High Voltage Engineering

Permittivity is usually used to describe properties of dielectrics. As metals are usually relatively good conductors, the idea of how much capacitance might be developed with a gap filled with a metal compared to a vacuum isn't really all that useful. Likewise, attempting to talk about the ability of a metal to concentrate lines of electrostatic flux isn't all that helpful either.

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I think it is safe to assume the reference you found suggesting mild steel has infinite permittivity is wrong. Perhaps undefined (true from my perspective at the moment), perhaps imaginary, but not infinite.

The term permittivity is as inapplicable to metals as the term conductivity is to insulators.

I would have thought steel would have a value of 0, since a dielectric of steel would short out the capacitor giving a capacitance of 0. Or did you mean a mixture of the two? What is your intent?

The problem is that, at zero frequency, permittivity does not really make much sense for a metal in that the charges are so free they just move to the surfaces to cancel the field. This is distinct from a dielectric with fixed charges where the polarization really describes a local condition and permittivity is a local property of the material.

At frequencies in a body large enough for the carriers not to run into the surfaces you can give a local meaning to the polarization in the medium. Damping gives an out of phase term so J dot E is not zero. This is generally written as an imaginary contribution.

Imagine that first you have two metal plates with a slab of metal (steel) between them but separated by an insulating medium between plates and slab. This is equivalent to two capacitors in series.

Now make the insulating layers thinner and thinner. The capacities increase. As the insulation approaches zero, the capacities approach infinity. So by this argument the permittivity of the slab approaches infinity.

What happens when the thickness of the insulators becomes zero, Auntie?

Simple! - Then the permittivity goes "to infinity & beyond!!"

"...Now make the insulating layers thinner and thinner. The capacities increase. As the insulation approaches zero, the capacities approach infinity. So by this argument the permittivity of the slab approaches infinity...."

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This slight of mind can only be convincingly illusory if you manage to forget (or convince yourself to ignore) the conductive property of metal slab in between.

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If you don't yet see it, consider: Is it really necessary to have two sheets of insulating medium so as to separate the slab from the plate on both sides?...Wouldn't a similar phenomena occur with only one side of the slab separated by an insulator from one of the plates?

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Once you admit that a similar increase occurs as thinner and thinner dielectric sheets are compared on just one side of the slab, the flaw should be apparent.

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Any measurement of permittivity of a dielectric between two plates is like the measurement of permittivity of a dielectric between a plate and a slab in contact with the other plate. In fact, you can just arbitrarily decide where the plate ends and the slab begins, there doesn't even need to be a separation at that point, or even a real plate separate from the slab at all. If you hold the dielectric constant and change the slab plate division, it will become apparent that it does not have infinite permittivity.

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Alternately you could just use your original set up with two dielectrics (so as to separate both sides of the slab from the plates on either side)and instead of decreasing the thickness of the dielectrics, hold those steady. Just put increasingly thin layers of the slab in. The capacitance will stay roughly the same, but when compared against the permittivity of the vacuum for corresponding decreasing total plate separation, the result will indicate that permittivity of the metal would be highly dependent upon things like the permittivity and thickness of the dielectrics on either side.... which should make you question how appropriate the experiment is to the question.

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Can you give us a context for what you are trying to do? For example, you could make a cermet of materials, including metals if they do not percolate and it will have a well defined polarization and bulk permittivity. Ultimately, you apply a field and see how the medium responds. The definition of permittivity sometimes helps and sometimes is not applicable. Regardless, nature always has an answer.