# Magnet: 1m iron ring with 1cm gap

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)").

• Hi there. "It is said" It is said by whom? Where did you get that formula? – 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, $$\mu_g=1$$