# Why is there no B-field parallel force on an electron orbiting a magnetic field line?

If an orbiting electron creates a toroidal magnetic field like a ring of current does, and this field is oriented opposite to the magnetic field line the electron is orbiting, then why is the electron not repelled by the applied field that it orbits (driven along the field line) like 2 end to end bar magnets of opposite polarity?

• @AaronStevens That's true, but that's not the reason for the phenomenon the question is asking about. You could have a uniformly charged ring rotating in a constant normal magnetic field (making the problem magnetostatic), and the question would be the same. – Buzz Jun 6 at 4:14
• electrons are quantum mechanical entities, and interact with macroscopic magnetic fields as single charge particles. If you are thinking of orbitals in atoms, then there exist corrections due to magnetic field interactions hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydfin.html – anna v Jun 6 at 4:17
• @Buzz Then I guess it depends on if the externally applied field is uniform or not. – Aaron Stevens Jun 6 at 4:18
• @annav A moving point charge will generate a magnetic field. It is more complicated than the field of a steady current. However, the qualitative problem the question is asking about still exists when there is just the field of a single charge in circular motion. – Buzz Jun 6 at 4:26
• @Matthew Is the externally applied field uniform? – Aaron Stevens Jun 6 at 4:27

If the applied magnetic field is uniform then there cannot be a net force on the current loop. This is because the magnetic force $$\text d\mathbf F$$ experienced by a small part of the loop of length $$\text dl$$ of current $$I$$ is given by $$\text d\mathbf F=I\text d\mathbf l\times\mathbf B$$

If we integrate this around the loop for a uniform field you will find that the net force is $$0$$ (and the loop will be either under stretching or compressive forces). So for the example given in the comments, a ring of current inside a solenoid would not experience a net force in the direction of the solenoid (it could experience a torque though)

However if the field is not uniform, then the net force will be non-zero in some direction. This is because around the loop the cross products will have non-zero components in the same direction. It is in this direction that the net force will be in. See the picture below • Thanks for the answer! I'm still trying to understand how the current loop is different from a bar magnet. In the figure above, could we not pretend that the current loop is a bar magnet with north pole on the right? And wouldn't that cause repulsion due to antiparallel field lines? – Matthew Jun 6 at 4:48
• @Matthew Are you referring to the left or right figure? Also keep in mind that the current direction on those figures represent positive charge motion. You can most certainly think of the current loop as a sort of bar magnet. In both figures the North Pole would be to the left. Just use your right hand rule. – Aaron Stevens Jun 6 at 4:51
• Sorry, I meant to say north pole on the left, which would apply to both figures since current direction is the same. But if it was a bar magnet in both figures with north pole on the left, I guess the net forces would be the same, so maybe I made an incorrect assumption. Would a bar magnet oriented along field lines feel any force in a uniform magnetic field? – Matthew Jun 6 at 4:57
• @Matthew No. A bar magnet in a uniform magnetic field would not experience a net force. – Aaron Stevens Jun 6 at 4:59
• I apologize. I was assuming that antiparallel magnetic field lines repel each other face-on in the along-line direction, so thank you for correcting me. Your image has helped me to understand that the repulsion is in the cross-line direction, which I guess is a purely magnetic way to think about things like Z-pinch between wires. – Matthew Jun 6 at 5:07