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This picture in Wikipedia is supposed to explain the solenoid formula for the magnetic field $$B l= \mu_0 N I$$ (assuming steady currents).

But this would be true if one consider only the part of the curve inside the solenoid. What about the part of the curve outside the solenoid. There the field is not null, since it must loop from pole + to pole -.

Thinking about, that's curious: we could extend the exterior part of the curve to infinity to make the formula exact. But then, Ampere law seems to be wrong since the integral of the magnetic field along the curve depends on the curve, while it should be always equal to $$\mu_0 N I.$$

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  • $\begingroup$ The final expression $B=\mu_0 n I$ (where $n=N/l$ is the number of turns per meter) is for an infinitely long solenoid. Otherwise one cannot use Ampère's law and there indeed is a field outside the solenoid. $\endgroup$
    – ZeroTheHero
    Commented Jul 6, 2022 at 14:06
  • $\begingroup$ @ZeroTheHero. I think this expression is said to be valid even for finite solenoid (according to the Wikipedia article). $\endgroup$
    – MikeTeX
    Commented Jul 6, 2022 at 19:12
  • $\begingroup$ No. It is used for finite solenoids if the solenoid is sufficiently long and/or if one is willing to neglect fringing (i.e. neglect the field outside) but the calculation based on Ampère's law is valid only for the infinite solenoid. One cannot apply Ampère's law for a solenoid of finite length as the field is not the same everywhere along the axis of the solenoid. $\endgroup$
    – ZeroTheHero
    Commented Jul 6, 2022 at 19:14
  • $\begingroup$ In addition, this calculations also assumes the solenoid is tightly wound else again the assumptions underlying Ampère's law fail. You cannot compute the field at the end of the solenoid using Ampère's law for instance, as this will give you a wildly inaccurate value. $\endgroup$
    – ZeroTheHero
    Commented Jul 6, 2022 at 19:16

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The reasoning based on Ampere's theorem effectively assumes an infinite solenoid. Otherwise, the field is not translation invariant and Ampere's theorem is not applicable. Obviously, as always in physics, the obtained field is assumed to be valid inside a finite solenoid, far from the edges. Even if we limit ourselves to the infinite case, it is not at all simple to justify carefully that the field is zero outside the solenoid. In particular, it is not obvious that the field must be zero at infinity: a uniformly charged infinite plane creates a finite field at infinity. This is a usual problem with Ampere's theorem applied to infinite distributions. If one wants to be careful, it raises more complex questions than a direct calculation.

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