No Magnetic Field on the Axis of a Coil?

Posted on 2006-07-22 as part of the Photonics Spotlight (available as e-mail newsletter!)

Permanent link: https://www.rp-photonics.com/spotlight_2006_07_22.html

Author: Dr. Rüdiger Paschotta, RP Photonics AG, RP Photonics AG

Abstract: Have fun with a tricky physics conundrum: an apparent proof saying that a coil cannot generate a magnetic field on its symmetry axis.

Admittedly, this is somewhat off-topic, but you may nevertheless enjoy to think about a tricky physics conundrum:

Consider a cylindrically symmetric coil with the z axis being the symmetry axis. The magnetic field caused by a current flowing through the coil must also reflect the cylindrical symmetry, so that its <$x$> and <$y$> components must vanish on the z axis. Therefore, Maxwell's equation <${\rm div} \: B = 0$> reduces to <$\partial B_z / \partial z = 0$>, so we see that <$B(z)$> must be constant along the axis. Now we understand that <$B(z)$> must be zero far away from the coil (which of course has a finite extent in z direction), so it must be zero everywhere! In conclusion, there cannot be any magnetic field on the symmetry axis.

This is surely contradicting textbook knowledge, but isn't it convincing anyway?

A hint: if your resolution looks very sophisticated, it does probably not address the crucial point.

Note that the resolution of this issue has been published on 2006-08-18.

This article is a posting of the Photonics Spotlight, authored by Dr. Rüdiger Paschotta. You may link to this page and cite it, because its location is permanent. See also the RP Photonics Encyclopedia.

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