As the title sais,
I got a 1meter iron ring with a 1cm gap and 10000 wire twists, it is said that following equation can be used for it:

$$B = \frac{µ_0 n I}{d+ \frac{l}{µ}}$$

My Question is, where does "+(l/µ)" come from. I understand that in this case it is significantly smaller than "d", hence it can be ignored. But I just cannot find any Information on why it is in the equation in the first place.

My understanding of the equation so far: d is the distance of the gap filled with air, n is the number of twists and I=current inside the wire. It is basicly the formula we were shown (minus "+(l/µ)").

  • $\begingroup$ Hi there. "It is said" It is said by whom? Where did you get that formula? $\endgroup$ – Gert Aug 28 at 22:10

assume that the fringing fields around the gap can be neglected then from the continuity of the flux you have $\Phi=BA$ through every cross-section $A$, be it in the gap or in the core.

Since $B = \mu_g\mu_0 H_g = \mu_c\mu_0 H_c$ where $\mu_c$ is the core's permeability and for air gap $\mu_g=1$; also from Ampere's law we get $NI=H_c\ell_c + H_g \ell_g$. Now let $\mu^*$ be an equivalent or average permeability that characterizes the core+gap so that $NI = \frac{B}{\mu^*\mu_0 }(\ell_c+\ell_g)$. Then $$\frac{B}{\mu_c \mu_0}\ell_c + \frac{B}{\mu_g \mu_0}\ell_g = \frac{B}{\mu^*\mu_0 }(\ell_c+\ell_g)\\ \frac{1}{\mu_c }\ell_c + \frac{1}{\mu_g }\ell_g = \frac{1}{\mu^*}(\ell_c+\ell_g)\\ \mu^* = \frac{\ell_c+\ell_g}{\frac{\ell_c}{\mu_c } + \frac{\ell_g}{\mu_g }}$$ Also $$ B=\frac{\mu^*\mu_0 NI}{\ell_c+\ell_g}\\ = \frac{\mu_0 NI}{\frac{\ell_c}{\mu_c } + \frac{\ell_g}{\mu_g }}$$

This is your formula for an air gap, $\mu_g=1$

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