PHY 417G: Review

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PHY 417G: Review

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Title: PHY 417G: Review

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

3
Boundary Value Problem (BVP)

Partial Differential Equation (PDE) BULK
Represents the physics of continuous media
General solution by separation of variables
Linear equation gt 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!

4
Magnetic scalar potential

Electrostatics Coulombs 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
5
L/T separation of EM fields
6
Formulations of E M PDEs

Electricity Magnetism
Note the interchange of flux and flow twisted
symmetry!

7
Electrodynamics

Faradays 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 Ampères Laws H(J), A(B), E(eB/dt)
31 lumped components capacitor, resistor,
inductor (reluctance)
Maxwells displacements current theoretical
prediction
Relativistic complete derivative chain gauge,
potential, fields, current
Completes Maxwell equations PDEs of
electrodynamics
Macroscopic equations 3 charges 5 currents
We could go back and create 5 formulations of
electrodynamics
I) Jefimenkos eqs, IIIII) Maxwells
integral/differential equationsIV) Retarded
potential Greens function of V) WAVE EQUATION

8
Polarization Magnetization

Chapter 4 electric materials gt Chapter 6
magnetic materials
Polarization chain gt Magnetization mesh

9
3 Materials gt 3 Components

Materials constants permittivity, resistivity,
permeability
Electrical components capacitor, resistor,
inductor
Each is a ratio of Flux / Flow !

10
Equations of Electrodynamics
11
Dynamics of EM

Maxwells equations dynamics of the field
Source equations charge (?,J) generates the
EM field
Force equations nature of EM force
conservation of (E,p)
Lorentz Force equation dynamics of charged
particles
Additional equation independent of Maxwell eqs.
Integrate to get energy EF?dx, momentum pFdt,
Conserved currents
Charge (current density)
Energy (Poynting vector)
Momentum (stress tensor)
Conservation principles can be used to simplify
problems

12
Electromagnetic waves

Homogeneous wave equation Helmholtz equation
Separation of variables / eigenfunctions Exp,
Legendre, Bessel
3 material properties (e, µ, s) gt 2 complex
medium properties
Dispersion relation k(?) propagation
(attenuation, wavelength)
Characteristic Impedance Z(?) boundary
(reflection, phase shift)
Boundary value problems
Across an interface Fresnel coefficientsreflect
ion / transmission impedance
Along a wave guide modes of propagation
standing transverse waves, kt2 affects
dispersion relation
Examples of waves
1-d String wave, telegraphers equations
2-d Surface waves, gravity waves, transverse
waveguide modes
3-d Seismic/acoustic waves, electromagnetic
waves