# Basic question about magnetic circuits

I am just learning about magnetic circuits and I'm trying to solve the simplest possible circuit you can have, but I seem to be getting a nonsense unphysical answer and I'm not sure why.

Say we have a magnetic core loop (with $$μ \gg μ_0$$) of length $$L_c$$ , with a small air gap of length $$L_g$$ , all with constant cross sectional area $$A$$. Now if we put an excitation coil with magnetomotive force $$F$$ on the core, there's gonna be some magnetic flux $$Φ$$ flowing through the circuit, where we assume there's no leakage. Here are my assumptions that lead to something that doesn't make sense:

1. The magnetic field $$B$$ has the same magnitude inside the core and in the air gap, since $$B=Φ/A$$ and the flux and area are the same everywhere.
2. The field inside the core is $$B=\frac{μF}{l}$$, generated by the solenoid with length $$l$$ (the standard formula, with $$F=NI$$ ampere-turns)
3. The reluctance of the circuit is $$\frac{F}{Φ}=R=R_c+R_g=\frac{L_c}{μA}+\frac{L_g}{μ_0A}$$

Now putting all this together, I want to find the actual length $$l$$ of the solenoid: $$l=\frac{μFA}{Φ}=μRA=L_c+\frac{μ}{μ_0}L_g$$.

And here's the problem, this will always be greater than the length of the entire apparatus, the coil simply doesn't fit! But it's also just a constant that only depends on the geometry, which I don't think makes sense. So what, if anything, went wrong?

Your second assumption is incorrect. $$\Phi$$ and $$B = \Phi/A$$ are determined by the whole magnetic circuit, not just the solenoid. Specifically, $$\Phi = F/R$$, where $$R$$ is as you have calculated. Everything else seems correct.
The length of the solenoid doesn't matter, since $$F$$ doesn't depend on it. It's determined by $$N$$, the diameter of the wire you wind around the core, and how many layers you wind. The solenoid is a bit like a battery in an electric circuit: the current depends on the voltage (electromotive force), not the battery length. You can find a bit more on the analogy between electric and magnetic circuits here.
• Thank you! So the formula $B=μNI/L$ for the magnetic field inside a solenoid doesn't hold in this model of a circuit? I mean if we just had a solenoid wrapped around a straight long piece of iron this would give us the value of the field inside it, right, so why does it stop being true if we basically "bend" the ends of the core into a closed loop? Commented Dec 16, 2023 at 20:21
• @FinnCoal As you might know, magnetic field lines form loops. In an iron loop, magnetic flux can has a low reluctance path through which it can "flow". In a solenoid, magnetic flux must complete the circuit through air, which has much higher reluctance, so the flux is lower. This is akin to having a massive air gap (very high $L_g$) in your magnetic circuit. I should also point out that $B = \mu NI/L$ assumes an infinitely long solenoid. It doesn't work very well at all for real solenoids with a high permeability core, as the return path through air is effectively neglected.