Is this a correct way of finding magnetic field of a magnet created with Wasilewski's method?

Question

$\quad$ Wasilewski's method (Or the method that could be seen in Dr Stone episode 9) consists of piece of iron being struck by a lightning, which then magnetises it. Let's say, we have a cylindrical piece of iron of length $d$ and radius $r$. Then we insulate the iron and wrap a copper wire of length $l$ and cross section area $S$ around the iron $n$ times. Then we let the lightning strike it. Now comes the calculation.

We have some given value of lightning voltage, so the current passing through the wire will be$$I=\frac{U}{R}=\frac{US}{l\rho}$$ $\rho$ being resistivity of copper. Thanks to the Ampere's law we know, that magnetic field inside a solenoid is$$B=\frac{\mu In}{d}=\frac{\mu USn}{dl\rho}$$ $\mu$ being magnetic permeability of iron. Then as a last step we can express length of the wire as $l=2\pi rn$, making the magnetic field of a our iron magnet$$B=\frac{\mu US}{2\pi d\rho}$$

Now I don't know, if this really is correct. Please, if I'm wrong, tell me where is my mistake. And in case this is correct, I know the field I just calculated is just the field inside of the magnet, not outside of it, but considering putting two same magnets very close to each other with opposite poles, the field between them should be around the same as the field inside of them, am I correct?

Answer 1

Probably not. Your method would be mostly^* correct (I think) to find the magnetic field induced inside a piece of iron if a solenoid carrying a steady current was flowing around it. But lightning is not a steady current^{[citation needed]}. Among various issues that would throw your answer off, perhaps substantially:

• The wire itself changes temperature in the process (in fact, it is vaporized). This means that its resistance is not constant during the magnetization, and Ohm's Law does not hold.

• The equation you used for the field of the solenoid is only really valid for static fields. In the case you're talking about, the current is changing rather rapidly. In particular, in the split second before the wire vaporizes, it will have a substantial back EMF due to Lenz's Law (or, if you prefer, due to its self-inductance.) This will reduce the net current flowing through it, and the net magnetic field experienced by the iron will be reduced accordingly.

• There is not a single value for the permeability of a substance. The one you commonly see in textbooks is the value for steady fields; but for dynamical fields, the response of the iron will differ. Physicists often think about this in terms of "permeability as a function of frequency", which can change rather dramatically at high frequencies.

In addition, there may be other effects due to dynamical fields (a changing electric field induces a magnetic field, etc.) I suspect that you'd need a full-fledged numerical simulation to get a reliable answer.

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^* There is one problem with your method, even in the static case. Ferromagnetic materials don't have a "permeability" in the sense that you don't have $ H = \frac{1}{\mu} B$ for such materials. Rather, $H$ is a non-linear function of $B$.