Calculations of a half loop electromagnet

[Tweej]

Homework Statement

This is a past exam question which I just for the life of me can't figure out.

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

Homework Equations

My guess is that the energy of the magnetisation is equal to the gravitational energy

Relevant Equations:
E_ms = Mass * g * δ

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

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.

 

[Hesch]

The energy-density in a magnetic field, E_d = 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 = E_d*(volume of airgap).

The force in the airgap is calculated by: F = dE/ds. ( s = "small delta", (lower case delta?) )

 

[Tweej]

So

E_ms = ρ * A * π * D/2 * g * δ = V_AirGap * E_d = 1/2 * δ * A * 2 (as there are two air columns) * E_d

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

 

[Hesch]

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

The circulationintegral: ∫ H⋅ds = N * I = 2π*R_mean⋅H , R_mean = ½(R+r).

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

Now assume that μ_r = 1000 as for iron. Then

B = μ₀ * μ_r * 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. ( H_air = μ_r * H_iron ).

(The flux in magnetizm is as current in an electric circuit: nothing will disappear, and nothing will be added: Kirchhoffs current law).

 

[Tweej]

That makes sense!

Thank you very much for the help Hesch!