Ferrite core

Do you have a link to the spec sheet?

Strictly speaking, it can be used at 500kHz. But you should linearly extrapolate from your data sheet to guess how much permeability will still be available at 500kHz. Without a link to your data-sheet, I cannot give you an estimate what will be a good nominal value. (Also I normally get paid when I do this. ) I would not be surprised if your relative permeability guess puts you very close to a value of 1, if not less than 1. This will mean that your ferrite core will actually be nothing more than a ceramic bobbin to hold your wires. The good thing with this is that you will not have to do all of the other calculations to figure out at what magnetic flux density will the core saturate, since you will already be saturating this core. This will also ease your inductor stored power calculation ,for you won't have to consider the higher harmonic roll off effect turning your square wave into a sign wave since the fundamental frequency will be outside of core's designed range. The draw back of doing this is that since you will be actually using the permeability of free space, you will be transmitting your power very well out into free space. So don't expect to get all of it back.

So now that I've shoveled enough to hopefully dissuade you, allow me to sum this up. Yes, you can use this core at 500kHz. But you're not going to like it.

What is the initial permeability?

Is that the value of the permeability at the point where the hysteresis loop cuts the horizontal axis( where B=0) ?

What is the initial permeability?

Of what, free space or a vacuum (μ0)? Well that's defined as 4π * 10^-7. (π is supposed to be pi.) If you're asking about the particular ferrite core material, like most materials that changes with the frequency of the magnetic field. At the frequency of interest, the manufacturer does not specify what should be the relative permeability. Whatever the permeability at DC or in the frequency band stated by the manufacturer almost doesn't matter. We have no idea how steep the cliff edge is so knowing the height of the cliff will not help identify what the permeability is at any frequency. We do know that the manufacturer does not recommend operation in these frequencies since no data is given for these frequencies.

Is that the value of the permeability at the point where the hysteresis loop cuts the horizontal axis( where B=0) ?

No, magnetic hysteresis is the effect of some magnetic material to retain a magnetic field after being externally magnetized. Until one reaches a magnetic flux density known as saturation, the permeability remains the same. Technically the permeability changes just before saturation, but since this is nearly a positive feedback scenario once the permeability starts to drop saturation soon follows.

A good site on the magnetic properties of materials.

GA... for an excellent link

At reduced performance.