Laplace's equation

The formula

∇²φ = 0        i.e.   ∂²φ/∂x² + ∂²φ/∂y² + ∂²φ/∂z² = 0

The curvatures of the field in the three directions sum to zero at every point. Wherever the field bulges up one way, it must sag an equal amount another way. Solutions are called harmonic functions.

∇²φ = f(r)

Poisson's equation — the same operator with something on the right side ^[2]. ∇²V = −ρ/ε₀ for electrostatics with charge present, ∇²T = −q̇/k for steady heat conduction with internal generation, ∇²Φ = 4πG·ρ for gravity inside matter. Laplace is the special case where the region is empty of sources.

φ(center) = average of φ over any sphere drawn around that point

The mean value property. A harmonic function carries no private opinions: its value anywhere equals the average of its values on any surrounding sphere that stays inside the domain. Everything below follows from this one fact.

φ[i,j] = ( φ[i+1,j] + φ[i−1,j] + φ[i,j+1] + φ[i,j−1] ) / 4

The same statement on a grid — the five-point stencil. Set every interior node to the average of its four neighbors, repeat until nothing moves, and you have solved Laplace's equation. This is the relaxation method, and it is how the equation was solved by hand for decades before FEM.

Where you meet it

Steady-state thermal on a cold plate or chassis. Once the transient dies out, temperature in source-free metal obeys Laplace; under a dissipating component footprint it's Poisson. The sanity check the equation hands you for free: with no internal heat generation, the hottest and coldest points must be on the boundary. If your steady-state sim shows an interior hot spot in bare metal, the model is wrong, not the aluminum.
Impedance field solvers. The 2D cross-section of every transmission line is a Laplace problem for the potential between conductors. Coax has the closed-form answer: Z₀ = (59.96/√εr) · ln(b/a) ohms, which is why a 50 Ω air line needs a shield-to-center diameter ratio of 2.30. Microstrip has no closed form, so the "field solver" button in your SI tool runs a numerical Laplace solve on the trace geometry.
Preliminary aero and panel codes. Incompressible, irrotational flow has a velocity potential that satisfies Laplace, so a panel method can predict pressure distributions over a fuselage by stacking sources and doublets — superposition works because the equation is linear. The catch is d'Alembert's paradox: pure potential flow predicts exactly zero drag. The lift number is usable once the Kutta condition is imposed; the drag number is fiction until viscosity comes back in.
Seepage under a dam or levee. Groundwater head under Darcy flow satisfies Laplace, which is why civil engineers spent a century drawing flow nets — the hand-sketched grids of orthogonal squares are a graphical solution of the equation, and the count of squares gives seepage rate and uplift pressure.

How it works

Laplace's equation is the archetype elliptic PDE, and elliptic means: no time, no wave, no preferred direction, no information in transit. Every interior point feels the entire boundary at once. Change the temperature on one edge of a plate and the steady solution changes everywhere, instantly — because "steady" means the transient already happened. That global coupling is also why the equation smooths: even with ragged, discontinuous boundary data, the interior solution is infinitely differentiable ^[1]. Kinks do not propagate inward; they melt.

The mean value property forces the maximum principle: a non-constant harmonic function takes its extremes on the boundary, never inside ^[1]. That single fact has a famous physical consequence. Since the electrostatic potential in empty space is harmonic, it has no interior minimum for a charge to rest in — so no arrangement of fixed charges or permanent magnets can hold an object in stable levitation. That is Earnshaw's theorem ^[13], and it is the reason every maglev demo on your bench uses feedback coils, spin, or diamagnetism rather than static fields alone.

Superposition is the working tool. The equation is linear, so solutions add. A square plate with one edge held at 100 °C and three at 0 °C has exactly 25 °C at the center — rotate the problem four ways, add them, and the sum is the uniform-100 solution, so the center takes one quarter. A 201-grid relaxation run confirms 25.000. Cheap symmetry arguments like that catch modeling errors before the review board does.

Uniqueness is where the practical gotchas live. Fix the field value on the whole boundary (Dirichlet) and the interior solution is unique ^[1]. Fix only fluxes (Neumann) and the solution floats by an arbitrary constant — and no solution exists at all unless the boundary fluxes net to zero. That is the compatibility condition behind two familiar solver complaints: the "singular matrix / rigid body mode" error in a thermal FEA with only heat-flux boundary conditions, and the model that never converges because 10 W goes in and only 8 W is allowed out. Steady state has no place to put the other 2 W. Pin one node or balance the fluxes.

The mistakes people actually make: applying the source-free equation where sources exist (a trace carrying current, a potted resistor — that's Poisson); calling a system "steady-state" before the thermal time constant has run out; and forgetting that ∇²T = 0 assumes uniform conductivity — the honest equation is ∇·(k∇T) = 0, and at a copper-to-FR4 boundary the difference is not subtle. Numerically the equation is a friendly one — the operator is symmetric positive definite, so solvers behave — but plain relaxation crawls: that 201×201 plate takes roughly 144,000 Jacobi sweeps to converge, which is why production codes use multigrid or direct factorization instead.

History

The equation carries Laplace's name, but it showed up first in fluid mechanics. Euler, working out the foundations of incompressible flow in a paper written in 1752 and published in 1761, assumed the velocity came from a potential and found himself constructing harmonic polynomials to represent it ^[7]^[8] — solving the equation before anyone had named it. MacTutor is blunt on the point: the equation was known before Laplace's time ^[4].

Pierre-Simon Laplace (1749–1827) earned the naming rights through gravity. His memoir on the attraction of spheroids and the figure of the planets, read to the Paris Academy in 1782 and published in 1785, attacked the general problem of the pull a spheroid exerts on an external particle ^[5]^[6], and the machinery he built — the potential function and the expansion coefficients now called Legendre polynomials — went on to anchor the Mécanique Céleste from 1799 ^[4]^[5]. Separate his equation in spherical coordinates today and the angular pieces are the spherical harmonics, still cataloged under exactly that role ^[3]. In empty space, his potential satisfied ∇²V = 0, and the equation became the backbone of celestial mechanics.

It had a bug, though. In 1813 Siméon Denis Poisson (1781–1840) examined the potential inside an attracting mass and showed the right-hand side is not zero there — it is proportional to the local density ^[9]^[10]. One short correction, and the source-free equation gained its sourced companion; the pair have divided the work between them ever since.

The subject's strangest chapter came from a windmill. George Green (1793–1841), a self-taught miller from Sneinton with about one year of schooling, self-published An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism in 1828 — 51 subscription copies — in which he coined the term "potential" and introduced the theorem and functions that carry his name ^[11]^[12]. The essay sank without a trace until William Thomson, the future Lord Kelvin, was handed a copy in 1845 and carried it to Paris, where it finally detonated ^[11]^[12]. Samuel Earnshaw added the no-levitation theorem in an 1842 Cambridge Philosophical Society paper ^[13]^[14], closing out the classical era of what is now simply called potential theory.

Related tools

/tools/heat-conduction — Fourier's law; Laplace's equation is what it becomes at steady state
/tools/coax-impedance-dimensions — the closed-form Laplace solution in an annulus
/tools/microstrip-impedance — the same 2D problem where no closed form exists
/tools/lift-force — potential-flow theory is where the lift coefficient comes from
/tools/drag-force — the part d'Alembert's paradox says potential flow cannot give you
/tools/lmtd — steady-state heat transfer bookkeeping one level up from the field equation

Sources

Written by HE in our own words from the cited sources — engineering judgment included, your stamp still required. All entries →