Summary of electromagnetics: time harmonic form of Maxwells equations

1
Summary of electromagnetics time harmonic form
of Maxwells equations

• summarizing everything we have so far, assuming
  time harmonic behavior, and using Ohms law for J
  
• dielectric displacement current dominates, w
  gtgt s/e
• conductor conduction current dominates, w ltlt
  s/e
• plus (time independent) material properties

2
Uniform plane wave solution to Maxwells equations

• the complete, time harmonic solution is
• E and H are perpendicular to each other
• g is called the complex propagation constant
• direction of propagation

3
Power flow

• is there anything more general we can say about
  what it means for a wave to be propagating?
• lets start with
• now take dot product with E on both sides
• left hand side
• so now we have

4
Power flow

• so far using Maxwells equations and a vector ID
• or
• lets look closely at and
• using the chain rule
• or
• so now we have

5
Power flow

• we now have, using the general form of Maxwells
  equations
• lets integrate over some volume of space
• the divergence theorem lets us convert the
  volume integral of div(ExH) into a surface
  integral of ExH

6
Power flow the Poynting vector

• the right hand side represents the power flowing
  into the volume
• so the left side must represent the same thing
• then getting rid of the minus sign tells us that
• so we interpret the Poynting vector P as the
  instantaneous power density
• units EH (V/m)(Amp/m) Watt/m2

7
Poynting vector in phasor form

• when using phasors we need to remember to take
  the real part to get a physically meaningful
  result (as opposed to a mathematically convenient
  result)
• for our time harmonic form we also have to do the
  time average over one period
• so the actual power flow would be
• where H is the complex conjugate of H
• the imaginary part of H is replaced by its
  negative
• notes
• the complex conjugate of a product is the product
  of the complex conjugates
• the complex conjugate of exp(ajb) is exp(a-jb)

8
Example our uniform plane wave

• for the case we looked at earlier we found for
  the wave propagating in the z direction

9
Example our uniform plane wave, low loss, good
dielectric

• in the good dielectric (high frequency) limit
  we found
• so for the case with E in the x direction,
  picking the wt bz solution, which gives H in
  the y direction, the Poynting vector is
• in the zero conductivity (zero loss tangent)
  limit a 0, so
• units V/m2 / ohm Watt / m2 power/area

10
Plane wave applets

• very nice, with ability to vary materials
  properties and frequency, includes power flow
• http//www.amanogawa.com/archive/PlaneWave/PlaneWa
  ve-2.html
• this site is pretty useful for other things like
  transmissions lines
• index http//www.amanogawa.com/archive/wavesA.htm
  l
• fairly elaborate combinations of sources, ability
  to add surfaces, show interference, etc
• http//www.falstad.com/emwave2/
• simple example http//lompado.uah.edu/EMWave.htm
  
• from an oscillating field source
  http//www.phy.ntnu.edu.tw/java/emWave/emWave.html
  

• now what? What happens at an interface between
  two media?