Will a electron be accelerated when it move along the non-uniform magnetic field?

Question

1. For the case that the magnetic field changing with time (i.e., $\frac{\partial{B}}{\partial{t}} \neq 0$), the vortex electric field will be created as a result of the change of the magnetic flux. If the magnetic field doesn't change with time (i.e., $\frac{\partial{B}}{\partial{t}} = 0$) but it is non-uniform, considering that a electron moves along magnetic field, will this electron fell a vortex electric field? cause the magnetic flux that this electron feel will change with time. ———————————————————————————————————————————— Supplement to xXx_69_SWAG_69_xXx: As you describe above, if the magnetic field is a no-time-variation, there should not exist electric field. But the following example still puzzles me. Given a permanent magnet and a electron placed near the magnet, the permanent magnet's magnetic field is a no-time-variation and this electron is static in our reference frame. If we move the magnet, the electron will feel a electric field without any doubt. This case suggests that although the magnetic field doesn't vary with the time, its spatial change will result in the change of the magnetic flux and further the electric field (in my understanding, the spatial change of the magnetic field can be converted to it temporal change. but I am not sure whether there exist some mistake in my understanding). Is this case equal to the case that the magnet is static but the electron is moving? If so, the electron will feel a electric field. This conclusion is so strange and opposite to what you described above. Following the Maxwell's equations, I will make the same conclusion as you. But if we consider the case I described above, what's the mistake?

Answer 1

If $\frac{\partial}{\partial t}B=0$, then $\nabla \times E=0 \rightarrow E=-\nabla \phi$. That is the general form an electro-static potential. And of course $B=\nabla \times A$. You can plug these equations into the expression for the Lorentz force to find the generalized dynamics of a point charge in a static field; however I believe your question is about how a non-uniform constant magnetic field affects the electric field experienced by a point charge (i.e. no electric potential; $\phi=0$)

The answer to this is simple; no time variation in the magnetic field means no electric field is generated by Faradays Law. An electron does not 'feel a flux', it experiences a force $F=q(E+ v \times B)$, if your question was about a loop moving through a non-uniform magnetic field then there in fact would be a current generated since the flux is changing.