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Solutions to Maxwells equations

lets try a simple guess at a solution time harmonic solutions

lets substitute into Maxwells equations, and see if this could work

left hand side of equation

? , , j tsE E x y z e ? , , j tsH H x y z e

BEt

, , , ,j t j ts sE x y z e e E x y z


right hand side of curl E equation

now substitute

if the time independent forms Es and Hs satisfy this equation, they will satisfy Maxwell


, , j tr o sr o

H H x y z eBt t t

, , , ,j t j tr o s r o sthis is just H

eH x y z j H x y z e

t

, , , ,j t j ts r o se E x y z j H x y z e

, , , ,j t j ts sE x y z e e E x y z

, , , ,s r o sE x y z j H x y z


lets keep going substitute into next Maxwell equation

left hand side of equation

left side


, , , ,j t j ts sH x y z e e H x y z DH Jt

, , , ,j tr o s j tr o r o sE E x y z eD j E x y z et t t

, , j tss

J E E x y z e

J

s s r o sH r J r j E r

notice I took the material properties out of the time derivative

Maxwells equations, time harmonic form

so assuming that none of the materials change in time, time independent fields Es and Hs that satisfy

when multiplied by exp(jt), the product will satisfy the full time dependent set of Maxwell equations

so now what??? can we use these equations to predict/understand any

other phenomena? lets try to solve this set of coupled partial differential

equations

s s r o sH r J r j E r s r o sE r j H r , , , , , j tsE x y z t E x y z e

, , , , , j tsH x y z t H x y z e

Summary of electromagnetics: Maxwells equations summarizing everything we have so far, valid even if things are

changing in time

plus material properties

0B

r o

B H

vD

r o

D E

J E

DH Jt

BEt

Faradays law

Amperes law

Gausss law

Summary of electromagnetics: Maxwells equations in time harmonic form

summarizing everything we have so far, assuming time harmonic behavior, and using Ohms law for J

plus (time independent) material properties

0B

r o

B H

vD

r o

D E

J E

s sH r j E r s sE r j H r

, , , , , j tsE x y z t E x y z e , , , , , j tsH x y z t H x y z e

Spatial forms that solve Maxwell lets assume the time dependence is taken care of using the

phasor approach (i.e., time dependence is exp(jt)) what about the Es and Hs (the spatial part)?

as usual, lets guess special case: lets assume there are no components of the fields in

the z direction, i.e., NOTE: we are NOT assuming that E & H are independent of z!! the fields are transverse to the z direction the field is contained in the x-y plane the only field components are x^ and/or y^ directed:

Ex(x, y, z), Ey(x, y, z), Hx(x, y, z), Hy(x, y, z) or ^ and/or ^ if its cylindrical

under these conditions, what happens to Maxwells equations? lets look at the curl first

0 0 y yx xz z

F FF FF Fcurl F F x y zy z z x x y

Maxwell for transverse-to-z fields

but from the first Maxwell equation

for the transverse-to-z case this reduces to

y yx xtrans zF FF Fcurl F F x y zz z x y

s sE r j H r

0y yx x

trans x ybyassumptionthat H is trans

E EE EE x y z j H x H y zz z x y

ytrans xx

EE j H

z

x

trans yy

EE j Hz

0y xtrans zE EEx y

Maxwell for transverse-to-z fields

from the second Maxwell equation

for the transverse-to-z case this reduces to

y yx xtrans zF FF Fcurl F F x y zz z x y

0y yx xtrans x ybyassumptionthat E is trans

H HH HH x y z j E x E y zz z x y

ytrans xx HH j Ez

xtrans yy HH j Ez

0y xtrans zH HHx y

s sH r j E r

Maxwell, time harmonic, transverse-to-z collecting all the terms,

assuming time harmonic solutions using Ohms law assuming there is no component of either E or H in the z direction

Maxwells equations reduce to

things to notice Ey is connected to Hx via d/dz and Ex is connected to Hy via d/dz and Hy is connected to Ex via d/dz and Hx is connected to Ey via d/dz and Ey and Ex are connected via d/dx and d/dy Hy and Hx are connected via d/dx and d/dy

yx

Ej H

z

xy

E j Hz

0y xE Ex y

y xH

j Ez

x yH j Ez

0y xH Hx y

A little more fiddling with time harmonic transverse-to-z Maxwell

time harmonic solutions, using Ohms law, assuming there is no component of either E or H in the z direction

can we get rid of the mixture or E and H?

