EM field of dipole derivation from Green's function

[krindik]

Hi all,

I know that the electric field generated by a dipole is given by

$\mathbf{E}= [1-i(\omega/c) r]\frac{3 (\mathbf{p}\cdot\mathbf{r})\mathbf{r}-\mathbf{p} }{r^3}+(\omega/c)^2\frac{\mathbf{p}-(\mathbf{p}\cdot\mathbf{r})\mathbf{r}}{r} e^{i(\omega/c)r}$
where $\mathbf{p}$ is the dipole's dipole moment proportional to $e^{-i\omega t}$.

I'm struggling to find out how this is derived from a Green's function approach. Can somebody help me with this or point me to somebook/reference that shows derivation?

Thanks in advance.

cheers,
Krindik

 

[Born2bwire]

Some references that deal with deriving the dyadic Green's function are Jackson's "Classical Electrodynamics" and Chew's "Waves and Fields in Inhomogeneous Media."

 

[krindik]

Hi,

Thanks for the response.
This is what I understood of its derivation. Hope u could clarify.

The electric field due to the dipole can be given by
$\mathbf{E}(\mathbf{r}) = k^2 \mathbf{G}(\mathbf{r}, \mathbf{r}')\mathbf{p}(\mathbf{r'})$ where $\mathbf{G}(\mathbf{r}, \mathbf{r}')$ is the dyadic Green's functions for the dipole source, which is located at $\mathbf{r}'$ and k is the wave number in the unbounded background.
$\mathbf{G}(\mathbf{r}, \mathbf{r}')$ can be determined from the scalar Green's function $G_0(\mathbf{r}, \mathbf{r}')$ by
$\mathbf{G} = \left[\mathbf{I} + \frac{1}{k^2}\nabla \nabla \right]G_0(\mathbf{r}, \mathbf{r}')$ where $\mathbf{I}$ is the unit dyad, and $G_0 = \displaystyle\frac{e^{ik|\mathbf{r}-\mathbf{r}'|}}{|\mathbf{r}-\mathbf{r}'|}$.


Is this correct? Really appreciate your response.

cheers,
Krindik

 

[Born2bwire]

Hi,

Thanks for the response.
This is what I understood of its derivation. Hope u could clarify.

The electric field due to the dipole can be given by
$\mathbf{E}(\mathbf{r}) = k^2 \mathbf{G}(\mathbf{r}, \mathbf{r}')\mathbf{p}(\mathbf{r'})$ where $\mathbf{G}(\mathbf{r}, \mathbf{r}')$ is the dyadic Green's functions for the dipole source, which is located at $\mathbf{r}'$ and k is the wave number in the unbounded background.
$\mathbf{G}(\mathbf{r}, \mathbf{r}')$ can be determined from the scalar Green's function $G_0(\mathbf{r}, \mathbf{r}')$ by
$\mathbf{G} = \left[\mathbf{I} + \frac{1}{k^2}\nabla \nabla \right]G_0(\mathbf{r}, \mathbf{r}')$ where $\mathbf{I}$ is the unit dyad, and $G_0 = \displaystyle\frac{e^{ik|\mathbf{r}-\mathbf{r}'|}}{|\mathbf{r}-\mathbf{r}'|}$.


Is this correct? Really appreciate your response.

cheers,
Krindik

Yep, that should be pretty much it. Actually you skipped the first step, the relationship between the electric field and the source currents is integrated over all space as
$$\mathbf{E}(\mathbf{r}) = C\int_V \overline{\mathbf{G}}(\mathbf{r},\mathbf{r}') \cdot \mathbf{J}(\mathbf{r}') d\mathbf{r}'$$
I can't remember what the constant C is off hand. But in the case of the dipole, the current source is a point source and so we can drop the integration. Relating the resulting expression to what you have in the OP is a different story. They should be equivalent but it probably would require some work to show that. Chew's text doesn't derive the expression you gave originally but it's a good resource about deriving the dyadic Green's function and his chapter 2 applies it to dipoles and dipoles in layered media. Jackson's would probably work with an expression more closely related to what you gave. It's just that I am much more familiar with Chew's and Kong's texts than say Jackson.

 

[krindik]

Thanks a lot Born2bwire, u were really helpful.