# EM field of dipole derivation from Green's function

1. Jul 3, 2011

### 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?

cheers,
Krindik

2. Jul 4, 2011

### 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."

3. Jul 4, 2011

### 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

4. Jul 4, 2011

### Born2bwire

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.

5. Jul 4, 2011

### krindik

Thanks a lot Born2bwire, u were really helpful.