I Boundary conditions and discontinuity of EM fields

[Unconscious]

Premise: everything that follows is done in the frequency domain.

Boundary conditions
If there are superficial currents (electric and magnetic) impressed on the boundary between two media, we have these discontinuities for the tangential components of the fields:

$$\mathbf{n}\times(\mathbf{H}^+-\mathbf{H}^-)=\mathbf{J}_S$$
$$\mathbf{n}\times(\mathbf{E}^+-\mathbf{E}^-)=-\mathbf{J}_{m_S}$$

Consequently in these cases I never know what the field is in the points that constitute the boundary.

Uniqueness theorem
Given a region of volume τ occupied by a linear, stationary, dissipative for conductivity and non-dispersive medium, bounded by a closed region S, the field is unique in every point of τ that meets the following conditions:

1. both solution of Maxwell's equations and constitutive relations;
2. at least one of the two tangential components of the electric or magnetic field is assigned to each point of S.

Love theorem (or equivalence theorem)
Given a field in a region of space V, solution of Maxwell's equations, it is possible to express the field inside a volume τ⊂V like that generated by fictitious surface current sources present on the surface S that surrounds τ. These sources on S are obtained from the knowledge of the field in V, in particular from the tangential components of electric and magnetic fields on S:

$$\mathbf{n}\times(\mathbf{0}-\mathbf{H}^-)=\mathbf{J}_S$$
$$\mathbf{n}\times(\mathbf{0}-\mathbf{E}^-)=-\mathbf{J}_{m_S}$$

where the normal is outgoing from S, and the only fields that appear (superscript '-') are those inside τ in the vicinity of S (they are obtained as a limit approaching a point of S from the inside of τ).
Personal doubt
The proof of Love's theorem, at least as an idea, is simple having in mind the uniqueness theorem. However I see a problem of principle. If instead of using these fictitious sources, I directly imposed the tangential components of the fields on S, then the Love theorem would be very clear to me.
Going through the use of equivalent surface currents, I no longer return because in this way I have not imposed the value of the tangential field on S, but I have only imposed a discontinuity between the tangential fields inside S and outside S But the field exactly on S? I do not know its values in this way, for what was said at the beginning (boundary conditions). Consequently the uniqueness theorem for the field inside τ can no longer be applied correctly, because hypothesis 2 fails.

Do you agree or am I missing something?

 

[Paul Colby]

Don't know if this helps but let me try anyway. Consider a perfect conducting closed spherical shell of infinitesimal wall thickness. There are frequencies where internal resonances exist. This is where the tangent E field vanishes while the tangent H does (need) not. Meanwhile for the exterior fields, a great variety of solutions can exist. The discontinuity at the shell is therefore not unique. One must specify the fields exterior to the sphere in order for the discontinuity to be unique. I think in the statement of the theorems that E=0, H=0 is assumed everywhere exterior to the region? If this is the case, then one is really specifying the tangent fields. Best I can do at 1 in the morning.

 

[Unconscious]

Hi,
$\mathbf{E}^-$ (or equivalently $\mathbf{H}^-$) is not the field on a point of S, but is the limit (tending to that point of S) of the field calculated in points inside τ, not on its boundary. This is the reason of the superscript "-".
So I'm not really specifying the tangential component of the field on S.
Do you agree?

Thanks for your reply.

 

[vanhees71]

That's right. The field value at the boundary itself is ill defined in such cases. It's where the "singular source distributions" are located. Of course that's just due to our mathematical simplifications leading to classical approximations. In fact there are no sharp boundaries (as there are no classical point particles) in nature. If you want to resolve things down to the microscopic realm quantum effects have to be taken into account.

 

[Unconscious]

So how could we formulate a consistent proof of the Love theorem, based on uniqueness theorem?

