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

2. 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

3. 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:

4. 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)

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

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

7. 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