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## Summary of electromagnetics: time harmonic form of Maxwells equations

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### Summary of electromagnetics: time harmonic form of Maxwell's equations. summarizing everything we have so far, assuming time harmonic behavior, and ... – PowerPoint PPT presentation

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Title: 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?
• 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
• 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?