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Faraday's Law: Conservation?

  1. Nov 3, 2007 #1
    Faraday's Law: Conservation??

    Hi there

    There seems to be a few related posts on this topic but I can't make sense of them.

    In the integral form, I think that Faraday's Law says that round-trip path integral of electric field is equal to the negative temporal rate of change of magnetic flux through the loop defined by the path. I.e. a changing magnetic flux induces emf around a loop.

    Ok, the mathematics is fine but I'm uncomfortable with the idea of the nonzero path integral of electric field...

    I gather that for some reason the idea that the "round-trip path integral of electric field equals zero" is not valid for electric fields created by changing magnetic fields...but why? If electric potential is not defined for this case, why?

    An emf is induced in some loop. So let us say that an electron that happens to be in this loop (some wire) receives some energy input as a result of the changing magnetic flux, and is accelerated through the wire. Where does the energy come from -- does energy stored in a field decrease? Is this the energy that was required to change the magnetic flux in the first place? I've seen some sites mentioning a nonconservative field. How does this fit into the picture - at a fundamental level, all forces are conservative right?

    I'm confused! thanks in advance...
  2. jcsd
  3. Nov 3, 2007 #2


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    Consider this question. What is causing the change in the magnetic field in the first place? Maybe it's a generator or a turbine. The turbine could be spun by water. As the turbine spins, the magnetic field through the loops of wire inside the turbine changes it's orientation with respect to the loops, causing a change in magnetic flux. This creates a current in the loops according to Faraday's Law. Now, here the energy is coming from the water! The kinetic energy of the water used to turn the turbine is transferred to the electrons through the induction process. This is how electricity is generated in a hydroelectric plant. Pretty cool huh?

    So, in conclusion, energy is still conserved, since the energy gained by the electrons in the wires comes from whatever is causing the field to change in the first place. Does this explanation help?
    Last edited: Nov 3, 2007
  4. Nov 3, 2007 #3
    You know that a Coulomb's electric field around a point charge is conservative. You can easily show that the sum of conservative fields is still conservative. So if the electric field is always a sum of electric fields of individual point charges, how can ever be non-conservative electric fields?

    Well that's the question! :smile:

    There are some very clear reasons why non-conservative electric fields can exist. If the Coulomb's electric field around point charges always propagated with infinite speed when the charge changes its position, then there would not be non-conservative fields. But we know that the field configuration updates itself with finite speed of light when the charges move. It is quite easy to convince oneself, by drawing some pictures, that there is no particular reason to assume that an integral of the electric field along some arbitrary loop, at some given instant, should be zero when the charges are moving and the electric fields are propagating with finite speeds.

    Besides this, the Coulomb's formula for point charges doesn't work in its simplest form, when individual charges are moving. So there's lot of problems going on.

    It is wrong to think that electric fields could be created by changing magnetic fields, because they are the charges that create the electric fields! (Edit: Okey. Even though it is true that the electric fields are always created by the charges, it is not wrong to think that they are also created by the magnetic fields. Attempt to define the "true source" can get philosophical.) Thinking that the curl in the electric field could be created by the changing magnetic fields, could be half true, but still dangerous anyway. It is true that the equation
    \nabla\times E = -\partial_t B
    is always true, but you should not rush to make the conclusion, that the electric field is somehow trying to be conservative naturally, and the magnetic field would be making it non-conservative. These thoughts are popular, but not very well justified. After all, it is not usually the purpose of the physics to tell why things are, but instead how they are. Now physics is merely telling, that that equation is true.

    The electric potential is still defined, but there is the vector potential A too. In general electric and magnetic field are given by the potentials [itex]\phi[/itex] and A like this
    E=-\nabla\phi - \partial_t A
    B = \nabla\times A
    Last edited: Nov 4, 2007
  5. Nov 4, 2007 #4
    @G01: Thanks, that does seem to make sense!

    @jostpuur: Aha, I forgot that Coulumb's law is not relativistically correct...as for the differential form of Maxwell's equations: unfortunately I haven't really had much dealing with those yet, I shall have to do some reading...

    Would you please explain what you mean when you say "It is wrong (...) because they are the charges that create the electric fields"? Are you saying that although the changing magnetic fields are related to the induced emf, they are not the actual mechanism causing the induced emf?

  6. Nov 4, 2007 #5
    I'll put it this way. This is something that nobody should complain about: If you use retarded potentials, and define the electric field around a loop as a sum of the electric fields created by each individual accelerating electron in the loop, you will get the non-conservative electric field also this way.

    But that is a very technical task. I must admit I haven't done it explicitly ever (not on paper, and not on computer), but it should work like that still.

    So in this sense even the non-conservative electric field is still coming from the charges, and not from the magnetic field.

    However, the terminology-"change in magnetic field induces electric field" seems to be specifically designed to create the impression that the change in the magnetic field is the true mechanism... so this is bit confusing topic :confused: My opinion on this matter is precisely as you understood it: The change in the magnetic field and the curl in electric field are related, but it is not the mechanism. There are probably people who disagree with me in this. Fortunately our scores in the exams depend on our ability to calculate, and not on our opinions about what are true mechanisms behind physical phenomena.

    hmhm... I just realized that the each individual accelerating electron in the loop is also contributing to the change of the magnetic field, so in fact one could still insist that it is the change in the magnetic field, that causes the curl in the electric field. So my previous argument wasn't perhaps as rigor as I though.
    Last edited: Nov 4, 2007
  7. Nov 21, 2007 #6
    I too have been confused for a very long time about this, since so many books mention that the fundamental forces are all conservative. However, the electromagnetic field, due to Faraday's Law, seems to be definitely nonconservative. For example, see post #4 of this thread. Faraday's Law also presents other problems. For instance, see post #51 of this thread.
    Last edited: Nov 21, 2007
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