Jackson's Electrodynamics: Green's function prefactor

Question

In Ch. 6 of Jackson's Classical Electrodynamics 3rd ed., the Helmholtz equation Green's function is written as satisfying the following inhomogeneous equation (Eqn. 6.36): $$ (\nabla^2 + k^2)G(\mathbf{x},\mathbf{x}^{\prime}) = -4 \pi \delta^3(\mathbf{x} - \mathbf{x}^{\prime})$$

However, I've seen other sources write the above equation without the factor of $4 \pi$. Where does this extra factor come from? Explicit math would be greatly appreciated.

Which sources? Of course you can just redefine $G\to G/4\pi$ - it is a matter of convention. — Tobias Fünke, Jul 11 '23 at 22:06
For example, Novotny's Principles of Nano-optics, Eqn. 2.73 doesn't have the $4\pi$. I was wondering if this has to do with the definitions of the Fourier transforms? If so, can someone show the math for this? — photonica, Jul 11 '23 at 22:09
You're overthinking this too much. You should realize that once you have a solution for the equation given here, you have a solution for the equation without this pre-factor and vice versa. — Tobias Fünke, Jul 11 '23 at 22:12
Don't pretty much all factors of $4\pi$ stem from angular symmetry (e.g., integrals over $\theta$ and $\phi$)? — Kyle Kanos, Jul 11 '23 at 22:16
For comparison, Jacksons' 2ed does not have the $4\pi$ (eq. 16.17) — Mauricio, Jul 11 '23 at 22:17
This may be a repeat of what others have explained, but $4\pi$ is the solid angle of a sphere. The difference is which definition includes the $4\pi$. Comparing the description of Coulomb's law in the cgs unit system with the description in the MKSA unit system, the $4\pi$ difference is noticeable. — HEMMI, Jul 12 '23 at 04:12
Related (???) : Divergence of $\frac{ \hat {\bf r}}{r^2} \equiv \frac{{\bf r}}{r^3}$ , what is the 'paradox'?. — Frobenius, Jul 12 '23 at 08:13

score 0 · Answer 1 · answered Jul 12 '23 at 13:44

Jackson is one of the authors that uses Gaussian units instead of metric units:

https://en.wikipedia.org/wiki/Gaussian_units

These usually appear in literature which deals with electromagnetic phenomena, where units become somewhat complicated due to the electric charge unit in SI having a somewhat "dimensionless" unit.

To counteract this, the electro-static units were devised by Gauss in the 19th century purely because it was more convenient for electromagnetic calculations, but it turned out to be quite inconvenient for matching with other measurement systems. In the end, for electromagnetic phenomena some calculations are simplified, but once you add any other physics you have to do some conversions.

https://en.wikipedia.org/wiki/Centimetre%E2%80%93gram%E2%80%93second_system_of_units#Electrostatic_units_(ESU)

You can also find further information in the wikipedia for inhomogeneous electromagnetic wave equation:

https://en.wikipedia.org/wiki/Inhomogeneous_electromagnetic_wave_equation

You can find the derivation in this paper as well:

https://pubs.aip.org/aapt/ajp/article/78/2/181/570645/On-the-derivation-of-the-Green-s-function-for-the

But you can see that they claim "the constant −4π is introduced by convention" but it generally comes from the expression of the gaussian units for electromagnetic fields.

The third edition of Jackson — which the OP refers to — uses primarily SI units, not Gaussian units. (There are a few chapters that still use Gaussian units, but Chapter 6 isn't one of them.) — Michael Seifert, Jul 12 '23 at 13:50
I went to my 4th edition of the Jackson (in German) to double check and the equation number is the same -- I think the "convention" that I mentioned in the equation does come from historical reasons from the Gaussian units and the solution chosen is simply not updated. I thought maybe it came from the fourier transforms but then it does not make a lot of sense if the \pi term is not squared (and I don't think denominators simply add up). — ondas, Jul 12 '23 at 15:55

Jackson's Electrodynamics: Green's function prefactor

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