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Imagine we have a uniform magnetic field, $\mathbf{B}$, and a single electron is moving normal to it, the electron will produce a magnetic field of its own which interacts with $\mathbf{B}$ and so electron experiences a force.

This is perfectly fine, but what troubles me is when we switch perspectives. If we are moving with the electron, then to us, the electron would be stationary, so it produces no magnetic field and hence no interaction with $\mathbf{B}$ making it experience no force.

How can this be possible? Clearly there should be something that I am missing allowing for a force to be exerted but all we see is a stationary electron in a magnetic field and it will somehow experience a force out of nowhere.

What's going on?

Quantum Zain
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    The electron's perspective is not an inertial reference frame. – aRockStr Dec 10 '19 at 14:50
  • First : in the system $S$ of the uniform magnetic field $\mathbf{B}$ the electron feels a force $\mathbf{f}\boldsymbol{=}q\left(\mathbf{E}\boldsymbol{+}\boldsymbol{\upsilon}\boldsymbol{\times}\mathbf{B}\right)\boldsymbol{=}q\boldsymbol{\upsilon}\boldsymbol{\times}\mathbf{B}$. There is no such interaction between magnetic fields. – Frobenius Dec 10 '19 at 15:07
  • Second : in the rest frame $S'$ of the electron we have electric field $\mathbf{E}'$ and magnetic field $\mathbf{B}'$ so the electron feels a force $\mathbf{f}'\boldsymbol{=}q\left(\mathbf{E}'\boldsymbol{+}\boldsymbol{\upsilon}'\boldsymbol{\times}\mathbf{B}'\right)\boldsymbol{=}q\mathbf{E}'$. – Frobenius Dec 10 '19 at 15:07
  • The electron in the lab in a uniform magnetic field makes a circle, from $Bqv=mv^2/r$. It is continuously accelerated. . If you go to it rest mass frame , the B field is no longer uniform , and for the electron to be at rest, the forces from the transformed B field and the E field should add up opposite, so the electron will be in its rest frame .could not find the solution by searching.. – anna v Dec 10 '19 at 15:51
  • the electron will produce a magnetic field of its own which interacts with B and so electron experiences a force Magnetic fields do not interact with each other. They simply superpose additively. – G. Smith Dec 10 '19 at 15:56
  • This video might help: https://www.youtube.com/watch?v=Ii7rgIQawko – rghome Dec 10 '19 at 16:10
  • You may find the first few pages interesting of Einstein 1905 "ON THE ELECTRODYNAMICS OF MOVING BODIES":

    http://hermes.ffn.ub.es/luisnavarro/nuevo_maletin/Einstein_1905_relativity.pdf

     – Jasoba Dec 10 '19 at 20:10

3 Answers3

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Electric and magnetic fields are in effect different views of a single electromagnetic field. That is, if we have an electromagnetic field then different observers moving at different velocities will see the electromagnetic field as different combinations of an electric field and a magnetic field.

And it is this that answers your question. We lab observers see a stationary magnetic field. However to the moving electron the same electromagnetic field appears as a combination of a magnetic field and an electric field. It is the electric field that appears in the electron's rest frame that exerts the force on the electron and makes it move in the trajectory observed in the lab.

John Rennie
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As mentioned by @Frobenius, the observer at rest WRT the electron asserts that the magnetic field is moving, and thus a moving magnetic field, applying Lorentz transformation for fields, produces an additional electric field that exerts force on the electron.

Mohammad Javanshiry
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    I'm a bit surprised this got downvoted as it is a correct answer. It is effectively the same answer as mine though in a more concise form. – John Rennie Dec 10 '19 at 17:28
  • @JohnRennie and Mohammad Javanshiry electromagnetic fields do not have the property of velocity. It is factually incorrect to assert that the magnetic field is moving. The sources may move, and waves in the field may propagate (which propagation can be considered movement), but the field itself has no velocity. – Dale Dec 10 '19 at 21:03
  • @Dale I meant the motion of the source indeed. – Mohammad Javanshiry Dec 10 '19 at 21:06
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Here is a thought experiment: 2 + charges and 2 - charges are arranged in a square. If the square is perfect, the forces balance. But it is unstable - the slightest inequality in the lengths of the arms will make + and - snap together. If you run by at relativistic speeds, two arms are foreshortened and two are not. You might think it is no longer balanced. But it is. Now there are magnetic forces. Forces transform in more complex ways than you might expect.

mmesser314
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