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§7.2 Maxwell Equations

§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

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§7.2 Maxwell Equations

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  1. §7.2 Maxwell Equations Christopher Crawford PHY 417 2014-02-27

  2. Outline • Review – TWO separate derivative chains (in space only) • ES and MS formulations: potentials and Poisson’s equation • THREEobservations: a) Coulomb, b) Ampere, c) Faradaythe 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

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

  4. Two separate formulations ELECTROSTATICS MAGNETOSTATICS • Faraday’s law stitches the two formulations togetherin space and time

  5. One unified formulation ELECTROMAGNETISM • Faraday’s law stitches the two formulations togetherin space and time • Previous hint: continuity equation

  6. 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

  7. Example: current through a capacitor • Which surface should oneuse 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)

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

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

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

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

  12. Unification of D and H Summary

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

  14. Conservation of Energy • Similar to other fluxes x flows

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