Christopher Crawford PHY 311 2014-02-07 §2.2 Div. and Curl of E Christopher Crawford PHY 311 2014-02-07

Outline 5 Formulations of electrostatics Derivative chain Flux and Flow of electric field Electric flux Gauss’ law Three applications Electric flow Irrotational / conservative electric field Electric potential Recovery of Coulomb’s law Helmholtz theorem

5 Formulations of Electrostatics All electrostatics comes out of Coulomb’s law & superposition Note: every single theorem of vector calculus! Flux and Flow: Schizophrenic personalities of E Integral vs. differential Purpose of each formulation Q  E  V Derivative chains:

Electric flux and flow FLUX FLOW Field lines counts charges inside surface E = flux density The flux counts the amount of charge inside the surface FLOW Equipotential surfaces counts flow along any path E = flow density The surfaces are closed: E is a conservative field Flow = change in potential ΔV B.C.’s: Flux lines bounded by charge Flow sheets continuous (equipotentials)

Gauss’ law Integral form Differential form – divergence theorem Direct calculation of divergence

Three applications of Gauss’ law Plane – surface charge – boundary conditions Cylinder – line charge Sphere – point charge

Curl of E Differential form – E is irrotational Inverse Poincaré theorem E derives from a potential Integration of flow E is a perfect gradient! (E is exact) Potential V = potential energy / charge