# Induced Electrical Field (Maxwell-Faraday's Law)

• DirecSa
In summary, the magnetic induction causes an electric current in a wire, and Maxwell-Faraday's equation is: ##\nabla \times E=-\frac{\partial B}{\partial t}##. And then Maxwell-Faraday's equation is: ##\nabla \times E=-\frac{\partial B}{\partial t}##, until now this was just an introduction. My confusion is how one could conclude that there is an electric field which is motive the charges in a conductor (wire) to move after exposure to time-varying magnetic field... What I mean is if I get the result as Faraday did then I will expect that there is a direct connection between the current and the time-f

#### DirecSa

As we know that the magnetic induction causes an electric current in a wire and Faraday has formulated his Electromotive equation ##\epsilon=-\frac{d\Phi}{dt}##. And then Maxwell-Faraday's equation is: ##\nabla \times E=-\frac{\partial B}{\partial t}##, until now this was just an introduction. My confusion is how one could conclude that there is an electric field which is motive the charges in a conductor (wire) to move after exposure to time-varying magnetic field... What I mean is if I get the result as Faraday did then I will expect that there is a direct connection between the current and the time-varying magnetic field and I start to research more about this, and not that the time-varying magnetic field produces an electric field and then this electric field makes the charges moves in the wire, and by the way I can't understand how this electric field causes the charges to move since this electric field is at the surface and can't be inside a conductor.

In short, how one can conclude from such experiment that there is an induced electrical field that is produced due to the time-varying magnetic field and not some another direct phenomena that should be investigated?.. In addition, the nature of this induced electrical field, how it makes the charges move?

Note: I know that this is non-conservative electrical field and the logic that electromotive in such case with no battery should exist and the closed path integral of electric field should be non-zero.. but still I have the confusion.. :/

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I will expect that there is a direct connection between the current and the time-varying magnetic field
There cannot be a direct connection between the current and the time-varying magnetic field because you can change the current by changing the material. More conductive materials have higher currents, which indicates that the changing B field produces an E field and not a current directly.

by the way I can't understand how this electric field causes the charges to move since this electric field is at the surface and can't be inside a conductor
This is incorrect. Perhaps you are thinking about the skin depth? https://en.m.wikipedia.org/wiki/Skin_effect But the skin depth is not zero even for superconductors.

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• vanhees71 and DirecSa
In short, how one can conclude from such experiment that there is an induced electrical field that is produced due to the time-varying magnetic field and not some another direct phenomena that should be investigated?.. In addition, the nature of this induced electrical field, how it makes the charges move?

I think this is because the current density in the conductor is proportional to the electric field, that is ## ~ J = α E ~## , where ## ~E~ ## is the electric field and ## ~α~ ## is the electrical conductivity.

Please note that in practical applications, the conductivity of the conductor will not be infinite, so for a fixed current density, the value of the electric field can only be close to zero, but it will not be zero.

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• DirecSa
I think the confusion is due to the idea that a time-varying magnetic field is the source of an electric field. One should rather interpret Faraday's law such that if there is a time-varying magnetic field than there is also an electric "vortex field". The sources of the electromagnetic field are charge and current distributions.

Of course you can measure the electric vortex field due to a time varying magnetic field by short-circuiting a voltmeter by a wire loop (for simplicity keeping the entire thing at rest). Then from Faraday's Law you get, using Stokes's integral theorem
$$\mathcal{E}=\int_{\partial A} \mathrm{d} \vec{r} \cdot \vec{E}=-\frac{\mathrm{d}}{\mathrm{d} t} \int_{A} \mathrm{d}^2 \vec{f} \cdot \vec{B}.$$
This "electromotive force" is what your voltmeter will measure. If you think of it as a good old Galvanometer what's in fact is measured is the (small) current induced by the electric field due to the time varying magnetic field.

• DirecSa, etotheipi and Dale
Thanks for all of you :)