1. Christopher Crawford PHY 416 2014-12-12
End of Semester Review
Christopher Crawford
PHY 416

2. Foundation of Electrostatics
Classical fields: combination of Linear and Differential spaces
a) Fundamental Theorem of Differentials (extension of FTC)
definite integrals: Gradient, Curl (Stokes), Div. (Gauss) theorems
Indefinite integrals: Potential theorem (Inverse Poincaré)
B) Helmholtz theorem (projection of fields)
Geometric interpretation of vector fields: Flux and Flow
5 formulations of electrostatics
Derivative chain – gauge, potentials, fields, sources
Structure of and relations between different formulations
Field calculation methods organized around formulations
Poisson’s formulation most powerful: Boundary Value Problems
Radial coordinate systems: Multipole expansion
Dielectric materials: Polarization flux

3. Final Exam Cumulative exam 50% longer than midterm exams
Similar problems as midterms
Proof – relation between formulations
Direct Integration – Coulomb’s law / Potential
Boundary value problems – with dielectrics, sources
Multipole – integrate over charge
Capacitance – either using Gauss’ law or BVP
Essay question – structure of electric fields in dielectrics

4. Linear spaces Linear combinations Bilinear products Linear operators
Projections into direct sums
Basis, components
Bilinear products
Dot product (Inner product, metric): symmetric, scalar: Length
Cross product: antisymmetric, [bi]vector: Area
Triple product (determinant), trilinear antisymmetric: Volume
Linear operators
Matrices / transformations
Symmetric: Eigenvectors
Orthogonal: Rotations
Continuous linear [function] spaces
Everything above applies

5. Summary of differentials / integrals

6. Fundamental Theorems
Fundamental Theorem of Differentials (extension of FTC)
Definite integrals: Gradient, Curl (Stokes), Div. (Gauss) theorems
Indefinite integrals: Potential theorem (Inverse Poincaré)
Helmholtz theorem (projection of fields)
Inverse Laplacian
What do they have to do with electrostatics?

7. 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 V  E  Q

8. Electrostatic derivative chain
ELECTROSTATICS
Coulomb’s law
MAGNETOSTATICS
Ampère’s law
E+B: Faraday’s law; b) rho + J: conservation of charge; c) space + time

9. Next semester: unified formulation
ELECTROMAGNETISM
Faraday’s law stitches the two formulations together in space and time
Previous hint: continuity equation

10. L/T separation of E&M fields