Understanding Green's Function in Electromagnetism

[quantum123]

How does Green's function work in electromagnetism?

 

[Born2bwire]

If you look at Jefimenko's Equations ( http://en.wikipedia.org/wiki/Jefimenko's_equations ), we can see that the electric and magnetic fields can be independently determined by the source currents and charges. A Green's function can be derived that relates a point source to the excited fields via an integral. Normally we are interested in electromagnetic waves and assume time-harmonic conditions. This means that we only have current sources that operate at a desired frequency. We can then derive a Green's function that relates a vector point source current at r' to the excited electric or magnetic fields at r. This Green's function turns out to be a dyadic Green's function.

The resulting electric field integral equation that relates a current source with an electric field by convoluting the dyadic Green's function with the currents is the foundation of many computational electromagnetic techniques like the method of moments. It also turns out that the dyadic Green's function of the magnetic field is a simple vector operation on the electric field's dyadic Green's function and thus we usually focus only on the electric field's Green's function in literature.

 

[quantum123]

How do you prove the Jefimenko's Equations ? In some textbooks , these are proved from the retarded potentials, which are in turn proved by Green's functions, but how do you prove the existence of Green's functions and that they work?

 

[Born2bwire]

How do you prove the Jefimenko's Equations ? In some textbooks , these are proved from the retarded potentials, which are in turn proved by Green's functions, but how do you prove the existence of Green's functions and that they work?

The Jefimenko's Equations are derived from Maxwell's Equations. As for Green's functions, they are mathematical tools. If you want to know more about their underlying theory you should look in a mathematics textbook. Most mathematics for physics textbooks should deal with the subject like Keener's "Principles of Applied Mathematics" (which to be honest is a rather horrid textbook for learning).

 

[kcdodd]

The simplest way to think of greens functions I have seen is as an inverse operator. For instance, creating an inverse to the laplace operator in electrostatics will give back the full potential.

 

[quantum123]

The simplest way to think of greens functions I have seen is as an inverse operator. For instance, creating an inverse to the laplace operator in electrostatics will give back the full potential.

Wow, it is that simple?
I have found something here, that seems to agree with what you said:-
http://www.hep.upenn.edu/~rreece/docs/notes/jefimenko_equations.pdf

 

[Born2bwire]

Wow, it is that simple?
I have found something here, that seems to agree with what you said:-
http://www.hep.upenn.edu/~rreece/docs/notes/jefimenko_equations.pdf

Yeah. Given some nth-order linear ordinary differential operator, the operator acting on its Green's function is a dirac delta function. This allows us to solve for the excitation to any known response to this operator. That is, if the operator acting on E(r) gives us J(r), then we can determine E(r) by integrating the Green's function and the known J(r).

This can be easily applied to electromagnetics by the fact that we can decompose the electric field into a vector wave equation that relates a differential operator acting on the electric field that gives the equivalent source current for that field. So the Green's function for this operator can allow us to calculate the electric field component of the wave due to the excitation currents. Likewise if we take the general case where we allows for charges we should be able to eventually derive Jefimenko's Equations.

 

[quantum123]

Strange, the way you explain it here seems so simple compared to some texts, which uses Green's theorem, Green's identity with strange boundary conditions in electrostatics to explain Green's functions. (eg Jackson).
And talking about Dirac delta , reminds me of bra-ket in , sorry to divert , QM. Is this somehow related to bra-ket in QM?