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This review covers the foundation of electrostatics, including Linear and Differential spaces, Fundamental Theorems, Field formulations, Derivative chain, and more. It also delves into the abstract concepts of linear spaces, projections, basis, and linear operators.
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End of Semester Review Christopher Crawford PHY 416 2014-12-12
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
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
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
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?
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
Electrostatic derivative chain ELECTROSTATICS • Coulomb’s law MAGNETOSTATICS • Ampère’s law
Next semester: unified formulation ELECTROMAGNETISM • Faraday’s law stitches the two formulations togetherin space and time • Previous hint: continuity equation