# Calculations of a half loop electromagnet

1. Apr 26, 2015

### Tweej

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.

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. Apr 26, 2015

### Hesch

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
3. Apr 27, 2015

### Tweej

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

4. Apr 27, 2015

### 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: ∫ 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
5. Apr 28, 2015

### Tweej

That makes sense!

Thank you very much for the help Hesch!