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Consider an infinite hollow cylinder with inner radius $a$ and outer radius $b$. The volume current density flows anti-clockwise across the surface of the cylinder ($\vec{J} = J\hat{\phi}$). The charge density is $0$ everywhere.

How can I compute the magnetic field $\vec{B}$ for all points in space? It's fairly simple when the current flows parallel with the axis of the cylinder, for you can then use Ampère's law with an Amperian loop that's perpendicular to the current. In this case, I can't seem to think of a good loop. What I do know is that the field inside the cylinder is in the positive z-direction and outside in the negative z-direction.

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Compare the scenario with that of an infinite solenoid.

enter image description here (Image source, note that $B \neq \mu nI$ in this case.)

Since you know the volume current density, you can calculate the current enclosed by the loop.

Also, the magnetic field outside (e.g. at point P) the infinite coil is zero, due to the following reason. enter image description here (image source)

For two loops far away, the resultant magnetic field at the point P in the figure is directed upward, and for nearby loops it directs downward. The net magnetic field becomes zero (This can be shown rigourously by integrating components of magnetic field from each loop).

So you can find the magnetic field inside the coil.

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  • $\begingroup$ I see, thank you ! But which symmetry leads to a zero field outside ? I can't seem to grasp that. $\endgroup$ – Einsteinwasmyfather Jun 23 at 17:00
  • $\begingroup$ I have updated my answer $\endgroup$ – Archisman Panigrahi Jun 23 at 17:16

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