Equivalence principle and the Uniqueness theorem

In summary, the equivalence theorem tells me that I can compute the field in each point of ## V ## as ( Stratton-Chu solution):$$\mathbf{E}(\mathbf{r})=\frac{1}{4\pi}\int_V\left( \frac{\rho}{\epsilon}\nabla'\psi-j\omega\mu\psi\mathbf{J} \right)dV+\frac{1}{4\pi}\int_{\partial V}(\mathbf{n}_0\cdot\mathbf{E})\nabla'\psi-j\frac{\omega
  • #1
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We work with Maxwell's equations in the frequency domain.
Let's consider a bounded open domain ## V ## with boundary ## \partial V ##.

1. The equivalence theorem tells me that if the field sources in ## V ## are assigned and if the fields in the points of ## \partial V ## are assigned, then I can compute the field in each point of ## V ## as ( Stratton-Chu solution):

$$\mathbf{E}(\mathbf{r})=\frac{1}{4\pi}\int_V\left( \frac{\rho}{\epsilon}\nabla'\psi-j\omega\mu\psi\mathbf{J} \right)dV+\frac{1}{4\pi}\int_{\partial V}(\mathbf{n}_0\cdot\mathbf{E})\nabla'\psi-j\omega\psi(\mathbf{n}_0\times\mathbf{B})+(\mathbf{n}_0\times\mathbf{E})\times\nabla'\psi)dS$$
$$\mathbf{B}(\mathbf{r})=\frac{1}{4\pi}\int_V\left( \mu\mathbf{J}\times\nabla'\psi \right)dV+\frac{1}{4\pi}\int_{\partial V}(\mathbf{n}_0\cdot\mathbf{B})\nabla'\psi-j\frac{\omega\psi}{c^2}(\mathbf{n}_0\times\mathbf{E})+(\mathbf{n}_0\times\mathbf{B})\times\nabla'\psi dS$$

where ##\psi=\frac{e^{-jkR}}{R}## is the Green function.

2. The uniqueness theorem tells me that if only the tangential component of only the electric field (or only the magnetic field) on ## \partial V## is assigned, then the field in the points inside ## V ## is uniquely determined.

I wonder then: why it seems that for the equivalence theorem it is necessary to know the entire field (both electric and magnetic and both normal and tangential components) on ## \partial V ##, while the uniqueness theorem needs much less information ? Is it just a question of calculation? In the sense that perhaps it is true that less information is needed to uniquely determine the field, but then to actually calculate it we do not know how to do it if we do not have all the information that the equivalence theorem requires on ## \partial V ##?
 
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  • #2
I think in this case you need the boundary conditions as well as initial conditions for the em. field. You can also work in the boundary conditions into the Green's function. The Green's function you quote should be the usual retarded Green's function (i.e., the out-going wave solutions of the Helmholtz equation, i.e., the temporal part in your ansatz should be ##\exp(+\mathrm{j} \omega t)##, i.e., the engineering convention, which is opposite from the physics convention! This Green's function holds if you take the entire ##\mathbb{R}^3##, where the complete sources ##\rho## and ##\vec{j}## are defined ("microscopic electrodynamics"), i.e., in your boundary-value problem you need to add an arbitrary solution of the source-free Helmholtz equation, which then has to be found by employing the initial and boundary conditions.
 
  • #3
Why do you talk about 'initial conditions'? We are working in frequency domain, so there is no 'initial time', there are only boundary conditions. No?

The eventually presence of an additional source-free solution will modify the field value on the boundary points, so this fact is automatically taken into account:
1. in the Stratton-Chu solution, by the terms in the surface integral (boundary conditions),
2. in the uniqueness theorem, by the boundary conditions again.

I don't know if you wanted to tell me anything else that I didn't understand.
 
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  • #4
I'm not so familiar with the engineering approach. So if you just look at the harmonic time dependence, i.e., the Helmholtz equations boundary values are sufficient.
 
  • #5
When you talk about Helmoltz equation, what I think is this:

$$\nabla^2 \mathbf{A}(\mathbf{r})+k^2\mathbf{A}(\mathbf{r})=-\mathbf{J}(\mathbf{r})$$

where ##\mathbf{A}(\mathbf{r})## is the magnetic vector potential and ##\mathbf{J}(\mathbf{r})## is the current density (the source).
Se, when you say that ' Helmholtz equations boundary values are sufficient ' I agree, because assigning the values of the magnetic vector potential on ##\partial V## is equivalent (in the sense that then the field is uniquely determined) to assign the values of the field on ##\partial V##.
But this does not solve the problem, it still seems that the extra-information needed by Stratton-Chu integration are only for effective calculus reasons.
 
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Related to Equivalence principle and the Uniqueness theorem

1. What is the Equivalence Principle?

The Equivalence Principle is a fundamental concept in physics that states that the effects of gravity are indistinguishable from the effects of acceleration. This means that an observer in a gravitational field cannot tell the difference between being at rest in a uniform gravitational field and being in an accelerated reference frame.

2. How does the Equivalence Principle relate to the Uniqueness Theorem?

The Uniqueness Theorem is a mathematical principle that states that a solution to a physical problem is unique if it satisfies certain boundary conditions. In the context of the Equivalence Principle, the Uniqueness Theorem can be used to show that the laws of physics are the same for all observers in a given gravitational field, regardless of their position or motion.

3. What are some practical applications of the Equivalence Principle and the Uniqueness Theorem?

The Equivalence Principle and the Uniqueness Theorem have many practical applications in physics and engineering. For example, they are used in the construction of accelerometers and gyroscopes for navigation systems, as well as in the development of theories of gravity such as Einstein's General Theory of Relativity.

4. Are there any exceptions to the Equivalence Principle?

While the Equivalence Principle holds true in most cases, there are some exceptions. For example, at very small scales, such as in the realm of quantum mechanics, the Equivalence Principle breaks down and the effects of gravity and acceleration can be distinguished. Additionally, the Equivalence Principle does not apply in situations involving extremely strong gravitational fields, such as near a black hole.

5. How has the Equivalence Principle and the Uniqueness Theorem been tested and confirmed?

The Equivalence Principle and the Uniqueness Theorem have been extensively tested and confirmed through various experiments and observations. One of the most famous examples is the Eötvös experiment, which demonstrated the equivalence of inertial and gravitational mass to a high degree of precision. Additionally, the predictions of the Equivalence Principle have been confirmed through observations of gravitational lensing and the behavior of objects in orbit around massive bodies.

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