 

[Paul Colby]

Hi,
$\mathbf{E}^-$ (or equivalently $\mathbf{H}^-$) is not the field on a point of S, but is the limit (tending to that point of S) of the field calculated in points inside τ, not on its boundary. This is the reason of the superscript "-".
So I'm not really specifying the tangential component of the field on S.
Do you agree?

Thanks for your reply.

Well, in cases where the limit is finite[1] (which is most often in the cases I have considered) I would have to disagree. There are idealized boundaries, for example the knife edge and the cone point, where tangent components become weakly singular, but these can be made finite with small inconsequential smoothing of the boundary in the neighborhood of problem points.

[1] Limit is understood as approaching the boundary along a curve contained within the region of interest excluding the boundary.

Edit: Here I'm referring to a material boundary such as a perfect electric conductor or surface of a dielectric volume. The Love Theorem discussion refers to an arbitrarily selected boundary which may or may not be a material surface.

 

[Paul Colby]

Okay, I've done a little read (perhaps too little) on the Love theorem. If I read correctly the sources of the fields are entirely within a region, $R$. One replaces these sources with surface currents on the boundary, $\partial R$ with zero fields inside $R$. Since in the first situation with the original sources, the fields are analytic functions at each boundary point, correct?

 

[Unconscious]

Yes, the fields are "well-behavior functions". So, on each point of the boundary we can define the normal unit vector and so we can talk about tangential field and normal field to S in each point.

 

[Paul Colby]

I don't know what "well-behavior" functions are exactly. For analytic functions of a single complex variable, there is a unique analytic continuation. For time harmonic solutions of Maxwell's equations, the field components are analytic in each of the coordinates, $x$, $y$ and $z$ at every source free point. Each component will have a unique analytic continuation about each source free point. This of course assumes a theorem about differential equations that I can't quote or reference off hand and therefore should be taken with some skepticism. None the less I recall that it is a theorem.

 

[vanhees71]

I guess with "Love theorem" you mean the boundary conditions of the electromagnetic field in matter.

You derive them from the integral form of the macroscopic Maxwell equations. Let's start from the local form. In Heaviside Lorentz units we have
$$\vec{\nabla} \times \vec{E}+\frac{1}{c} \partial_t \vec{B}=0,\\
\vec{\nabla} \cdot \vec{B}=0,\\
\vec{\nabla} \times \vec{H} -\frac{1}{c} \partial_t \vec{D}=\frac{1}{c} \vec{j},\\
\vec{\nabla} \cdot \vec{D}=\rho.$$

There are equations of the "curl type" and of the "div type". The latter ones are simpler to treat.

Take the fourth equation above and integrate it over the red "pill box". We assume that $\vec{D}$ has at most a jump across the boundary but not something like a $\delta$-distribution like singularity. The latter may be the case for $\rho$, if there's a surface-charge distribution $\sigma_{Q}$ along the boundary. Now use Gauss's theorem, and you get
$$\vec{n} \cdot (\vec{D}_1-\vec{D}_2)|_{\text{boundary}} =\sigma_{Q},$$
where I've first made $\Delta h \rightarrow 0$ and then shrunk $\Delta f$ to a point on the boundary. This means that the jump of the electric excitation $\vec{D}$ equals a possibly presend surface charge.

Using the same figure for the 2nd equation you get
$$\vec{n} \cdot (\vec{B}_1-\vec{B}_2)|_{\text{boundary}}=0.$$For the "curl equations" look at

The the first equation (Faraday's Law of induction) and integrate it over the red rectangle with the boundary $C$. The choice of the tangent vector $\vec{t}$ on the boundary is thereby arbitrary. Using Stokes's theorem you get, assuming that $\partial_t \vec{B}$ has no $\delta$-function like singularities at the surface
$$\vec{t} \cdot (\vec{E}_1-\vec{E}_2)=0.$$
This holds true for any tangent vector, and this implies for the tangential component of $\vec{E}$, and thus you can write the equation thus also in the form
$$\vec{n} \times (\vec{E}_1-\vec{E}_2)=0.$$
Analogously you get from the 3rd equation
$$\vec{n} \times(\vec{H}_1-\vec{H}_2)=\vec{K}_Q,$$
where $\vec{K}_Q$ is a possibly existing surface current along the boundary. If you have (hard) ferromagnetica, this also includes corresponding jumps of magnetization across the boundary.

