# 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.