yx

Ej H

z z x y

H j Ez

2

2y xE Hj

z z

2 2y yE j j Ez 2

2 0y

y

Ej j E

z

xy

E j Hz z

y xH

j Ez

2

2yx HE j

z z

2 2x xE j j Ez 2

2 0x

xE j j Ez

A little more fiddling with time harmonic transverse-to-z Maxwell

time harmonic solutions, using Ohms law, assuming there is no component of either E or H in the z direction

can we get rid of the mixture or E and H?

x yH j Ez z

yx

Ej H

z

2

2yx EH j

z z

2 2 x xH j j Hz

2

2 0x

xH j j Hz

y xH j Ez z

xy

E j Hz

2

2y xH Ej

z z

2 2 y yH j j Hz 2

2 0y

y

Hj j H

z

Wave equation form of the time harmonic transverse-to-z Maxwells equations

assuming time harmonic solutions, using Ohms law, and that there is no component of either E or H in the z direction Maxwells equations reduce to

2 2 0x xH j j Hz

2

2 0y

y

Hj j H

z

2 2 0y yE j j Ez

2

2 0x

xE j j Ez

0y xE Ex y

0y xH Hx y

Wave equation form of the time harmonic transverse-to-z Maxwells equations

to get a better handle on all this, lets make another simplifying assumption E points in the x^ direction only

then the only thing we have left is

do we recognize a solution to an equation of this form?

2 2 0x xE j j Ez

22

2 0F Fz

( )zF constant e

Transverse plane waves plane waves

all the fields are contained in a plane perpendicular to a particular direction here we made that direction z^

this still looks pretty abstract lets try a simple guess at a solution, and check to see if we can

satisfy these new equations proposal: uniform plane wave

E points in the x^ direction only Exo is a CONSTANT, independent of time, place, frequency!

this is what makes it a uniform plane wave is also a constant, independent of time and place, but Im not

sure about substitute and see what we get

??ztrans xoE E e x

Uniform plane wave substitution

in the transverse-to-z version of Maxwell we needed the derivative

and one of Maxwells equations was

so we have the y component of H, what about the x component?

xy

E j Hz

zxo zxo

E eE e

z

assumed

ztrans xoE E e x

zxo yE e j H

yo

zzxo

y xo

H

E eH j E ej

yo xoH j E

xEz

Uniform plane wave substitution

the x component of H is given by

but by assumption, Ey = 0, so Hx = 0!

so far we have assumed

and from this Maxwell requires

so the presence of an electric field requires the presence of a magnetic field!

we still have a couple of equations to check! maybe we can figure out what the constant is?

yx

Ej H

z

assumed

ztrans xoE E e x

assumed

ztrans xoE E e x

z ztrans yo xoH H e y j E e y

Uniform plane wave substitution:what is ?

another of Maxwells equations was

y xH j Ez

assumed

ztrans xoE E e x

z ztrans yo xoH H e y j E e y

y

x

H

zxo

zxoE

j E e

j E ez

z zxo xoj E e j E e 21j j

2 j j

Uniform plane wave solution to Maxwells equations

so we now have the whole spatial solution!!!

and the complete, time harmonic solution is

is called the complex propagation constant note that E and H are perpendicular to each other

this is really a big deal!!

assumed

ztrans xoE E e x

ztrans xoH j E e y

2 j j

assumed

j t zplane xowave

E E e x j t zplane xowave

H j E e y

j j

Lossless uniform plane wave solution to Maxwells equations

to understand what this thing looks like, lets make an additional assumption: the region of space we are in is an insulator = 0

then we get a simplification in

gamma is purely imaginary, = j !

assumed

j t zplane xowave


H j E e y

j j

20j j j

j t zplane xowave

E E e x j t zj t zplane xo xowave

jH j E e y E e y

Lossless, uniform plane wave solution to Maxwells equations

this is also a really big deal!! what does this thing look like???? is called the phase or propagation constant

note the units on are inverse length every time you move a distance of 2/ along z, the exp function

repeats itself

so this function is periodic in space the spatial repeat interval of a periodic function is usually called

the wavelength

j t zplane xowave


H E e y

2exp exp 2 expj t z j t z j t z

2 2


what does this thing look like???? what if we wanted to follow a plane of constant phase as the

clock ticks in other words, if time t is increasing, how should we move along

the z axis to keep the phase of the exponential a constant? lets look at the case first, where zo is the observation point

needed to keep the total phase at the constant value o

the constant phase point speed (the phase velocity vp) is just dzo/dt

j t zplane xowave


H E e y

2 2

exp o o oj t z t z ooz t

1op

dz vdt


lets continue to follow a plane of constant phase as time t is increasing, but this time consider the + case

