The Photonics Spotlight
No Magnetic Field on the Axis of a Coil?
Posted on 2006-07-22 (revised on 2006-08-25) as a part of the Photonics Spotlight (available as e-mail newsletter!)
Permanent link: https://www.rp-photonics.com/spotlight_2006_07_22.html
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 div B= 0 reduces to ∂Bz/∂z= 0, so we see that Bz must be constant along the axis. Now we understand that Bz 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 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 Encyclopedia of Laser Physics and Technology.
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