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The Photonics Spotlight

No Magnetic Field on the Axis of a Coil?

Dr. Rüdiger Paschotta

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, because its location is permanent. See also the Encyclopedia of Laser Physics and Technology.

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