Meaning of Dyadic Green Function in Electromagnetism

[Apollo2010]

I know that in the vector potential formulation one can use a scalar Green function (to find the said potential and from then on the electric and magnetic fields), and that this works because the components of the potential are in the same direction as those of the source - i.e. a current in the x-direction will give you the x-component of the potential.

I also know that this is no longer true if instead of a differential equation involving the vector potential, we use equations involving the fields, and this is why we have to use a dyadic Green function in this case rather than a scalar one.

My question is: what do the columns and rows of the dyadic represent?

I assume one index represents the direction of the component of the Green's function itself, i.e. the direction of the component of the field due to a unit idealized differential bit of current? And the other index represents the component of the differential bit of current responsible for that dyadic element? For instance, Gxy would be the x component of the field due to the y component of the source current (or the other way around?)? Is that right, and if so which index represents what?

 

[Meir Achuz]

"Gxy would be the x component of the field due to the y component of the source current" is correct.

 

[Born2bwire]

You should take a look at tensor products to see how the dyad acts, though Meir Achuz pretty much explains it, but it isn't that scary of an operation. I always looked at it like a matrix-vector product (as scary as that may be to mathematicians).

 

[Apollo2010]

Thank you very much!

 

[sardar]

The dyadic Green function in electromagnetism is a mathematical tool used to solve Maxwell's equations for the electric and magnetic fields in the presence of sources, such as charges and currents. It is a tensor quantity that relates the fields at a given point to the sources at another point in space.

To answer your question, the columns and rows of the dyadic represent the directions of the source and field components, respectively. For example, Gxy would represent the contribution of the y-component of the source to the x-component of the field. This is because the dyadic Green function takes into account the vector nature of the fields, allowing for the possibility of a cross-coupling between different components.

In the vector potential formulation, a scalar Green function is sufficient because the potential and the source are both vector quantities in the same direction. However, when working with the fields, a dyadic Green function is necessary to properly account for the vector nature of the fields.

Overall, the dyadic Green function is a crucial tool in solving complex electromagnetic problems and understanding the relationship between the fields and sources in a given system.