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Exploring Vector Potential in Electromagnetism: Geometry, Gauge Invariance, and Boundary Conditions

This outline covers the geometry of vector potential, A-sheets bounded by B-flux lines, and the picture of gauge invariance. It delves into derivative chains, existence and non-uniqueness of A, application of the Fundamental Theorem of Differentials, potential momentum, boundary conditions of A, and the source and non-uniqueness of A. Various gauge transformations, such as flow sheets bounded by charge and current, and the analogy with Ampère's law are discussed. Additionally, the text examines the solenoid, derivative chain, gauge condition, unique gauge transformation, Coulomb and Lorenz gauges, gauge symmetry (U(1)), and the Fundamental Theorem of Differentials. The concept of potential momentum and boundary conditions on A are also addressed.

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Exploring Vector Potential in Electromagnetism: Geometry, Gauge Invariance, and Boundary Conditions

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  1. §5.3: Vector Potential Christopher Crawford PHY 417 2015-02-04

  2. Outline • Geometry of vector potential A-sheets bounded by B-flux lines Picture of gauge invariance • Derivative chain – gauge invariance Existence and non-uniqueness of A Gauge invariance and U(1) group • A is for anti-derivative of B Application of Fundamental Theorem of Differentials Potential momentum • Boundary conditions of A

  3. Geometry of Vector Potential Source and Non-uniqueness [gauge transformations] of A A-sheets B.C.’s:Flux lines bounded by charge Flux lines continuous Flow sheets continuous (equipotentials) Flow sheets bounded by current Solenoid: Cos-theta coil:

  4. Analogy with Ampère’s law • Same curl form

  5. A for a solenoid (B-flux tube)

  6. Derivative chain • Gauge condition and unique gauge transformation • Coulomb (radiation gauge) • Lorenz Gauge (covariant) • Gauge symmetry U(1)

  7. Fundamental Theorem of Differentials • Given a star-like [spherical] coordinate system,

  8. A is the anti-derivative of B • Application of FTD • Potential momentum

  9. Boundary Conditions on A

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