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PHY 417G: Review
Christopher Crawford
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
Boundary Value Problem (BVP)
• Partial Differential Equation (PDE)
– Represents the physics of continuous media
– General solution by separation of variables
– Linear equation –> inf. dim. linear solution function space
• Boundary Conditions (BC)
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!
Magnetic scalar potential
Electrostatics – Coulomb’s law
Magnetostatics – Biot-Savart law
Flux lines bounded by charge
Flow sheets continuous (equipotentials)
Flux lines continuous
Flow sheets bounded by current
L/T separation of E&M fields
Formulations of E & M PDEs
• Electricity
• Note the interchange of flux and flow: twisted symmetry!
• Faraday’s law: 3rd 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
Polarization & Magnetization
• Chapter 4: electric materials –> Chapter 6: magnetic materials
• Polarization chain
–> Magnetization mesh
3 Materials –> 3 Components
• Materials constants: permittivity, resistivity, permeability
• Electrical components: capacitor, resistor, inductor
• Each is a ratio of Flux / Flow !
Equations of Electrodynamics
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
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, kt2 affects dispersion relation
• Examples of waves
– 1-d: String wave,
– 2-d: Surface waves, gravity waves,
– 3-d: Seismic/acoustic waves,
telegrapher’s equations
transverse waveguide modes
electromagnetic waves
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