Christopher Crawford PHY 311 2014-04-28 §7.2 Maxwell Equations Christopher Crawford PHY 311 2014-04-28

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 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 charges and FIVE 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?

Example: potential momentum

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

Maxwell’s equations Field equations Boundary conditions: integrate Maxwell Eq.’s across surface

Electrical properties of materials Same old THREE charges Now: FIVE currents, include displacement!

Duality rotation