§7.2 Maxwell Equations the wave equation Christopher Crawford PHY 311 2014-04-30

Outline Review – electromagnetic potential & displacement current consequences of Faraday law and conservation of charge Physical interpretation – unified symmetry in space and time Displacement current – capacitor Electromagnetic momentum – inductor Electromagnetic waves – oscillation of energy between the 2 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

Two separate formulations ELECTROSTATICS Coulomb’s law MAGNETOSTATICS Ampère’s law E+B: Faraday’s law; b) rho + J: conservation of charge; c) space + time

TWO cracks in the foundation Faraday’s law appears to violate conservation of energy? Unified gauge transformation for V and A Continuity equation vs. Ampère’s law

Example: current through a capacitor Which surface should one use for Ampère’s law?

Example 7.8: potential momentum Angular momentum from turning off the field Potential momentum associated with the field Magnetic field energy acts as “electromagnetic inertia”

Magnetic field energy Work against the “back-EMF” is stored in the magnetic field It acts as “electrical inertia” to keep current moving

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

One unified formulation ELECTROMAGNETISM Faraday’s law stitches the two formulations together in space and time Previous hint: continuity equation

Unification of E and B Projections of electromagnetic field in space and time That is the reason for the twisted symmetry in field equations

Unification of D and H Summary