Ambiguity of Curl in Maxwell-Faraday Equation

[greswd]

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?

 

[dextercioby]

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.

 

[greswd]

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.

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?

 

[Dale]

By applying some boundary conditions as well. This is what @dextercioby was referring to

http://faculty.uml.edu/cbaird/95.657(2012)/Maxwell_Uniqueness.pdf

 

[greswd]

By applying some boundary conditions as well. This is what @dextercioby was referring to

http://faculty.uml.edu/cbaird/95.657(2012)/Maxwell_Uniqueness.pdf

What sort of boundary conditions are present in the problem I have specified in #3?

 

[vanhees71]

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.

 

[Dale]

What sort of boundary conditions are present in the problem I have specified in #3?

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

http://web.mit.edu/6.013_book/www/book.html

 

[greswd]

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

http://web.mit.edu/6.013_book/www/book.html

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?

 

[vanhees71]

Well, doesn't this violate energy conservation? It's pretty unphysical at least.

 

[greswd]

The source of the trouble is a typical misconception...it's unnatural and misleading to think as some components as the source of other components of the tensor.

Yes indeed. That's why I've made my problem as static as possible, in order to avoid all the tricky Jefimenko concepts.

Well, doesn't this violate energy conservation? It's pretty unphysical at least.

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 realized that it would lead to much more conceptual difficulties than a circular loop.

Of course, to get unique solutions you need initial and boundary conditions,..
...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...

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.

 

[Dale]

Is this specific enough?

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

What sort of boundary conditions should I specify?

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.

 

[greswd]

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.

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.

 

[Dale]

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

Call it what you will. Boundary conditions are needed. That is the mathematical nature of partial differential equations. You cannot avoid it.

 

[greswd]

Call it what you will. Boundary conditions are needed. That is the mathematical nature of partial differential equations. You cannot avoid it.

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.

 

[Dale]

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.

 

[greswd]

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.

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.

 

[vanhees71]

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.

 

[Dale]

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

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.

 

[greswd]

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.

but the current is flowing in a loop.

 

[greswd]

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.

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.

 

[Dale]

B is zero, ... and E is zero.

There you go! That is the kind of thing you need to specify. Once you do that enough (usually you have to specify spatial boundaries also, like the behavior at infinity) then the solution to your partial differential equation becomes unique.

 

[greswd]

There you go! That is the kind of thing you need to specify. Once you do that enough (usually you have to specify spatial boundaries also, like the behavior at infinity) then the solution to your partial differential equation becomes unique.

Thanks! Now how do I go about solving it?

 

[greswd]

I have an idea. Since we have a circular current loop, let's assume that the E field has rotational symmetry about the axis of the loop.
The axis of the loop is also the direction of the z-axis.

Let's assume that the E field has no z component, it has only x and y components. This field circulates around the axis. So the field lines are concentric circles.

Next, we construct circular line integrals, which are perpendicular to the axis and centered on the axis. Using the integral form of Faraday's law, we can easily find the voltage. Then, in order to find the magnitude of the E-field that encompasses the line integral, we can just divide the voltage by the radius of the circle.

It's quite hard to calculate. But I wonder if such an E-field will satisfy the Maxwell-Faraday curl equation.
We can imagine three components of the E field. One circulates the axis. The second radiates outward from the axis. The third lies parallel to the axis.The circulating and radiating fields have no z components. The parallel field has no x nor y components.

 

[Dale]

Thanks! Now how do I go about solving it?

Personally, I would use the retarded potentials formulation. They assume all "external" fields are 0, so that is a little more strict than your boundary condition, but it is probably what you had intended.

 

[greswd]

Well, doesn't this violate energy conservation? It's pretty unphysical at least.

Personally, I would use the retarded potentials formulation. They assume all "external" fields are 0, so that is a little more strict than your boundary condition, but it is probably what you had intended.

Its meant to be a classical problem though.
Also, what do you think of my idea in #23?

 

[Dale]

Its meant to be a classical problem though.
Also, what do you think of my idea in #23?

The retarded potentials are part of classical EM.

The idea in 23 was unclear to me, I would just use the retarded potentials.

 

[greswd]

The retarded potentials are part of classical EM.

I mean uber-classical, when information speed is not limited by c, and fields are instantaneous. Anyway, let's put this argument on hold first.

The idea in 23 was unclear to me...

I'll be happy to clarify.

There is rotational symmetry about the loop's axis for the B-field. We assume that this extends to the E-field as well.

We assume that the E-field circulates the axis:

The 3rd image is not so accurate to the problem I described, but its the best I could find.

 

[Dale]

I mean uber-classical, when information speed is not limited by c, and fields are instantaneous.

Oh, for that you would want to use the magneto quasi static approach outlined in chapter 8 of the textbook I cited earlier. See especially equation 8 in section 8.1. It is similar in form to the retarded potential, but does not involve a retarded time, so it is much easier to use.

 

[greswd]

Oh, for that you would want to use the magneto quasi static approach outlined in chapter 8 of the textbook I cited earlier. See especially equation 8 in section 8.1. It is similar in form to the retarded potential, but does not involve a retarded time, so it is much easier to use.

Thanks Dale. I'll put this on hold for the moment though.
I wonder if the idea of circulating E-fields will work.

 

[vanhees71]

Again, a vector field is only completely specified when giving it's curl and its divergence together with the boundary conditions. Look for Helmholtz's fundamental theorem of vector calculus!

 

[greswd]

Again, a vector field is only completely specified when giving it's curl and its divergence together with the boundary conditions. Look for Helmholtz's fundamental theorem of vector calculus!

Do you need any clarifications with the notions in #23 and #27? (and I'm not trying to be rhetorically snarky)

 

[vanhees71]

As I said, I don't see the physics behind your artificial assumptions. You need to solve the complete set of Maxwell equations, not just one!

 

[Dale]

You need to solve the complete set of Maxwell equations, not just one!

I agree completely with this and all of my comments are assuming that we are looking for solutions to all of Maxwell's equations.

 

[rude man]

You don't need relativity or tensors to get a grip on this problem. You do need to use dynamic rather than static potentials; for example, the familiar E -= -∇V changes to E = -∇V - μ ∂A/∂t
where A is the vector potential such that H = ∇ x A = B/μ
and V is the scalar electric potential;
and Poisson's equation changes to ∇²V = -ρ/ε - μ ∂/∂t (∇⋅A) etc.
If you try to use static potentials you get that the E field around even a time-varying current is zero.
EDIT: I was assuming a long wire but what I said can be applied to your coil also.
The idea of retardation potentials is included in what I said FYI.