the phase velocity vp is still dzo/dt

so if we consider the (t + z) solution, the phase front is moving in the -z^ direction

if we consider the (t z) solution, the phase front is moving in the +z^ direction

regardless, the speed is

j t zplane xo

waveE E e x 2 2

exp o o oj t z t z ooz t o

pdz vdt

1pv

j t zplane xowave

H E e y


so in a region of space that has zero conductivity, one possible solution to Maxwells equations is

this is called a uniform transverse electromagnetic plane wave a wave because it is periodic in time and space a plane wave because a surface of constant phase is a flat plane,

and is propagating in the z^ direction electromagnetic because E and H are intimately connected transverse because the E and H fields are contained completely in

the x-y plane, transverse to the direction of propagation uniform because the magnitude of the field is constant wrt x and y

j t zplane xowave

E E e x

j t zplane xowave

H E e y

2 2

1pv

Units check we have

: 1/sec : farad/meter : henry/meter

: (henry)(farad)/m2 = (sec)(sec/)/m2 = sec2/m2 : [sec/m]/[sec] = 1/m = 1/distance

we have

: 1/sec : [sec/m]/[sec] = 1/m = 1/distance vp: [1/sec]/[1/m] = m/sec (speed) : farad/meter : henry/meter

: (henry)(farad)/m2 = (sec)(sec/)/m2 = sec2/m2 vp: 1/[sec/m] = m/sec

numbers: free space, o = 8.854x10-12 F/m, o = 4x10-7 H/m vp = 3x108 m/sec @ = 2x1010sec-1 (f = 10GHz): = 3 cm = 0.03m

1pv

2 2


lets look at the relationship between E and H a little more closely

we found that if we pick the (t - z) the wave is traveling (or propagating) in the +z^ direction

for this choice, assuming Exo is positive E points in the +x^ direction H points in the +y^ direction note that the direction of propagation is x^ y^ = z^

i.e., its the same as

j t zplane xowave


H E e y

j t zplane xowave


H E e y

o oE H


if we pick the (t + z) the wave is traveling (or propagating) in the -z^ direction

for this choice, assuming Exo is positive E points in the +x^ direction H points in the -y^ direction note that the direction of propagation is x^ (-y^) = -z^

i.e., its still the same as

this type of solution to Maxwells equations, with no field components in the direction of propagation, is a TEM (transverse electromagnetic) wave

j t zplane xowave


H E e y

o oE H


lets look at the magnitudes of E and H

notice that the ratio of E to H is

units check: E: volt/meter : farad/meter H: amp/meter : henry/meter E/H: volt/amp = ohm : henry/farad = (ohm)2

j t zplane xo xowave

E E e x E j t zplane xo xowave

H E e y E

planewave xo

plane xowave

EE

H E


the ratio of E to H is

where is called the wave impedance

if the = o and = o then we are in free space, and the impedance of free space is

377oplane plane owave wave o

E H

planewave

planewave

E

H

Uniform plane wave summary

the direction of propagation is given by

the propagation constant is

the wavelength is inversely dependent on frequency inversely dependent on square root of and

the phase velocity is independent of frequency inversely dependent on square root of and

and the fields are

o oE H

j t zplane xowave


H E e y

2 2

1pv

plane planewave wave

E H

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, >> conductor: conduction current dominates,

Uniform plane wave solution to Maxwells equations

the complete, time harmonic solution is

E and H are perpendicular to each other

is called the complex propagation constant

direction of propagation

2 j j

assumed

j t zplane xowave


H j E e y

j j j 2 pv

o oE H

1

gd dvd d

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

DH Jt

DE H Jt

vectorID

E H H E E H

DH E E H E J E tBt

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

H EH E H E J Et t

E HE H E J E Ht t

EEt

2E E E E E E E Et t t t

21 12 2E E E E Et t t

HHt

2 21 12 2E H E J E Ht t

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

2 21 12 2E H E J E Ht t

2 21 12 2volume volume volumeE H dv E Jdv E H dvt

2 21 12 2surface volume volumeenclosing energy energyohmic powerV stored storeddissipation in E field in H field

E H dS E Jdv E H dvt

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

2 21 12 2surface volume volumeenclosing energy energyohmic powerV stored storeddissipation in E field in H field

E H dS E Jdv E H dvt

surfaceenclosing

V

E H dS power flow into the volume

surfaceenclosing

V

E H dS power flow out of the volume

P E H

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(a+jb) is exp(a-jb)

, 1 Re2s ave S SP E H

a jb c jd a jb c jd a jb a jbe e

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