Here, I used only the usual Maxwell equations. If you add (as far as we know fictitious) magnetic monopoles, you get also different boundary conditions (like the 2nd Eq. in the OP #1).

 

[Unconscious]

I don't know what "well-behavior" functions are exactly. For analytic functions of a single complex variable, there is a unique analytic continuation. For time harmonic solutions of Maxwell's equations, the field components are analytic in each of the coordinates, $x$, $y$ and $z$ at every source free point. Each component will have a unique analytic continuation about each source free point. This of course assumes a theorem about differential equations that I can't quote or reference off hand and therefore should be taken with some skepticism. None the less I recall that it is a theorem.

I can't understand this answer at all, because I have not studied complex analysis.
However, it seems to me that the key to solving the problem lies in "extending" the value of the field near the boundary, even on the boundary itself. Am I wrong? If what I said is true, I don't understand how this "extension" could be justified.
Thank you very much.

I guess with "Love theorem" you mean the boundary conditions of the electromagnetic field in matter.

No, the equivalence theorem doesn't need two real different means on which apply boundary conditions. It is just an equivalent formulation of the same EM problem, that substitutes real sources with other ficticious sources.
Thank you for your answer.

 

[Paul Colby]

I can't understand this answer at all, because I have not studied complex analysis.

That would limit things. I don't recommend following that path I was trying to hack through the weeds because it isn't the standard way of looking at these types of questions.

I do have a question with your statement of the uniqueness theorem in #1. I would benefit from some clarification. In #1 you seem to claim that specifying one tangent component on S is enough to uniquely determine the other. Let's say S is a perfect conducting sphere and the interior of the sphere is vacuum. At a resonance frequency, the tangent E field on S is zero while the tangent H field is given by a suitable sum of spherical bessel functions which are regular at the origin, $H(R,\theta,\phi)$. It's also clear that $2H(r,\theta,\phi)$ also supplies a different solution which also has a vanishing $E$-filed on $S = R$. I must be missing something as this seems to violate the theorem?

Edit: Okay, quick google says both tangent components must be specified for the solution to be unique within the region. Is this your understanding?

 

[Unconscious]

No, for every point of S, one between tangent E component or tangent H component is sufficient.
This is one condition, the other is that the field must be a solution of Maxwell equations and the medium equations.
If this two simple conditions are satisfied, then the field in the volume inside S is unique.

 

[Paul Colby]

Yes, this is also my understanding. The surface currents required to make the fields inside the region zero are found by solving the the first pair of equations in #1 assuming $E^+ = 0$ and $H^+=0$ for $J_S$ and $-J_m$ yielding the second pair of equations you provided in #1. Wherein lies your doubt?

Going through the use of equivalent surface currents, I no longer return because in this way I have not imposed the value of the tangential field on S, but I have only imposed a discontinuity between the tangential fields inside S and outside S But the field exactly on S?

Okay, the first set of equation are true for all situations as explained in detail in #10. Requiring $E^+=0$ and $H^+=0$ while leaving $E^-$ and $H^-$ unchanged from the initial tangent field values yields the surface currents of Love's theorem. All quantities then being unique with the field discontinuities at their original values.

 

[Unconscious]

Wherein lies your doubt?

In the fact that I cannot conclude that

'the field inside the volume bounded by S is unique because: 1. the field is solution of Maxwell equations and 2. the tangential component of the EM field are assigned on S'

because 'the tangential component of EM field are assigned on S' is false. The tangential component are not assigned to S. On S currents are assigned, not fields.

Okay, the first set of equation are true for all situations as explained in detail in #10.

Ok, I agree.

