Christopher Crawford PHY 416 2014-10-08 §2.3 Electric potential Christopher Crawford PHY 416 2014-10-08

Outline Electric flow Meaning of Electric potential 5 Formulations Continue with flow derivations Recover Coulomb Green’s functions Helmholtz theorem

Meaning of electric potential Analogy with gravitational field Potential energy per mass or charge Equipotential contours

Flow: Circulation vs. Conservative fields

Curl of E Direct calculation – E is irrotational Potential V Inverse Poincaré Stokes’ theorem Flow from a to b is unique Fund. thm. of vector calc. What is V ?

Integration of V Direct calculation – Coulomb’s law is “exact” (1/r2 is a perfect differential) Inverse: gradient of V

Poisson’s equation Combine div. & curl eq’s Into 2nd order differential equation Only scalar fields: V, ρ Formal solution of Poisson’s eq Expand as delta fn’s Invert Laplacian to Green’s fn’s

Putting it all together… Solution of Maxwell’s equations