# Calculate superconductor's magnetic susceptibility from Inductance

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1. Mar 27, 2016

### J.Sterling47

1. The problem statement, all variables and given/known data
Hi, I have some data taken on voltage and resistance of a coil part of a yttrium barium copper oxicde (ycbo) superconductor. I have no information about the coil itself. A thermocouple attached to the superconductor also measured the temperature of it as it was cooled. This data was used to calculate the inductance of the coil as a function of temperature. I need to convert this somehow to susceptibility as a function of temperature, but since I lack other data such inductance of the coil in a vacuum, I do not know if there is a direct way to do this.

2. Relevant equations
L is inductance, V is voltage across the coil, and R is the resistance.

L = sqrt( (V/I)2 + R2)

3. The attempt at a solution

If the inductance of the coil in a vacuum L0 could be determined (probably use online value if I can find it) then magnetic susceptibility X is

X = (L / L0 - 1)/α

Where α is the fraction of the coil occupied by the sample. This value I don't know but assuming I could find L_0 could I just approximate it to 1?

There is a similar equation using magnetic permeability

µ = µ0(1 + X)

but once again I do not have µ.

Another attempt was taking the derivative of my graph of inductance over temperature, and it gave a plot closer to the real plot of magnetization vs temperature, but I have not taken advanced E&M classes and do not know how to justify this.

So is there anyway to directly get X without knowing other things besides I, L, R and V? Or do I use these to find other quantities which I then use to get X?

2. Mar 28, 2016

### jwinter

I think you need to tackle the problem by considering the fraction of the area of the coil that is shielded by the superconductor - that is use your "X = (L/L0 - 1)/α" equation. I don't think you can get away without knowing this even if you wanted to try the permeability approach (which looks more difficult).

You will need to find the inductance of the coil with none of its area shielded L0, and you will either need to know (a) what fraction of its area the superconductor sample covers, or you will need to (b) assume that at a low enough temperature the susceptibility of the sample approaches -1 (complete blocking). In which case you can calculate what fraction of the area of the coil it covers from the ratio of the low temperature inductance at low temperature to the inductance at high temperature (or with the sample removed).

You can know the inductance of the coil from your measurement using your first equation L = sqrt( (V/I)2 + R2). You simply have to measure V and I in a situation where you know that the YCBO sample is not affecting the measurement - ie either removed completely or at such a high temperature (relative to its Tc) that it is just a piece of ordinary ceramic. This gives you L0.

So the simplest approach is to use your highest temperature readings to give L0 (by assuming X=0 at that temperature). Then determine the area shielding ratio α from your lowest temperature reading (by assuming X=-1 at that temperature). Then plot all your other points using these constants. You should obviously get an interesting curve for X which starts from 0 at high temperature and ends at -1 at low temperature.

Good Luck!

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