Requiring E+=0E+=0E^+=0 and H+=0H+=0H^+=0 while leaving E−E−E^- and H−H−H^- unchanged from the initial tangent field values yields the surface currents of Love's theorem.

I agree again.

All quantities then being unique with the field discontinuities at their original values.

How can you say that all the field quantities are unique? How have you calculated the field on S? On S we set some currents, but how can we say who is the field on the points of S (where that currents flow)?

 

[Paul Colby]

Well, for a -perfect- conductor there are exactly no fields on the interior of the conductor. There is in general a surface current given by

$J(x) = \hat{n}(x) \times H(x)$

where $H(x)$ is evaluated at the surface point, $x$, and $\hat{n}(x)$ is the outward directed normal at the same point. From #10 currents and surface fields may be used interchangeably. One may even show that $J(x)$ given above obeys the usual continuity/conservation relation

$i\omega \rho + \nabla \cdot J = 0$

by applying the field equations.

 

[Paul Colby]

On S currents are assigned, not fields.

These are the very same thing.

The fields on S are obtained by solving the field boundary value problem. If one impresses a current on a surface, then the tangent component on the boundary is provided by that current at each point.

Another point to leave you with is the first 2 equations in #1 define the total current in the boundary. This total is the sum of a current from each side of the boundary.

Sorry for the frequent editing. You're asking an important question.

 

[Paul Colby]

Another thought occurs. Solution uniqueness is not the same as solution existence which may be a source of some confusion. Uniqueness says there is only one solution to a well posed boundary value problem. The existence of a solution with a given (arbitrary) assignment of boundary values on the surface isn't guaranteed.

 

[Unconscious]

If one impresses a current on a surface, then the tangent component on the boundary is provided by that current at each point.

No, a current on a surface defines the jump, between two values: the field inside and the field outside (tangential component). Surface current says nothing about field on the surface.

Maybe, in the Love theorem, one can assign the field also on the surface, forcing it to the desired value to make uniqueness theorem applicable.
This would be the answer to the question, it seems that there is no problem in the mathematical steps involved in the Love theorem proof if one does this other assignment.

 

[Paul Colby]

No, a current on a surface defines the jump, between two values: the field inside and the field outside (tangential component). Surface current says nothing about field on the surface.

Then we disagree. These currents are at the level of mathematical idealizations. The field discontinuity determines the net or total current. For example, a surface where $H(x)$ is continuous everywhere in a neighborhood of a surface point, $x$, has two surface currents,

$J_1(x) + J_2(x) = \hat{n}_1(x)\times H(x) + \hat{n}_2(x)\times H(x)$

where $\hat{n}_1(x)$ is the surface normal on side 1 and $\hat{n}_2(x)$ the surface normal on side 2. Clearly, $\hat{n}_1(x) = -\hat{n}_2(x)$. This leads to,

$\hat{n}_1(x)\times (H(x) - H(x)) = 0$

So, it's only when there is a jump discontinuity at, $x$, that there is a net surface current. The surface in Love's theorem may appear anywhere in space, not just at material boundaries. That's my take.

 

[Unconscious]

In your last equation, the first $H(x)$ is not the same as the second $H(x)$, because they are not calculated in the same $x$. In fact, if this were the case, then $H(x)-H(x)=0$ independently of the fact that H is continuous or not at S.

 

[Paul Colby]

In your last equation, the first $H(x)$ is not the same as the second $H(x)$, because they are not calculated in the same $x$. In fact, if this were the case, then $H(x)-H(x)=0$ independently of the fact that H is continuous or not at S.

Not in the case of an infinitesimally thin surface for which the Love theorem is formulated.

okay, there are two sides, $x^+$, and $x^-$. The case considered in #20 clearly and carefully considers $H(x^+)=H(x^-)$.

 

[Paul Colby]

then H(x)−H(x)=0H(x)−H(x)=0H(x)-H(x)=0 independently of the fact that H is continuous or not at S.

You don't really mean this, right?