I'm trying to understand better magnetic induction, and I came across a situation where I find an apparent contradiction between the local form ($\nabla×E=-\partial B/\partial t $) and the corresponding integral form ($e=-d\phi/dt$, which of course corresponds to the integration of the local form) of induction law.
[EDIT : in order to make the problem easier to visualize in space, I found a simpler situation with a similar problem]
Take these two coaxial coils :
We provide an increasing current $i_1$ to coil 1, and want to examine the induced current $i_2$ in coil 2.
The question is : in which direction does $i_2$ flow?
Surface of reference : the cross section of coil 2 (in green).
- The magnetic field $B_1in$ in coil 1 is oriented up.
- There's also a downward-pointing magnetic field $B_1out$ outside coil 1, in the small annular zone between coil 2 and coil 1.
- But globally the integral $\phi$ of the field over S is positive.
So, as $i_1$ increases, $\phi$ is more and more positive. Therefore $e=-d\phi/dt <0$ : $i_2$ is such that the induced field points downward, in opposition to the increase of $\phi$ ($i_2$ < 0)
the wire of coil 2 is subject to $B_1out$ : the wire doesn't "know" what happens inside coil 1. If, instead of coil 1, we had an externally-imposed flux pointing downward everywhere, then coil 2 couldn't tell the difference.
As $i_1$ increases, $B_1out$ gets stronger, so $\nabla×E=-\partial B/\partial t $ points up at each point of the wire loop. If the curl points up, it means that the induced electric field is such that $i_2$ is positive (the field induced in coil 2 points up).
This is obviously a contradiction with the integral reasoning. I know that the correct solution is the first one, but I don't see where the mistake is in the local reasoning.
[old description] The situation is simple: a permanent magnet rotates counter-clockwise (the axis of rotation is z) in a wire loop (the loop is in the y-z plan).
I want to examine the induced current at instant t when the magnet "crosses" the plan of the loop (that is to say : the moment seen on the picture above).
If I use the integral form of Faraday's law, I have an integrated flux that is positive at t-dt, and negative at t+dt. So, the variation of the flux is negative, and the induced current is positive. The induced magnetic field points in the +x direction (the induced magnetic field opposes the rotation of the permanent magnet, which is expected).
So far, so good.
Now, let's have a look at the magnetic field that is applied to the wire, to use local form of the equation. If we look from above, the magnetic field at t-dt points slightly to the left. At t+dt, it points slightly to the right (it's crucial to notice that the field lines aren't quite parallel, and that they curve out of the magnet axis faster that the magnet rotates). So, $\partial B/\partial t $ has a positive x component, which means that $\nabla \times E$ points to the left (negative x component)... which is in total contradiction with the prediction of the integral form.
We have the same problem if we look at the horizontal portions of the loop: the magnetic field points in the opposite direction of that of the magnet, due to the fact that the magnetic lines form a closed loop.
So, when the magnet rotates, the magnetic field variation in the horizontal wire points to the right, and the curl of the induced current points to the left (as well as the induced magnetic field).
You can drive the same analysis at any point of the wire : the induced magnetic field points to the left, because the magnetic field outside the magnet actually isn't aligned with that "inside" the magnet.
We see therefore that the integral form predicts something different from the local examination.
I know that the local analysis I made is wrong, because it predicts that the induced field helps the magnet to rotate (so, any slight pulse at the beginning would lead to a never-ending self-acceleration of the magnet, creating mechanical energy from nowhere). Yet, I don't see what the flaw is in my local analysis. I mean : the wire actually is subject to the local magnetic field, right? And this magnetic field variation points towards the positive x, so...
I don't get what I'm missing.