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When an additional loop is added to a generator the magnetic flux through the loops does not change yet the current increases. This seems counter-intuitive and has been described as 'non-conservative.' What causes this unexpected phenomena known as Faraday's Law?

Qmechanic
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Dale
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1 Answers1

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Faraday's law doesn't say that the current changes; the current is whatever it is – it is treated as an independent variable here.

Instead, it says that if we keep the current fixed, the voltage – more precisely the electromotive force (EMF) – is proportional to the number of loops (times the time derivative of the magnetic flux through one loop). That's because the EMF is the potential difference and the potential differences from each loop simply add: $${\mathcal E} = V_N - V_0 = (V_N-V_{N-1})+(V_{N-1}-V_{N-2})+\dots + (V_1 - V_0)$$ Here, $V$ is meant to be the electrostatic potential. If all $N$ terms on the right hand side are the same, you may just write the right hand side as $N$ times one term, therefore the direct proportionality.

Equivalently, you may imagine that the collection of $N$ loops is one convoluted loop encircling a complicated surface that looks like $N$ disk-shaped surfaces on top of each other. In all cases, you will get $$ \mathcal{E} = -N {{d\Phi_B} \over dt} $$

Luboš Motl
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  • You are saying that voltages add because each loop creates another surface for the magnetic flux to flow through. How do the electrons in the wire know that there is another surface? – Dale Jul 13 '13 at 18:37
  • @JoeHobbit - You're thinking about this backwards. The electrons in the wire don't even know (to a first approximation) that there is a loop! The surface is simply a geometrically equivalent way to look at the superposition of all the local effects. – Rex Kerr Jul 13 '13 at 21:14
  • @RexKerr What local effects? – Dale Jul 13 '13 at 23:30
  • Dear @JoeHobbit, Rex is telling you that the electrons are the causes who just freely behave in the way they like (in the wire). It's the job of someone else who measures the magnetic field around the coil (i.e. a device to measure the magnetic field, a compass etc.) to "know" something. And this device will see a magnetic field that was constructed by co-operation of "all pieces" of the wire. You may think of the wire as a union of individual loop, then the magnetic field is the sum from each loop separately. You may divide the N-loop wire to different pieces, too. – Luboš Motl Jul 14 '13 at 05:33
  • By the N-times copied surface, I meant something like the mesh surface here: http://www.nevron.com/gallery/FullGalleries/chart/meshsurface/images/mesh-surface-3d-spiral-chart.png - Although I would prefer a better picture to make it clear that it doesn't have to have any "inside" boundary, just the wire outside. It's effectively N sheets of paper inside the wire. By Stokes' theorem, the boundary of this mesh surface is the whole N-loop wire, so the flux through this surface is approximately N times the flux through one disk (disk is the boundary of 1 loop, neglecting the rise). – Luboš Motl Jul 14 '13 at 05:35
  • @LubošMotl You say; "Electrons are the causes who just freely behave in the way they like (in the wire)." If that is true, then what stops electrons from deciding, "today I will move in the opposite direction as Faraday's law says I will." You essentially say that the loops increase the surface area, but I remain confused since the surface can only be incidentally related to the EMF. Force times distance, where distance is the length of the wire loop and force is the force of the magnetic field on the electrons makes a lot more sense as increasing the loop size increases the distance and EMF. – Dale Jul 15 '13 at 16:15
  • @LubošMotl Perhaps you might comment on the section on Faraday's Law here. – Dale Jul 16 '13 at 21:15