End of Semester Review

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

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 • 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 singletheorem ofvector calculus! • Flux and Flow:Schizophrenicpersonalities of E • Integral vs. differential • Purpose of eachformulation V  E Q

8. Electrostatic derivative chain ELECTROSTATICS • Coulomb’s law MAGNETOSTATICS • Ampère’s law

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

10. L/T separation of E&M fields