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I am speaking, here, not in the more usual sense of a gap in the continuity of the core material. I mean, in reference to a rod of core material, that the outside diameter of the rod is sufficiently smaller than the inside diameter of the coil, that a uniform separation exists between the core and the coil.

The question is, does such a gap affect the inductance of such a system. I suppose a more general question would be how inductance would be affected if the coil were wound directly on a core whose permeability varies with diameter - that is, would such a system exhibit inductance different from a system of the same dimensions but with a core whose permeability is an average of the variable one.

To give a practical example, suppose I wind a coil on the outside of a piece of (nominal) 1/2 inch pvc pipe, whose outside diameter is 0.84 inches. I insert a rod of given permeability with an outside diameter of 0.622 inches (the inside diameter of nominal 1/2 inch pvc pipe).

Clearly, the inductance of such a system will be different than the case in which the coil is wound directly on such a core (smaller diameter coil, lower inductance), or in the case of a larger diameter rod (&c).

I have not found any material referencing such a system (possibly because the effects of such a system are so negligible as to be of no interest).

A further question might be of a system in which a layer of impermeable material fills the gap between the coil and a core of permeable material.

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    \$\begingroup\$ but with a core whose permeability is an average of the variable one. - now you're getting into effective permeability, and you can define that in a way that the two inductances are equal. Which means you're playing around with your defintion of 'average'. \$\endgroup\$ – Neil_UK Dec 30 '19 at 15:45
  • \$\begingroup\$ Mu * increased air gap / core ratio reduces inductance and force. Ferrite is a distrbuted core particles. \$\endgroup\$ – Tony Stewart EE75 Dec 30 '19 at 15:57
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If you do the maths on a coil wound on a full core it has this inductance: -

$$L = \dfrac{N^2}{\mathcal{R}}$$

Where R is the reluctance of the core and N is the number of turns. L of course, is inductance.

So, if instead of using the reluctance of the ferrous core you regarded it as two reluctances in parallel (one a tube of air and one a core of ferrite) you would get your answer.

Reluctances in parallel work just like resistances in parallel so, if the tube of air has 100 times the reluctance of the ferrite core then the net reluctance compared to a full ferrite core might increase by only 1%.

See also my answer to the question coil area versus core area. It seeks to justify the equations.

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