7 2 Maxwell Equations the wave equation Christopher

§ 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 2

Two separate formulations ELECTROSTATICS • Coulomb’s law MAGNETOSTATICS • Ampère’s law 3

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 4

Example: current through a capacitor • Which surface should one use for Ampère’s law? 5

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” 6

Magnetic field energy • Work against the “back-EMF” is stored in the magnetic field • It acts as “electrical inertia” to keep current moving 7

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 8

One unified formulation ELECTROMAGNETISM • Faraday’s law stitches the two formulations together in space and time • Previous hint: continuity equation 9

Unification of E and B • Projections of electromagnetic field in space and time – That is the reason for the twisted symmetry in field equations 10

Unification of D and H Summary 11