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For example, could you determine the resonant frequency of slag in an ore sample and melt it by putting it in an electric resonator at the appropriate frequency?

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Can an electric resonator (like a microwave) be used to effectively heat things other than water at different frequencies?

MAYBE as long as it's not explosive like petroleum-based.

Microwave is not the most efficient, but it conveniently fast and does have the advantage of a rotisserie and good penetration by radio waves less sensitive to the insulation of air. However, it does work well with any other mediums when there is an ionic transport dissolved in it. i.e. salt

e.g. Pizza or leftover salty chicken heats up very fast while bee's wax stays cold for a long time and is also an excellent electrical insulator like deionized water which has very extremely conductance at DC and 60 Hz.

There are also variable frequency induction heaters in large industry that use very large coils and capacitors perhaps the size of oil drums with electrodes filled with oil to generate very high voltages and currents at relatively low frequencies (100kHz).

could you determine the resonant frequency of slag in an ore sample

Yes, you can determine the RLC properties yes with a vector impedance meter for a known geometric shape. But rarely, would it have a sharp (high Q) resonant frequency even highly crystalline coal (graphite). High electrical Q's are derived by insulators more than conductors and I guess slag is in this middle zone.

For example, pure Quartz is highly insulating but between gold plated electrodes with a critical angle of cut will appear as a lossy inductor of several Henries and fempto-farads of motional piezo effects that create the high Q's of 10k. Several Henries is like a good solenoid or relay but then electronic crystal or MEMs resonator can only handle 10uW typ. of heat in the crystal lattice due to the nanosize and large power density

But I hope someone proves me wrong. That doesn't mean there isn't a resonant frequency of the industrial arc furnace due to the plasma gas electrode gap to slag, because there will be and that's about the best radio jammer you can find. Also, the megawatt transformers used for smelting must be designed in a special way to tolerance these negative resistance high arc currents.

and melt it by putting it in an electric resonator at the appropriate frequency?

Unless purified and strictly controlled by geometric portions, that won't work. I believe, the melting process is done by conduction and the plasma gas air buring in contact after an arc is struck.

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