Classical Electrodynamics Jingbo Zhang Harbin Institute of Technology

Chapter 3 Electromagnetic Potentials Section 1 Electrodynamic Potentials Section 2 Gauge Transformations

Jingbo Zhang Section 2 Gauge TransformationsChapter 3 Sept 9, 2004 Classical Electrodynamics Review Field Equations Fields and Potentials Potential Equations

Jingbo Zhang Section 2 Gauge TransformationsChapter 3 Sept 9, 2004 Classical Electrodynamics 1 Lorentz Gauge coupled imhomogeneous wave equations In Lorentz gauge uncoupled imhomogeneous wave equations

Jingbo Zhang Section 2 Gauge TransformationsChapter 3 Sept 9, 2004 Classical Electrodynamics  Introducing d’Alembert operator Lorentz equations for potentials

Jingbo Zhang Section 2 Gauge TransformationsChapter 3 Sept 9, 2004 Classical Electrodynamics  Retarded Potentials the solutions of Lorentz potential equations The solutions at time t at field point x are dependent on the behaviour at an early time t ’ of the source point x ’.

Jingbo Zhang Section 2 Gauge TransformationsChapter 3 Sept 9, 2004 Classical Electrodynamics 2 Coulomb Gauge coupled imhomogeneous wave equations In Coulomb gauge Poisson equation and imhomogeneous wave equations

Jingbo Zhang Section 2 Gauge TransformationsChapter 3 Sept 9, 2004 Classical Electrodynamics  solution of Poisson equation The scalar potential is only dependent on charge density source at time t. The retardation effects occur only in the vector potential.  solution of imhomogeneous wave equation

Jingbo Zhang Section 2 Gauge TransformationsChapter 3 Sept 9, 2004 Classical Electrodynamics 3 Gauge Transformations  Turning the potentials simultaneously into the new one according to the following transformation, where, is an arbitrary differentiable scalar function is called the gauge function. A transformation of the potentials which leaves the fields and Maxwell’s equations invariant is called a gauge transformation. A physical law or quality which doesn’t change under a gauge transformation is said to be gauge invariant.

Jingbo Zhang Section 2 Gauge TransformationsChapter 3 Sept 9, 2004 Classical Electrodynamics  Gauge invariant of Lorentz wave equations, If we require the gauge function itself be restricted to fulfill the following equation, such transformation would keep the Lorentz equations invariant.

Jingbo Zhang Section 2 Gauge TransformationsChapter 3 Sept 9, 2004 Classical Electrodynamics Homework 3.2  (Textbook page 46)Example 3.1 In Dirac’s symmetrised form of electrodynamics, derive the inhomogeneous wave equations with introducing the potentials.