Potential Theory

PDEs · Axiom Academy

Understanding scalar potentials, conservative fields, and equipotential surfaces in physics and PDEs 1. Conservative Fields and Scalar Potentials A vector field F is conservative if it can be expressed as the gradient of a scalar function φ (the potential): This means the field is path-independent : work done moving along any path depends only on the endpoints. Equivalently, the curl of the field vanishes: The gravitational field around a point mass M is conservative. The gravitational potential φ satisfies: where G is the gravitational constant and r is the distance from the mass. The gravitational field is then: The potential satisfies Laplace's equation everywhere except at the mass location: Similarly, the electric field E around a point charge q is conservative with potential: where ε₀ is the permittivity of free space. The electric field is: In regions with charge density ρ, the potential satisfies Poisson's equation : 4. Velocity Potential for Irrotational Flow In fluid dynamics, an irrotational flow has zero vorticity (∇ × v = 0). Such flows can be described by a velocity potential φ: For incompressible flow (∇ · v = 0), the velocity potential satisfies Laplace's equation: An equipotential surface is a surface where the potential φ is constant. Key properties: Orthogonality: Field lines are always perpendicular to equipotential surfaces No work: Moving along an equipotential surface requires zero work

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