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# 12.2: Magnetic Circuits and Ohm's Law

Some people find it helpful to see an analogy between a system of solenoids and various magnetic materials and a simple electrical circuit. They see it as a "magnetic circuit". I myself haven't found it to be particularly useful – but, as I mentioned, my experience in this field is less than extensive. I think it may be useful for some readers, however, at least to be introduced to the concept.

The magnetic field inside a long solenoid is given by $$B=\mu nI= \mu NI/l$$. Here, $$n$$ is the number of turns per unit length, $$N$$ is the total number of turns, and $$l$$ is the length of the solenoid. If the cross-sectional area of the solenoid is $$A$$, the $$B$$-flux is $$\Phi_b = \mu NIA/l$$. This can be written

$NI = \Phi_b \times \dfrac{l}{\mu A} \tag{12.2.1}\label{12.2.1}$

The analogy which some people find useful is between this and Ohm's law:

$V= I\,R \tag{12.2.2}\label{12.2.2}$

The term $$NI$$, expressed in ampere-turns, is the magnetomotive force MMF.

The symbol $$\Phi_b$$ is the familiar B-flux, and is held to be analogous to current.

The term $$l/(\mu A)$$ is the reluctance, expressed in H-1. Reluctances add in series.

The reciprocal of the reluctance is the permeance, expressed in H. Permeances add in parallel.

Although the SI unit of permeance is the henry, permeance is not the same as the inductance. It will be recalled, for example, that the inductance of a long solenoid of N turns is

$\dfrac{\mu AN^2}{l} \nonumber$

Continuing with the analogy, we recall that resistivity = $$(A / l)\, \times$$ resistance;

Similarly reluctivity = $$(A / l)\, \times$$ reluctance = $$1/\mu .\, (\text{m H}^{-1})$$.

Also, the reciprocal of resistance is conductance.

Similarly, the reciprocal of reluctance is permeance. (H)

And conductivity is  $$(l / A)\, \times$$ conductance.

Likewise $$(l / A)\, \times$$ permeance  is – what else? – permeability $$\mu .\, (\text{H m}^{-1})$$.

I have mentioned these names partly for completeness and partly because it's fun to write some unusual and unfamiliar words such as permeance and reluctivity. I am probably not going to use these concepts further or give examples of their use. This is mostly because I am not as familiar with them myself as perhaps I ought to be, and I am sure that there are contexts in which these concepts are indeed highly useful. The next section introduces some more funny words, such as magnetization and susceptibility – but these are words that you will need to know and understand.