PHY 417G: Review

Christopher Crawford2015-04-29

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

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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