phy 417g: review christopher crawford 2015-04-29
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PHY 417G: Review
Christopher Crawford2015-04-29
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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
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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) SURFACEUse orthogonality to calculate components of gen. solutionInterior BCs – continuity– Derives directly from the PDEExterior 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
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Magnetic scalar potentialElectrostatics – Coulomb’s law Magnetostatics – Biot-Savart law
B.C.’s: Flux lines bounded by charge Flux lines continuous Flow sheets continuous (equipotentials) Flow sheets bounded by current
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L/T separation of E&M fields
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Formulations of E & M PDEs• Electricity Magnetism
• Note the interchange of flux and flow: twisted symmetry!
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Electrodynamics• 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 equationsIV) Retarded potential: Green’s function of V) WAVE EQUATION
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Polarization & Magnetization• Chapter 4: electric materials –> Chapter 6: magnetic materials
• Polarization chain –> Magnetization mesh
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3 Materials –> 3 Components• Materials constants: permittivity, resistivity, permeability• Electrical components: capacitor, resistor, inductor• Each is a ratio of Flux / Flow !
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Equations of Electrodynamics
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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=Fdx, momentum p=Fdt,
• Conserved currents– Charge (current density)– Energy (Poynting vector)– Momentum (stress tensor)
• Conservation principles can be used to simplify problems
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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, telegrapher’s equations– 2-d: Surface waves, gravity waves, transverse waveguide modes– 3-d: Seismic/acoustic waves, electromagnetic waves
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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
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