To find the relationship between the deviation of a suspended mass and the magnetic field

Relevant Equations

The used potential (this might not be accurate but part of the experiment is seeing how accurate this is) is: V_mag=kB/r^a. B is the 'magnetic strength' and a is a constant.

My set-up is the following: i have an iron bolt suspended on a string next to an electromagnet, of which I steadily increase the voltage and thereby the magnetic field. Supposing the force is linear with the magnetic field and dependent on the distance between the bolt and magnet. The exact dependence is what I seek, but assume that it is ~ 1/r^a with a an undetermined constant. My goal is to see if it is an inverse square force (which would mean a= 2) or an inverse cubic force (a=3) What is the relation between the deviation of the bolt (𝑠𝑖𝑛(𝜃)) and the strength of the magnetic field? The data seems to point to a cubic relationship (𝑠𝑖𝑛(𝜃) ~ 𝐵^3) but I cant find why, any approximations ( even wild ones) are welcome.
I have tried Lagrange and force diagrams and that gave me sin(\theta)(z-lsin(\theta))^a=B (z is horizontal distance at equilibrium), but this doesnt seem to make sense. Thank you!

 

[haruspex]

The force would correspond to ##\tan(\theta)## rather than to ##\sin(\theta)##, but I assume the angles were small.
Please show your working for the equation you got.

 

[Sidsid]

I changed it so its only horizontal distance that matters, which is quite a wild approximation. This example is for a=2. But im beginning to think my problem is more that I dont know how to fit with this function, as B is my independent variable, not theta.

 

[Sidsid]

and yes there is a tan but under small angles tan approximates to sin which is handy for a B/sin plot. Also my equation now has a minus? Very annoying.

 

[kuruman]

The exact dependence is what I seek, but assume that it is ~ 1/r^a with a an undetermined constant.

What is distance ##r##? As a distance between two points, one of the points is, of course, the hanging bolt. At what point is the other end and why did you specifically pick that point? The electromagnet sets up a central field that goes as ##1/r^3## in the approximation that its size is much smaller than the distance to the point of interest. Otherwise, you are looking at a field like the one shown on the right. What is the equilibrium distance ##d## and why?

 

[Sidsid]

I am using a bar electro-magnet so there is indeed a bit of issue of what r is. I used the center of the closest side. To make it more simple i neglected the vertical distance and only focused on thevertical distance. The horizontal distance between the magnet and the bolt when the magnet is turned off is the equilibrium distance. As for the $1/r^3$ that is exactly what im trying to test experimentally. I have two approximations of the electromagnet one where it goes with $1/r^3$ and one with $1/r^2$ and I want to show with a fit comparison that one is correct and the other isnt.

 

[Sidsid]

and to add: my measurements are of the horizontal deviation (sin) at certain voltages. And I verified that the field (and I assume force) goes linearly with the voltage. this is the data, the fit is a simple cbuic fit, because initially I thought it would be a simple model, but then I found flaws in my reasoning.

 

[kuruman]

I used the center of the closest side.

You know that the on-axis magnetic field due to a circular ring of radius ##R## and current ##I## is $$B(x)=\frac{\mu_0IR^2}{2(R^2+x^2)^{3/2}}$$where ##x## is the distance from the center of the ring to the point of interest. If you have a coil with ##N## turns and length ##L##, the contribution of a coil element of length ##dx## is $$dB=\frac{Ndx}{L}\frac{\mu_0IR^2}{2(R^2+x^2)^{3/2}}.$$Integrate this from ##x## to ##x+L## to get a better expression for the dependence of the B-field on ##x## in terms of the geometry of your coil and the number of turns.

Also, it is better to use current instead of voltage because the magnetic field is proportional to current according the Ampere's law. The current could be proportional to voltage but only if the resistance is constant. That is not necessarily the case when the current is high enough to raise the temperature of the coil.

 

[Sidsid]

I appreciate your help, but it is not exactly what I need. I am trying to investigate what the field is, so I can't really start from the equation of the field. That is why I approximate it to a force that goes like $$ \frac{k B}{r^a}.$$ For r I use the horizontal distance ##(z-lsin(\theta))##, B is just a quantity that I assume I can vary linearly with the voltage (sorry there was no meter for the current, but I checked and it does go linearly for my set-up). So B is my independent variable. And importantly, I just want to see if a=2 works better than a=3. I am aware, that these are big simplifications but as my goal is just: which one works better I think it'll work. One of the other reasons why I dont want to overcomplicate things, is that I dont have that much space for my paper and not many datapoints. I hope this clarifies my goal.

 

[kuruman]

I appreciate your help, but it is not exactly what I need. I am trying to investigate what the field is, so I can't really start from the equation of the field. That is why I approximate it to a force that goes like $$ \frac{k B}{r^a}.$$ For r I use the horizontal distance ##(z-lsin(\theta))##, B is just a quantity that I assume I can vary linearly with the voltage (sorry there was no meter for the current, but I checked and it does go linearly for my set-up). So B is my independent variable. And importantly, I just want to see if a=2 works better than a=3. I am aware, that these are big simplifications but as my goal is just: which one works better I think it'll work. One of the other reasons why I dont want to overcomplicate things, is that I dont have that much space for my paper and not many datapoints. I hope this clarifies my goal.

So you are throwing out a perfectly good theory that you already have, invent your own and see if it matches your experiment. OK. Your expression is $$F~\text{~}~\frac{k B}{r^a}.$$Now ##B## depends on position and is not constant. How are you going to separate what part in the dependence of the fraction on ##x## is in ##B## and what part is in ##r^a~##? Also, why do you think that the exponent ##a## is an integer?

 

[Sidsid]

okay B was a bad choice of letters, I simply meant that as a quantity that I'm going to vary with the voltage, equivalent to electric charge in Coulomb's law. And what I concretely am doing is approximating the magnet as a monopole (F~##\frac{1}{r^2}##) and as a dipole (B~F~##\frac{1}{r^3}##). So that's why I choose a=2 and a=3. Again, I know this is quite incorrect, but this is the path I've chosen. I'm just comparing which one would work better.

 

[kuruman]

I'm just comparing which one would work better.

And how have you defined "better"? What quantitative criterion are you going to use for that?

Again, I know this is quite incorrect, but this is the path I've chosen.

Choosing a path that one knows is "quite incorrect" is not the way science progresses. What useful information is gained by knowingly determining whether one incorrect path works "better" than another incorrect path? This is different from narrowing down the scope of one's research by experimentally determining what cannot possibly work so as not to waste time on it.

Anyway, good luck with you project. I have offered all the help I can.

 

[nasu]

You assume that the field is proportional to the current (or voltage). But this will be the case only if the point of interest is always the same relative to the magnet. But here you have the bolt at various distances from the magnet, don't you? Is the suspension point of the bolt fixed as you increase the current? If this is the case you should change the conditions do that the distance between the bolt and magnet at equilibrium is the same for all values of current. Only the angle will be different. The field of the magnet is not uniform.

 

[Sidsid]

You have all mentioned important problems, which have caused me to step back and review my set-up. One note, I meant that 'magnetic charge' is proportional to current or voltage, but I know that that is a dated concept. Thanks.