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The standard argument about the vanishing of the magnetic field outside an infinite solenoid pertains to the longitudinal component, parallel to the solenoid axis. But there must be a non-zero toroidal field. However tightly the solenoid is wound, there is a net current in the longitudinal direction. Thus, by Ampere's law, there is a toroidal $B$ field whose magnitude dies off as $1/r$. Why is this field never mentioned? Or am I wrong?

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I'm fairly sure you're correct, and in fact, it's mentioned on Wikipedia:

Of course, if the solenoid is constructed as a wire spiral (as often done in practice), then it emanates an outside field the same way as a single wire, due to the current flowing overall down the length of the solenoid.

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  • $\begingroup$ In his pioneering experiments in the 1820s, Ampère knew about this field due to the longitudinal current in a coil, and corrected for it. $\endgroup$ Oct 8 '20 at 19:27
  • $\begingroup$ See Example 5.9 in the fourth edition of "Introduction to Electrodynamics" by David J. Griffiths. $\endgroup$ Oct 8 '20 at 22:00
  • $\begingroup$ These were interesting and instructive answers. $\endgroup$ Oct 11 '20 at 0:29

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