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Need some review, about magnetic fields

  1. Oct 31, 2008 #1

    I've been doing some things for the past couple of weeks just for fun, but I have no way of knowing if I'm on the right track or not, so if anyone is willing to take a glimpse, feel free to point out any errors or say harsh thing, or throw tomatoes at me, because I need some objectivity here.

    Before I say anything, I'm using SI units, Teslas for B (induction/field), A/m for H (field strength) , space in meters etc.

    Ok, first thing is a calculation of force within a finite solenoid, on a diamagnetic target anywhere inside or around of it.

    I started from the basic equation for the field around a coil which is:

    this one here:

    Then I used numerical integration to integrate that over the length of the solenoid
    (assuming the turns are small enough to be considered a smooth surface of even current distribution). The coordinate system uses bottom end of the solenoid as the z=0, and
    solenoid axis as the r=0.

    Here is how I did that:

    a bit of clarification on the letters used:
    z is obviously z coordinate, and r is r coordinate in a cylindrical system, a is radius of coil,

    both z, r and a are helper variables for integration, during which they are defined by other parameters, which are:

    d1 , is the distance from bottom end of the solenoid (equivalent to z coordinate of the point where the target is), d is the length of the solenoid.
    (so integration goes from one end of the solenoid to the other)

    I have also included the thickness of the coils, because a lot of these magnets (like Bitter magnets) have thick disks instead of just thin wires, so thickness needs to be included.
    This is integration over a (radius of infinitesimal coil), which goes from R which is the inner radius to R+x which is the outer radius.

    i[z] is the current (I have made it into a function to have more flexibility in case needed, but usually it is just a constant)

    Don't be confused by the 1.1/1.1 , I have included that because of some kind of a bug in Mathematica which refuses to calculate elliptic integrals in some instances (doesn't recognize the fraction), this is the only way I could get it to work for all values of m. Just ignore it.

    Ok, I hope its all clear so far.

    Let's move on.

    To calculate force I've used these expressions:


    To clarify further...

    I've assumed a very small diamagnetic target of volume V (so I don't have to integrate over its dimensions, but I'm not ruling that out later, once I establish that this approach is accurate (or not)), and its suceptibility of X (greet letter..)

    We finally get to the input parameters for position of the target, which are pr and pz.
    So in a nutshell , pr dictates r .
    To get 4 gradients, I'm using numerical derivative (ND) of r and d1 (remember d1 is the equvalent of z coordinates of the target), around pr and pz.
    I have to use these replacement variables because every time I use numerical derivative or integration functions, I need a free undefined variable to work with.
    This is why I first input pr which dictates r, and pz which dictates d1 which dictates z.

    Ok, I hope I was clear enough, but if not, please ask specific questions about which part does what if its unclear.

    I have no way of testing this. The field results are in the right ballpark.

    For example, I tried testing this using the famous floating frog experiment. Now the authors gave some rough dimensions of the Bitter magnet they used, and current on their site, so I was able input that to see if I get the claimed 20T from that magnet. I got 26T, which I think is near enough considering I didn't take into account all the deformations of the magnet geometry due to cooling holes and screw holes and who knows what else. Also they might have used slightly less than specified current (they roughly specified that they had a 20kA power source, but the current used could have been slightly less)

    But the radial force I get at say half of the radius, is not even enough to levitate
    a piece of Bismuth (I get around 10 times less force than I need) , much less a frog. So I'm suspicious about my equations, and wonder if I made some errors.

    So any help would be appreciated
    Last edited: Oct 31, 2008
  2. jcsd
  3. Nov 1, 2008 #2
    ...just an update... while comparing this to the frog experiment I mad a silly error, I assumed they were levitating the frog using radial force in a horizontal tube, but the tube was vertical and the put the frog in the "sweet spot" and used vertical component near the end of the solenoid.
    So I repeated the test for a piece of bismuth and I get about equal force to gravity for a piece of bismuth (slightly more I think) somewhere near the end of the solenoid.
    But still, Bismuth has an order (or two) of magnitude higher susceptibility than anything that could be found in a frog, or a drop of water or a nut, so I'm still suspicious.
    If at that spot a strawberry can levitate, a piece of bismuth should shoot out like a popcorn.
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