How is Hodge theory of harmonic forms related to maxwell's equations.Atiyah says that Hodge was directly motivated by considerations of maxwell's equations while commenting on donaldson.
In the absence of electric current, Maxwell's equations say precisely that the electromagnetic potential is a harmonic 1-form; see Wikipedia. This is in a space-time manifold, so it isn't the usual Hodge theory. Being harmonic in space-time is a wave equation, not an electrostatics equation. But if your space-time is a product of a Riemannian space manifold and a time direction, then time invariance of the electromagnetic potential reduces the equations of motion to harmonicity in the sense of the Hodge Laplacian.