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I Ambiguity of Curl in Maxwell-Faraday Equation

  1. Oct 27, 2016 #1
    This is an old problem, but one that may confuse many beginners.

    ##\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}##

    Let's say that we're trying to find the electric field produced by a changing magnetic field.

    We could take the inverse curl of the RHS, but the curl product is not injective, so the inverse curl can have more than one solution.

    However, there can only be one electric field produced under certain conditions, not two or three etc.

    How did physicists solve this problem?
  2. jcsd
  3. Oct 27, 2016 #2


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    Did you hear about the boundary or initial conditions in the study of differential equations (in this case PDEs)? They are the crucial concept behind the "uniqueness of the solutions" theorems which appear in the mathematics books.
  4. Oct 27, 2016 #3
    Imagine we have a current loop, in which flows a current that is increasing at a constant rate. To simplify the problem, the current moves unrealistically on its own volition.

    However, such a current would generate a magnetic field that changes at a constant rate, which would generate a constant electric field. How do we find the equations for that electric vector field?
  5. Oct 27, 2016 #4


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  6. Oct 28, 2016 #5
  7. Oct 28, 2016 #6


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    The source of the trouble is a typical misconception. This couldn't happen, if one would finally stop teaching E&M in terms of 19th century non-relativistic physics. The relativistic formulation clearly shows that the electromagnetic field with its 6 tensor components is an entity, and thus it's unnatural and misleading to think as some components as the source of other components of the tensor. Indeed the only sources of the electromagnetic fields are charge-current distributions, and then you get well-defined and physically easy to interpret solutions in terms of the retarded potentials (or equivalently Jefimenko's retarded solutions for the electromagnetic field). Of course, to get unique solutions you need initial and boundary conditions, which are also physically well motivated and can be found in any textbook.

    For the original problem this makes clear that the electric field is not uniquely determined by Faraday's Law alone. To specify a vector field (in the 3D sense!) you need both its curl and its divergence together with appropriate boundary conditions (Helmholtz's fundamental theorem of vector calculus). In other words you need the complete set of the Maxwell equations to find the correct field for given charge-currents distributions and boundary conditions dictated by the matter around.
  8. Oct 28, 2016 #7


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    You didn't specify. See chapters 4, 8, and 13 for a good description of how to specify and use boundary conditions.

  9. Oct 28, 2016 #8
    Ok, but while I'm trying to understand that, let me reiterate my problem:

    Imagine a circular current loop, of a certain radius R, floating in the vacuum of space, and lying perfectly still. We'll only use classical considerations. To keep things simple, we'll ignore electrical resistance and heating and other real world considerations of this sort.

    The loop is of zero thickness. There is a current in the loop which is increasing at a constant rate, through its own volition without the application of any electric field.
    Such a magical current can still generate a magnetic field.

    We specify that ##\frac{dI}{dt}=k##, a constant. The changing magnetic field is described by the vector field ##\frac{∂B}{∂t}##. In this scenario, it is easy to see that ##\frac{∂B}{∂t}## will always be a scalar multiple of B. The magnitude of ##\frac{∂B}{∂t}## is directly proportional to k.

    Although unrealistic, there are no EM waves of any sort. ##\frac{∂B}{∂t}## is entirely static, as is the E field generated by the changing B field.

    Is this specific enough?
    Last edited: Oct 28, 2016
  10. Oct 28, 2016 #9


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    Well, doesn't this violate energy conservation? It's pretty unphysical at least.
  11. Oct 28, 2016 #10
    Yes indeed. That's why I've made my problem as static as possible, in order to avoid all the tricky Jefimenko concepts.

    Yes it is, but so are frictionless surfaces, and we always see those in kinematics problems.

    The objective is to make finding the electric field as easy as possible, without real world complications. I just need an easy example.
    For instance, I thought of using an infinite straight wire, but I realised that it would lead to much more conceptual difficulties than a circular loop.

    What sort of boundary conditions should I specify? I think the problem I've written in #8 is specific enough, at least specific enough to derive whatever boundary conditions we might need.
  12. Oct 28, 2016 #11


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    No. You are specifying the wrong kinds of things. Are you familiar with initial conditions for an ordinary differential equation?

    For instance, the fields are all 0 at time, t=0, when the current is also 0, and at spatial infinity. You may need to specify more boundary conditions, but that is an example.
    Last edited: Oct 28, 2016
  13. Oct 28, 2016 #12
    I did think about that, and that's actually a realistic consideration, but I didn't want to include it as it would complicate the problem. I'm trying to make the problem as static as possible.

    Assuming that the E field is zero at infinity is common sense, but that's kinda like begging the question?

    Also, we can assume that we're looking at the situation a long time after the current started running. The E field is supposed to be static, and largely independent of initial conditions.
    A static ##\frac{∂B}{∂t}## field gives rise to a static E field, and the curl of E is also static.

    We don't need additional conditions to construct closed line integrals for the voltages of this static E field, as described by Faraday's Law. And the voltages remain as static as the ##\frac{∂B}{∂t}## field.

    To use other analogies, if we have a static charge density scalar field ρ(x,y,z), that's all we need to find the E field. And a 'static' ('static' and 'current' are somewhat oxymoronic) current density vector field J is all we need to find the B field.
  14. Oct 28, 2016 #13


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    Call it what you will. Boundary conditions are needed. That is the mathematical nature of partial differential equations. You cannot avoid it.
  15. Oct 30, 2016 #14
    hmm, but the problem doesn't seem to depend on even the current value of the current (what a lame pun), much less so the initial conditions.
  16. Oct 30, 2016 #15


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    I don't know what more to tell you.

    That is just the math of partial differential equations. You have to specify the boundary conditions if you want a unique solution. That applies to any physical law written as a partial differential equation.

    This is well discussed in the literature and I pointed you to specific references.
  17. Oct 30, 2016 #16
    I'm not denying that DEs don't require conditions, I'm just confused about the type of conditions and whether certain conditions are relevant or not.

    Let me specify some initial conditions anyway. I = kt. At t = 0, I = 0. Then I starts to increase at a constant rate.

    We examine the situation a long time after the current has started running, when the fields have become more stable.
  18. Oct 30, 2016 #17


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    I have no clue what physical situation this should describe. It doesn't sound compatible with the continuity equation either, and then there's no solution for Maxwell's equations at all.
  19. Oct 30, 2016 #18


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    A boundary condition is a specification of the fields on a given boundary (eg over all space at an initial time).

    For analogy consider the ordinary differential equation ##x'(t)=f(t)##. An initial condition would be a specification of something like ##x(t_0)=x_0##. What you are doing above is specifying f rather than x.
  20. Oct 30, 2016 #19
    but the current is flowing in a loop.
  21. Oct 30, 2016 #20
    Ok, at first, the current in the loop is zero. B is zero, ##\frac{∂B}{∂t}## is zero and E is zero. At a certain point in time, the current is switched on and starts increasing from zero at the constant rate of k.
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