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Calculations of a half loop electromagnet

  1. Apr 26, 2015 #1
    1. The problem statement, all variables and given/known data
    This is a past exam question which I just for the life of me can't figure out.

    2eq6ryo.png
    http://tinypic.com/r/2eq6ryo/8


    2. Relevant equations
    My guess is that the energy of the magnetisation is equal to the gravitational energy

    Relevant Equations:
    Ems = Mass * g * δ

    Where
    Mass = ρ * ACross Section * π * r
    (not 2πr as we are only using half the loop, also cross sectional r is negligible.)



    3. The attempt at a solution

    I don't know what the energy of magnetisation is, and definitely don't know how to get to it via the current. Any relevant equations would be much appreciated.
     
  2. jcsd
  3. Apr 26, 2015 #2

    Hesch

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    Gold Member

    The energy-density in a magnetic field, Ed = ½*B*H [ J/m3 ].

    If the (constant) B-field along the toroid is known, you calculate the H-field in the airgap from: B = μ0*H.

    Total energy in the airgap is E = Ed*(volume of airgap).

    The force in the airgap is calculated by: F = dE/ds. ( s = "small delta", (lower case delta?) )
     
    Last edited: Apr 26, 2015
  4. Apr 27, 2015 #3
    So

    Ems = ρ * A * π * D/2 * g * δ = VAirGap * Ed = 1/2 * δ * A * 2 (as there are two air columns) * Ed

    D/2 * g * ρ * π = 1/2 B * H
    D = (B * H) / (g * ρ * π)

    Then at this point I have one more question

    Do we have H = M, B = mu_0 H

    Or the Biot Savart law for B from i

    Thank you for the help
     
  5. Apr 27, 2015 #4

    Hesch

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    Gold Member

    You cannot use Biot-Savart as you don't know how the turns of the coil are placed. Use instead Amperes law:

    The circulationintegral: ∫ Hds = N * I = 2π*RmeanH , Rmean = ½(R+r).

    I don't quite understand the text below the figure in #1, but I think that Hiron = 1.7 A/m is meant. So you don't have to calculate the above at all.

    Now assume that μr = 1000 as for iron. Then

    B
    = μ0 * μr * H

    in the whole toroid (airgaps included). In the airgap you can find: H = B / μ0, so now B and H are known in the airgap. ( Hair = μr * Hiron ).

    (The flux in magnetizm is as current in an electric circuit: nothing will disappear, and nothing will be added: Kirchhoffs current law).
     
    Last edited: Apr 27, 2015
  6. Apr 28, 2015 #5
    That makes sense!

    Thank you very much for the help Hesch!
     
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