Poisson's and Laplace's Equations

Poisson's and Laplace's equations are simply the rule the voltage has to follow at every point: in empty space, the voltage at each point is the average of the voltages all around it. They're named after two French mathematicians, Siméon Denis Poisson and Pierre-Simon Laplace, and they're how you find the voltage everywhere when all you know is the voltage on the metal parts. The equation $\nabla^2V = -\rho/\varepsilon_0$ looks intimidating, but it's just three pieces: $\nabla^2V$ (how much $V$ at a point differs from the average around it), $\rho$ (how much charge is sitting there) and $\varepsilon_0$ (the usual constant). No charge means no difference from the average, and that's Laplace's equation. Bear with the derivation; the idea really is that simple.

Know the voltages, not the charges. Why do we need this? In a real device you know what each piece of metal is connected to: this electrode at 100 V, that shield grounded. You don't know where the charge on them ends up, so Coulomb's law and superposition can't get started. You need an equation for $V$ itself, with the voltages as boundary conditions.

Poisson's equation. Substitute $\mathbf{E} = -\nabla V$ into $\nabla\cdot\mathbf{E} = \rho/\varepsilon_0$ from Divergence and Gauss's Law:

In charge-free space this is Laplace's equation, $\nabla^2V = 0$, the same equation as steady heat flow in Laplace's Equation on a Rectangle.

One dimension first. Between parallel plates with no charge, $V'' = 0$: the potential is a straight line and $E$ is uniform, just as we found in the last unit. Fill the gap with uniform charge $\rho$ and $V'' = -\rho/\varepsilon_0$: a parabola, bulging toward higher potential for positive charge. That's the averaging idea at work. A straight line is the one curve where every point is the average of its neighbors, and charge is what pushes the curve off that average.

Averages and extremes. $\nabla^2V$ measures how $V$ at a point differs from the average of its neighbors. Expanding in Taylor series on a square grid of spacing $h$ gives $V_E + V_W + V_N + V_S \approx 4V_P + h^2\nabla^2V$ in two dimensions. So where $\nabla^2V = 0$, each value is the average of its neighbors, and in the continuum, $V$ at any point equals its average over any sphere around it: the mean value property, as in Laplace's Equation on a Disk. Think about what that rules out. A point can't be higher than all its neighbors if it's their average, so in a charge-free region $V$ has no local maximum or minimum, only saddles. A charge would need a minimum of potential energy to sit stably, so no arrangement of static charges can trap a charge in empty space (Earnshaw's theorem). Ion traps have to cheat, with oscillating or magnetic fields.

Uniqueness. Suppose $V_1$ and $V_2$ both satisfy Poisson's equation with the same $\rho$ and the same boundary voltages. Their difference $W$ satisfies Laplace's equation with $W = 0$ on the boundary. It has no interior maximum or minimum, so it can't rise above or dip below the 0 on its boundary, and it's 0 everywhere. That's a remarkably freeing result: any solution you find, by any trick at all, even a lucky guess, is the solution.

The method of images. Here's that freedom in action. A charge $q$ sits at height $d$ above a grounded metal plane. Guess: replace the plane with a charge $-q$ at depth $d$. Every point of the plane is equally far from $q$ and $-q$, so $V = 0$ there, and above the plane the guess satisfies Poisson's equation with the right charge. By uniqueness it's the true potential above the plane. The induced charge follows from $\sigma = \varepsilon_0E$ at the surface:

which adds up to exactly $-q$, and the charge is pulled toward the plane with force $\frac{kq^2}{(2d)^2}$, as if the image were real.

Beyond tricks. For general shapes, separation of variables or a computer takes over. Relaxation methods sweep a grid, replacing each value with the average of its neighbors (plus $\frac{h^2\rho}{4\varepsilon_0}$ where there's charge) until nothing changes: a giant linear system solved one equation at a time.

A common mistake. For a potential that depends only on $r$, it's tempting to write the Laplacian as just $V''$, since that's what it is in one dimension. It isn't: in spherical symmetry it's $\frac{1}{r^2}\frac{d}{dr}\left(r^2\frac{dV}{dr}\right)$, and the extra terms are what turn the straight-line solution into $\frac Ar + B$. Also keep the minus sign in $\nabla^2V = -\rho/\varepsilon_0$, and only use Laplace's equation in regions that really hold no charge.

Every electrostatic design tool solves these equations. Chip designers extract the capacitance between millions of wires by solving Laplace's equation around them; device simulators solve Poisson's equation inside every transistor; high-voltage engineers shape insulators and grading rings from computed equipotentials; electron microscopes focus their beams with electrode shapes found the same way. Electrical impedance tomography runs the problem backward, inferring what's inside the body from voltages measured on the skin. And Earnshaw's theorem is why the ion traps in mass spectrometers and quantum computers need radio-frequency fields.