I revisited electrostatics and I am now wondering what the big fuzz about Poisson's equation $$\nabla^2 \phi = -\frac{\rho}{\varepsilon_0}$$ is. Wiki says

One of the cornerstones of electrostatics is setting up and solving problems described by the Poisson equation. Solving the Poisson equation amounts to finding the electric potential φ for a given charge distribution.

while the text I am currently studying [1] defines $\phi$ via

$$\phi(\mathbf{r}) = \frac{1}{4\pi\varepsilon_0}\iiint_{\mathbf{r}' \in \mathbb{R}^3} \frac{\rho(\mathbf{r}')}{\|\mathbf{r}-\mathbf{r}'\|} dV' \ .$$

So why would you go all the way to a complicated partial differential equation when there is already a closed-form formula available (which solves exactly what the Poi.Eq. is good for according to wiki)? Is that integral not generally valid or what am I missing?

EDIT: I found out that the integral above is called the d'Alembert solution of Poisson's equation, but that doesn't answer my question about the application-importance of the latter.

[1] Equation (1.21) in http://users.ox.ac.uk/~math0391/EMlectures.pdf


The integral you wrote integrates $\rho$ over the whole space. This is impossible to calculate if $\rho$ is not known in the whole space.

For example, when the charge $\rho$ is known only inside some finite region enclosed by a metallic shell, the shell is known to have constant potential $\phi$ on its inner surface. This information is useless in calculating the integral because the potential does not occur in it and the charges outside are still unknown.

But together with the Poisson equation, the boundary condition determines the potential $\phi$ at all points inside uniquely. There are methods to find $\phi$ based on this, both analytical and numerical.


I just stumbled upon a perfect explanation in The Feynman Lectures Vol.II, Chapter 4: http://www.feynmanlectures.caltech.edu/II_04.html

We will begin with the simplest situations—ones in which the positions of all charges are specified. If we had only to study electrostatics at this level, life would be very simple — in fact, almost trivial. Everything can be obtained from Coulomb’s law and some integration, as you will see. In many real electrostatic problems, however, we do not know, initially, where the charges are. We know only that they have distributed themselves in ways that depend on the properties of matter. The positions that the charges take up depend on the $\bf E$ field, which in turn depends on the positions of the charges. Then things can get quite complicated. If, for instance, a charged body is brought near a conductor or insulator, the electrons and protons in the conductor or insulator will move around. The charge density $\rho$ may have one part that we know about, from the charge that we brought up; but there will be other parts from charges that have moved around in the conductor. And all of the charges must be taken into account. One can get into some rather subtle and interesting problems.


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