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

2. 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

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 together in space and time

5. One unified formulation
ELECTROMAGNETISM
Faraday’s law stitches the two formulations together in 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 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)

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