This example given bellow intrigued me, since it is really simple but I still can't find it coherent with experimental results.

Description : We have a static magnetic field pointing in the $\hat{x}$ direction (see picture below) and a charge moving in the $-\hat{x}$ direction. We choose two frame references : one on the magnet , and the second on the moving charge (primed) so that ($\hat{x}\parallel \hat{x'}$ and $\hat{y}\parallel \hat{y'}$ but different origins O and O')

The static magnetic field is in the $\hat{x}$ direction and increases when coming closer to the magnet.

The charge is moving at a constant velocity towards the magnet ($\vec{B}\parallel \vec{v}$). Now let us compute the Lorentz forces in each frame :

$$\vec{F} = q\vec{E} + q\vec{v}\times\vec{B}$$

$$\vec{F'} = q\vec{E'} + q\vec{v'}\times\vec{B'}$$

In the magnet frame :

There is no $\vec{E}$ ($\vec{B}$ static) and since $\vec{B}\parallel \vec{v} \implies \vec{F}=0$ so the charge should not experience any force , so it should keep it's velocity, in the $\hat{x}$ direction.

In the charge frame :

$\vec{v'}= 0$ and $\vec{E'} = \vec{v}\times\vec{B}$ (using a Galilean transformation ) so once again the force is zero, (expected since the change of frame reference should not create a force)

Conclusion : The charge won't experience any force, and its motion will be rectilinear in the $\hat{x}$ direction, and NOT a circular motion around the $\hat{x}$ axis.

**Lenz's Law!!!! **: If one now take many charges forming a closed circular loop (basically closed conductor loop), and send them down the same trajectory in the $-\hat{x}$ direction, then how come we see an induced current if we replace the charge (in the example) by the loop?

Lenz's law guarantees this result :

$$\text{EMF} = -\frac{\partial \Phi_B}{\partial t}=-S \frac{\partial B}{\partial t}$$

where $S$ is the area of the close loop.

Which means the charges in the conductor will be circulating around the $\hat{x}$ axis. enter image description here Thank you in advance.

  • $\begingroup$ Part of the problem stems from the fact that electromagnetism isn't invariant under gallilean transformations, it is inherently relativistic $\endgroup$ – Triatticus Mar 21 '18 at 4:44
  • $\begingroup$ Thank you for your answer. I hope was clear in describing my problem. Lenz's Law gives birth to an induced current regardless of the fact that v//B and the example above shows the opposite. $\endgroup$ – Edwardo Newagte Mar 21 '18 at 7:59
  • $\begingroup$ You were clear enough, I just am answering on mobile and hope someone can expand on what I mean about the gallilean noninvariance of e&m. $\endgroup$ – Triatticus Mar 21 '18 at 13:07

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