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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

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

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

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

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

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

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)

Example our uniform plane wave

- for the case we looked at earlier we found for

the wave propagating in the z direction

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

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?