Timeline for Can someone show me how Green's function would apply for this simple case?
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| Aug 29, 2014 at 2:37 | comment | added | hyd | There are various ways to show that explicitly. You may take the Fourier transform of the Poisson equation. Then you see $\tilde{G}(\vec{k}) = \varepsilon^{-1}_0k^{-2}$, where $\tilde{G}$ is the Fourier transform of $G$ and $k=|\vec{k}|$. Transforming back , you obtain $G(\vec{r}) = \varepsilon^{-1} \int d^3\vec{k} \frac{e^{i\vec{k}\vec{r}}}{k^2}$. To perform this integral, you can write $d^3\vec{k} = k^2\sin(\theta)dk d\theta d\varphi$ and $\vec{k}\vec{r} = kr\cos(\theta)$, where we have chosen the $\vec{r}$ along z-axis. Now you can use standard formula to get the final result. | |
| Aug 29, 2014 at 1:35 | comment | added | Fraïssé | Thanks, physically it makes sense, however, how would you go about deriving $G(|r-r'|)$ from scratch without invoking coloumb's law? | |
| Aug 29, 2014 at 1:33 | vote | accept | Fraïssé | ||
| Aug 28, 2014 at 11:43 | comment | added | gj255 | Ah OK, I see what you're saying. | |
| Aug 28, 2014 at 10:30 | history | edited | hyd | CC BY-SA 3.0 |
added 12 characters in body
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| Aug 28, 2014 at 10:29 | comment | added | hyd | (1) By 'obviously satisfying ...', I mean it on the physics not mathematics: Coulomb's law is a solution to Possion's equation (thanks for correcting me for using Laplace equation) in the presence of a point charge. (2) I would have used the symbol by the questioner, but I do not know how to make it in Latex. (3) Thanks for pointing out that the missing $d^3\vec{r}'$. | |
| Aug 28, 2014 at 9:25 | comment | added | gj255 | Do you mean 'obviously satisfies Poisson's equation'? I don't personally find this obvious --- first you must show that the Laplacian of $G$ vanishes for $r \neq r'$, and then you must show that the integral of the Laplacian of $G$ over a ball of radius $\epsilon$ centred on $r = r'$ gives you the $1/\epsilon_0$ that you seek. I would also say that the notation $\nabla^2$ is more common than $\partial^2$, and since the questioner used the former in their answer, I would favour it. As a final point, you're missing the $d^3 r'$ on one of your integrals. | |
| Aug 28, 2014 at 7:08 | history | answered | hyd | CC BY-SA 3.0 |