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The magnetic field in the middle of a solenoid is :

$$B = \frac{\mu_0 N I}{L}$$

where L is the length of the solenoid

The magnetic field in the air gap of a magnetic circuit with constant cross section like this one

enter image description here

is (I think):

$$B = \frac{\mu_0 N I}{l}$$

where l is the length of magnetic circuit.

So, if the total length of the magnetic circuit is roughly 3 times that of a solenoid with the same number of turns, its magnetic field would be 3 times smaller ?

I thought the magnetic circuit was supposed to focus the field in the gap and make it stronger... what gives ?

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This is a typical electromagnet.

Let's refer to your figure and say that $g$ is the width of the gap and $A_c$ the section of the core.

If $g << \sqrt A_c$,the magnetic field $\vec B$ inside the core will be approximately the same as the magnetic field $\vec B_0$ in the air gap (this is because in this case the magnetic field lines will stay approximately parallel to each other in the air gap and the flux $\Phi = B \ A_c$ will be conserved).

If this is the case, you will find using Ampere's law that

$$B = \frac{\mu_0}{g} (N i - H l_c)$$

with $l_c$ mean core length. If moreover $\vec H = \vec B / \mu$, then

$$B \left( 1 + \frac{l_c}{g \mu_r} \right)=\frac{\mu_0 N i}{g}$$

Notice again that this value is the same in the air gap and inside the core (solenoid included)!

For most ferromagnetic materials, $\mu_r$ is of the order of $10^3-10^5$ (https://en.wikipedia.org/wiki/Permeability_(electromagnetism)), so we can approximate the previous relation neglecting the term $\frac{l_c}{g \mu_r}$:

$$ B \simeq \frac{\mu_0 N i}{g} > \frac{\mu_0 N i}{l_c}$$

since of course $g < l_c$. So the field is indeed stronger than the field inside an empty solenoid of length $l_c$ with the same number of turns $N$.

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  • $\begingroup$ So my formula was wrong. I thought: the reluctance is $\mathfrak{R} = \frac{l_c}{\mu_0 A_c}$ and the flux $\Phi = \frac {N i}{\mathfrak{R}}$ from Hopkinson's law. So $B = \frac{\Phi}{A_c} = \frac{\mu_0 N i}{l_c}$ . What's wrong with that ? $\endgroup$ – mwa1 May 8 '16 at 7:36
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    $\begingroup$ You didn't consider the reluctance of the air gap. Moreover, the magnetic permeability of the core is not $\mu_0$ but $\mu_0 \mu_r$. The total reluctance is $R=R_{(core)}+R_{(gap)} = \frac{l_c}{\mu_0 \mu_r A_c} + \frac{g}{\mu_0 A_c}$. If you use this, you will obtain the same expression I obtained for the magnetic field. Notice that the reluctance of the air gap can be much higher than the reluctance of the core, because it doesn't have the term $\mu_r$ at the denominator! $\endgroup$ – valerio May 8 '16 at 8:35

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