Download Fundamental nuclear symmetries meet classical electrodynamic

§7.2 Maxwell Equations
the wave equation
Christopher Crawford
PHY 311
2014-05-02
Final Exam
• Based on 5 formulations of electromagnetism
– Derivative chain – gauge, potentials, fields, sources
– Structure of and relations between different formulations
– Field calculation methods organized around formulations
• Cumulative – uniform weighting through whole semester
1.
2.
3.
4.
5.
6.
– Will be 50% longer than midterm exams
– Similar problems as midterms
Essay question – structure of EM fields / media
Proof – relation between formulations
Integration – Coulomb / Biot-Savart / Potential
Integral – Gauss / Ampère [or modified versions]
Boundary value problem – see examples
Components – capacitor, resistor, inductor
2
Outline
• Review – electromagnetic potential & displacement current
propagate electromagnetic waves
– Capacitive ‘tension’ vs. inductive ‘inertia’
• Unification of E and B – filling in the cracks
– Derivative chain – different representations of fields
– Wave equation and solution – Green’s fn. and eigenfn’s
3
Electromagnetic Waves
• Sloshing back and forth between electric and magnetic energy
• Interplay: Faraday’s EMF  Maxwell’s displacement current
– Displacement current (like a spring) – converts E into B
– EMF induction (like a mass) – converts B into E
• Two material constants  two wave properties
4
Review: Two separate formulations
ELECTROSTATICS
•Coulomb’s law
MAGNETOSTATICS
•Ampère’s law
5
Review: One unified formulation
ELECTROMAGNETISM
• Faraday’s law stitches the two formulations together
in space and time
• Previous hint:
continuity equation
6
Unification of E and B
• Projections of electromagnetic field in space and time
– That is the reason for the twisted symmetry in field equations
7
Unification of D and H
Summary
8
Wave equation: potentials
9
Wave equation: gauge
10
Wave equation: fields
11
Wave equation: summary
• d’Alembert operator (4-d version of Laplacian)
12
Homogenous solution
• Separate time variable to obtain Helmholtz equation
• General solution for wave Boundary Value Problems
13
Particular solution
• Green’s function of d’Alembertian
Wikipedia: Green’s functions