E = -L di/dt => - 224x10^-6(0.036) => 8.064 micro volts.

That's the instantaneous Voltage (E(t)) across an Inductor (L) through which a current I(t) is circulating. That's not the rms voltage which is what you are after. In order to get the rms voltage you would need to perform complicated calculations with that E(t) including integration in time (http://en.wikipedia.org/wiki/Root_mean_square). If it is a periodic signal then the rms of the signal is equal to that of one period. Since all that is not very practical to do, then what you do, is to measure the rms voltage. To do that with the Oscilloscope, you measure the peak to peak Voltage (Vpp) and Vrms=Vpp/sqrt(2) if it is a sine wave signal, as I suspect. If its not a sine wave, then that's not the equation. Another way is to use a true rms multimeter capable of operating at that frequency (20KHz in your case) and it will read the rms voltage directly for you. That could be the reason for the differences you are observing, depending on which multimeter you are using. You can read the specs and find out if it can operate at that freq. I studied these things long ago and might not remember the details very well though.

I = Vac/ XL = 10 / 2*Pi*20*10^3*224*10^-6 => 0.36;

That looks somehow true, except for the fact that the 10Vac is applied to the serial combination of the multiplexer and the coil. That creates a voltage drop across the multiplexer also depending on its ON resistance and the current passing through. Therefore the 10Vac is not applied in its totality to the coil, part of it is lost in overheating the multiplexer. If too much, multiplexer blows.

So in reality it looks like:

I=Vac/(ZL+Ron)

and

VL=Vac*ZL/(ZL+Ron) VL-(Inductor Voltage) Ron-(Multiplexer ON Resistance) ZL=R+XL (Inductor Impedance)

The multiplexer datasheet should tell you its ON Resistance. In a normal case, the multiplexer Ron should be way less than the Impedance of the load you are driving through it, so the voltage drop across it is negligible and most of it gets applied to the load where you want it.

Furthermore, you are not considering the Ohm Resistance of the coil and calculating its impedance (Z=R+XL). That could be acceptable, if it is very low compared to its XL at the operating freq; but I have no idea about the wire gauge they are made of and the amount of turns which directly affect its Ohm Resistance.

Anyways, at f=20KHz, XL=2*PI*20*10^3*224*10^(-6)~28 Ohms. That is low and very well in the same order of magnitude of the inductor Ohm Resistance, therefore its Z may differ substantially from its XL. Please measure the coil resistance with a regular Ohmmeter to check how much it is. Furthermore, that low XL value could be also in the same order of magnitude of the multiplexer ON Resistance (I don't know); please check that also. If that's the case then most of the Vac is lost in the multiplexer and not in your useful load (the coils)

I have the impression the 224 uH value you are mentioning may not be right though. That looks like and odd value (particularly the 4 at the end). Like caps and resistors they are "mostly" manufactured in series of standard values (from what I have seeing). If that's what you are reading on the coil itself, then the 4 may mean the multiplier and the value could be 0.22uH or 22uH instead (I don't know). If you have not done it yet, I suggest you use an inductance or RLC meter and double check the inductors' real values. Can you post a picture of the actual coil you are using?

EDIT:By the way, I made a mistake when explaining to you how to measure the rms voltage with the Oscilloscope .

To do that with the Oscilloscope, you measure the peak to peak Voltage (Vpp) and Vrms=Vpp/sqrt(2)

Wrong, it should be the "peak voltage" (Vp) and not the "peak to peak voltage" (Vpp). Peak voltage means from zero to peak. Therefore:

Vrms=Vp/sqrt(2).

Sorry about that, too many things going on at the same time, I guess...