1. PHY 417G: Review Christopher Crawford 2015-04-29

2. Classical Electromagnetic Field action at a distance vs. locality field ”mediates “carries force extends to quantum field theories field is everywhere always E (x, t) differentiable, integrable field lines, equipotentials PDE – boundary value problems solution to physical problems 2

3. Boundary Value Problem (BVP) Partial Differential Equation (PDE)BULK – Represents the physics of continuous media – General solution by separation of variables – Linear equation –> inf. dim. linear solution function space Boundary Conditions (BC)SURFACE Use orthogonality to calculate components of gen. solution Interior BCs – continuity – Derives directly from the PDE Exterior BCs – physics input – Uniqueness theorem: one BC per surface (elliptic) 1 or 2 initial conditions (diffusion, hyperbolic wave) Now we just have to know the PDE to solve! 3

4. Magnetic scalar potential Electrostatics – Coulomb’s law Magnetostatics – Biot-Savart law B.C.’s:Flux lines bounded by chargeFlux lines continuous Flow sheets continuous (equipotentials)Flow sheets bounded by current 4

5. L/T separation of E&M fields 5

6. Formulations of E & M PDEs ElectricityMagnetism Note the interchange of flux and flow: twisted symmetry! 6

7. Electrodynamics Faraday’s law: 3 rd experimental law – Motional EMF equivalent to truly moving or changing magnetic field – Basis of special relativity – electromagnetic field F = E dt + B – 3 “Ampère’s Laws”: H(J), A(B), E(eB/dt) – 3+1 lumped components: capacitor, resistor, inductor (reluctance) Maxwell’s displacements current: theoretical prediction – Relativistic complete derivative chain: gauge, potential, fields, current – Completes Maxwell equations – PDE’s of electrodynamics – Macroscopic equations: 3 charges + 5 currents – We could go back and create 5 formulations of electrodynamics: I) Jefimenko’s eqs, II+III) Maxwell’s integral/differential equations IV) Retarded potential: Green’s function of V) WAVE EQUATION 7

8. Polarization & Magnetization Chapter 4: electric materials –> Chapter 6: magnetic materials Polarization chain –> Magnetization mesh 8

9. 3 Materials –> 3 Components Materials constants: permittivity, resistivity, permeability Electrical components: capacitor, resistor, inductor Each is a ratio of Flux / Flow ! 9

10. Equations of Electrodynamics 10

11. Dynamics of E&M Maxwell’s equations – dynamics of the field – Source equations – charge (ρ,J) generates the E&M field – Force equations – nature of E&M force: conservation of (E,p) Lorentz Force equation – dynamics of charged particles – Additional equation independent of Maxwell eq’s. – Integrate to get energy E=F  dx, momentum p=Fdt, Conserved currents – Charge (current density) – Energy (Poynting vector) – Momentum (stress tensor) Conservation principles can be used to simplify problems 11

12. Electromagnetic waves Homogeneous wave equation – Helmholtz equation – Separation of variables / eigenfunctions: Exp, Legendre, Bessel – 3 material properties (ε, μ, σ) –> 2 complex medium properties Dispersion relation k(ω): propagation (attenuation, wavelength) Characteristic Impedance Z(ω): boundary (reflection, phase shift) Boundary value problems – Across an interface: Fresnel coefficients reflection / transmission [impedance] – Along a wave guide: modes of propagation standing transverse waves, k t 2 affects dispersion relation Examples of waves – 1-d: String wave, telegrapher’s equations – 2-d: Surface waves, gravity waves, transverse waveguide modes – 3-d: Seismic/acoustic waves, electromagnetic waves 12

13. Final exam: Integration – Biot-Savart, vector potential – Ampère’s law H(J), Potential A(B), Faraday’s law E(dB/dt) – Calculation of Resistance, Inductance, Reluctance Dynamics and Conservation – Derivation of magnetic formulations, potentials, wave equations – Derivation of conservation principles: charge, energy, momentum Boundary value problems – Magnetostatic with materials – Interface reflection/transmission – Waveguide modes Essay questions – long and short – Flux, flow, Maxwell equations, displacement currents, waves – Properties of materials: magnetization, dispersion, impedance 13