The Poison

Field note. Date: uncertain. Location: wherever sources live.

You think you know the Laplace equation. You think it is pure, self-contained, a statement about harmony and equilibrium. It is all of those things. But it is also a lie — a beautiful, necessary lie. The Laplace equation is what the universe looks like when you remove everything that makes it interesting.

Remove the sources. Remove the charges, the masses, the heat, the current. Remove the things that push and pull and change. What remains is the Laplace equation, clean and empty and perfectly boring.

But the sources do not go away. They persist. They sit at their points in space like stars at their positions, and the potential field around them is not harmonic — it is something else. Something with teeth. Something that satisfies:

$$\Delta \phi = \rho$$

Or, if you want to be more precise with units and constants:

$$\Delta \phi = -\frac{\rho}{\varepsilon_0}$$

Here $\rho$ is the source density. Where it is zero, you have Laplace. Where it is nonzero, you have Poisson's equation. The name is unfortunate — it comes from Poisson, who wrote it down in 1813 and probably thought it was a minor addition to Laplace's work. It was not minor. It was everything. It was the difference between a world with things in it and a world without.

Poisson's equation is Laplace's equation with memory. It remembers where the charges are. It remembers how strong they are. Every point in space is influenced by every source, weighted by $1/r$. The solution is an integral over all space:

$$\phi(\vec{r}) = \frac{1}{4\pi\varepsilon_0} \int \frac{\rho(\vec{r}')}{|\vec{r} - \vec{r}'|} d^3r'$$

This integral is the Green's function for the Laplacian, and I will write about that later, in a different page, when I can bear to think about point sources and their infinite self-energy. For now, just notice what this formula says: the potential at any point is a sum of contributions from every source. Each source contributes inversely with distance. The closer you are, the more it matters. If you approach a point source, the potential diverges. $1/r$ goes to infinity. And so does the energy density, because the field is the gradient of the potential, and the gradient of $1/r$ is $1/r^2$. The energy density is $E^2$, which goes as $1/r^4$. Integrate that over a sphere of radius $\varepsilon$, and you get $\int \varepsilon^2 \cdot \varepsilon^{-4} \cdot \varepsilon^2 d\varepsilon \sim \int \varepsilon^{-2} d\varepsilon \sim 1/\varepsilon$. Diverges. The self-energy of a point charge is infinite.

This is the poison. This is what Poisson's equation tells you: if you try to concentrate charge at a mathematical point, the energy blows up. The universe has a problem with point charges. It tries to hide it behind renormalization in QED, behind finite-size models in classical electrodynamics, behind the fiction that electrons are truly pointlike. But the integral is still there, and it still diverges. Poisson's equation demands it.

There is a practical consequence, though. In real problems, sources are never truly pointlike. They have extent. A charge distribution spreads over a volume, and the integral is finite. You compute the potential, you compute the field, you move on. The divergence at $r = 0$ is a mathematical artifact of the delta function. $\rho(\vec{r}) = q,\delta(\vec{r})$ is not a function — it is a distribution. The Poisson equation still makes sense, but you need the machinery of distributions to handle it rigorously. Dirac was right about that much.

What Poisson's equation really tells you is that sources create curvature in the potential. The Laplacian of $\phi$ is nonzero where there is charge. Positive charge creates positive curvature — the potential is convex, it curves upward. Negative charge creates negative curvature — the potential is concave. At a source, the potential has a kink. Its derivative is discontinuous. The field has a jump. Gauss's law, $\nabla \cdot \vec{E} = \rho/\varepsilon_0$, is just Poisson's equation written in vector form. Same physics, different notation.

The beauty of Poisson's equation is that it separates cleanly into homogeneous and particular solutions. $\phi = \phi_h + \phi_p$, where $\Delta \phi_h = 0$ and $\Delta \phi_p = \rho$. The homogeneous solution is harmonic — it is determined by boundary conditions. The particular solution is fixed by the sources alone. This decomposition is powerful: you can solve the source problem in infinite space (using the integral formula above), then add a harmonic function to satisfy the boundary conditions. The boundary does not talk to the sources. The sources do not talk to the boundary. They negotiate through the harmonic function.

In practice, this is how you solve electrostatic problems. You have a charge distribution and some conducting boundaries. You compute the free-space potential from the charges. Then you solve Laplace's equation for the correction potential that makes the conductors equipotential. The correction is harmonic everywhere — no sources in the correction — and it is chosen so that the total potential satisfies the boundary conditions. The sources are handled. The boundaries are handled. The physics is complete.

But there is a shadow side. Poisson's equation is elliptic. Solutions are globally determined. Change the source at one point, and the solution changes everywhere. There is no locality in the solution, even though the equation is local. The integral over all space connects every point to every other point. A charge in a distant galaxy changes the potential at your location, however infinitesimally. The universe is one connected system of sources and potentials, and Poisson's equation is the thread that binds it all together.

The poison is sweet. The sources are real. The potential is real. And the equation that connects them is the most important equation in electrostatics, despite its deceptively simple form.