Christopher Crawford PHY

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Christopher Crawford PHY 417 2014-02-27 §7.2 Maxwell Equations Christopher Crawford PHY 417 2014-02-27

Outline Review – TWO separate derivative chains (in space only) ES and MS formulations: potentials and Poisson’s equation THREE observations: a) Coulomb, b) Ampere, c) Faraday the third ties the derivative chains of the other two together TWO+1 cracks in the foundation – patching up space and time Scalar potential, Maxwell’s displacement current Example: potential momentum associated with a B-field Example: the displacement current through a capacitor Materials: THREE+1 charges and SIX currents Maxwell Equations – unified symmetry in space and time Differential & integral fields, potentials, boundary cond’s Space-time symmetry – ONE complete derivative chain Duality rotations – magnetic monopoles revisited

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

Two separate formulations ELECTROSTATICS MAGNETOSTATICS Faraday’s law stitches the two formulations together in space and time

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

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? Maxwell’s displacement current Fluid mechanical model Elasticity of medium –> EM waves On Faraday's Lines of Force (1855) On Physical Lines of Force (1961) The Dynamical Theory of the Electromagnetic Field (1865)

Example 7.8: potential momentum Charges moving in magnetic field Charges in abruptly changing magnetic field Magnetic field energy acts as “electromagnetic inertia”

Maxwell’s equations Integral & differential Potentials & wave eq. Boundary conditions Constitution equations Continuity equation Lorentz Force Field energy

Electrical properties of materials Same old THREE charges (plus one magnetic) Now: SIX currents, including displacement!

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

Duality Rotation (ε,1/μ) tensor behaves like i : converts between flux and flow Compare (E,B) to (x,y) in the complex plane

Conservation of Energy Similar to other fluxes x flows