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Electrical Boundary Conditions

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3an2 is a unit vector normal to the interface from region 2 to region1

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Electric Field Boundary Conditions:

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Magnetic Field Boundary Conditions:

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K=Js

K=Js


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Dielectric- dielectric boundary conditions

Dielectric materials are dominated by “bound” rather than “free’’charges (E-fields causes +ve and – ve charges of molecules to separateand form dipoles throughout the material interior

Therefore, the free charge density sand the surface current density Jsare zero

1En1= 2En2

•The normal component of B is

continuous across the interface while the

tangential component of E is continuous

across the interface

2

2

1

1

t t B B

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Conductor-dielectric boundary conditions

sn Dn 1

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Conductor-free space boundary conditions

011 0 t t E D

snn E D 101

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11 12


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(Not Yet)

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Electrodynamics

• Electrostatic charges electrostatic fields

• Steady currents (motion of electric charges with uniform velocity

magnetostatic fields

• Time varying currents electromagnetic fields

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Changing Magnetic Field Current and

Voltage

Current

N S

B, H

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In summary: Faraday’s Law- Integral Form

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Faraday’s Law-Differential Form

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Time Harmonic fields and their phasor

representation

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• In general, a phasor could be a scalar or vector.

• If a vector E (x, y, z, t) is a time-harmonic field, the phasor form of

E is Es (x, y, z); the two quantities are related as E = Re (Es e jt)

If E = Eocos(t -x)ay, we can write E as: E = Re (Eoe-j x ay e

jt )

Es = Eo e-j x ay phasor form

Notice that

j

E t E

E jt

E

e E je E t t

E t j s

t j

s )Re()Re(

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• Maxwell’s equations in terms of vector field phasors (E, H) and source phasors (, J) in a simple linea r, isotropic and homogeneous medium are:

00

dS H H

dvdS E E

dS E jdS J dl H E j J H

dS H jdl E H j E

S

s s

v

vs

S

svs

s

L s s

s s s s s s

L s

s s s s

From the table, note that the time factor e jt disappears because it is

associated with every term and therefore factors out, resulting in time

independent equations22

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Plane Wave Equations

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Electromagnetic wave equation in free

space (coupling between E and H)


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Waves in General

• A wave is a function in both space and time.

• The variation of E with both time and space variable z, we may plot E as a function of t by keeping z constant and vice versa.

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The possible solution in free space is of the form:

)cos()cos(

]Re[

)Re(),(

)()(

z t E z t E

e E e E

e E t z E

omom

z t j

m

z t j

m

t j

x x

oo

A negative sign in (t oz) is associated with a wave

propagating in the +z direction (forward traveling or

positive going wave) whereas a positive sign indicates that

a wave is traveling in the – z direction (backward travelingor negative going wave)

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A plane wave traveling in the positive zdirection

)cos(

]Re[

])(Re[),(

0

)(0

z t E or

e E or

e z E t z E

o x

z t j

x

t j

x x

o

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What do Faraday and Ampere mean?

. .t

dlE s

Bd

. .C t

sH

Ddl J d

“a changing magnetic field causes an electric field”

“a changing electric field/flux causes an magnetic field”

Question : If we put these together, can we get electric andmagnetic fields that, once created, sustain one another?


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Cross-breed Ampere and Faraday!

C t

t t

t

... all in terms of E and H

... all in terms of

D

E

EH J

aB H

E

E nd H

dt d t

d

t

d

t

... differentiate both sides

... curl of both

EH E

HE

sides

2

2

d

d

d d

dt dt

d

t

d

t

H E

HE

E

2

2

d

dt

d

dt

EE

E

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C t

t t

t

... all in terms of E and H

... all in terms of

D

E

EH J

aB H

E

E nd H

t

t

t

... curl of b

HE

oth sid sE

E E eE

H

2

2t t

H HH

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Now some simplifications …

E = (0,EY,0) only

x

y

z

EY = EY0sin(ωt-βx)

Align y-axis with electric field and the x-axis with the direction of(wave) propagation (a travelling wave propagating in the x-direction, with only a y-component of E-field)

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Travelling Waves

EY = EY0sin(ωt)

EY = EY0sin(ωt)

EY = EY0sin(ωt-)


Take a time-varying electric field,E, at a point …

Add a second one with a smallphase difference, nearby …

Now let’s have a lot of them,with a sinusoidal variationof phase with direction x.

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Plane WaveWe will also look for a plane wave solution – where the field EYis the same (at an instant in time) across the entire zy plane.

Here is an animation to seewhat this means - looking at theyz plane, down the direction oftravel

Lookdownhere

E = (0,EY,0) only

x

y

z


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,0 ,

0 0

y y

y

dE dE d d d

dx dy dz dz dx

E

i j k

E

2 2 2 2

2 2, ,

0

y y y y

y y

d E d E d E d E d d d

dx dy dz dxdy dzdy dz dx

dE dE

dz dx

i j k

E

And, as we have simplified down to E=(0,Ey,0), with |EY| constantin the zy plane, this reduces to …

2

2y

y d E dx

E


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• Plane wave equation for E – describes the variation in time and space of an electric plane wave

• With a y-component only (we have aligned the y-axis with E)• propagating in the x-direction.

• There is an exactly equivalent equation for H – Eliminate E, not H, from the combination of Ampere and Faraday.

• rather a waste of our time.• We can, however, infer that whatever behaviour we get for Ey will apply to

H, although we do not yet know the direction of H.

2 2

2 2

y y y d E dE d E

dt dx dt

Becomes the 1D equation

2

2

d d

dt dt

E E

ESo (in 3D)

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What have we here?

2 2

2 2

y y y d E dE d E

dt dx dt

Variation of Ey in space(x=direction of propagation)

Variation of Ey with time

Magnetic permeability(4px107 in vacuum, larger in a magnet)

Conductivity(0 in an insulator, much larger in a conductor)

Dielectric constant(8.85x10-12 in a vacuum, larger in a dielectric)

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Start with an insulator to make life easy (=0)

2 2

2 2

y y d E d E

dx dt

( )0

j t x y y E E e

Look for a solution of the form

Where and depend upon and … the characteristics of the insulator

2 2

2 2

y y y d E dE d E

dt dx dt becomes

2 2 22 2

2 2 2

1

,y y

y y

d E d E

E E dx dt

2

2 1

, what does this mean??

,2 2

2 2

Remember, = =waveleng

f t

requency dh

an= f v f p

p

p p

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Still don’t know what it means …

• Travelling wave ofthe form

( )0 0 cos j t x

y y y E E e E t x

2

12It travels with a velocity f v

p

p

In a vacuum, =0=4px10-7, =0=8.85x10

-12

8

0 0

1 3 10 / ... a familiar speed?v m s

In (eg) glass, =0=4px10-7, =r0=5x8.85x10

-12

8

0 0

11.43 10 / ... light slows down in glass

r

v m s

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This is why lenses work …

V=3x108m/s V=1.43x108m/s V=3x108m/s

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What is H up to?( )(0, ,0) j t x y E e

E

( )00,0 , 0,0, , j t x y y dE

d j

x E

t e

HFaraday says E E

0 0(0,0, ) , 0,0,H

H j t j x

z z z t x

z H H H et

j H e

So and if

( )0 0

j t x j t x z y H e E e

H E time-phaand are in in a non-conduse ctor

0 0 0 0

1 1 Also, z y y y H E E E

(0, 0, ) (0, , 0)So and are at 90 to one another ... andz y H E H E

i Z

, the intrinsic impedance ( )of t realhe medium, is for an insulator


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Summary so far : Insulator

• H and E both obey e j(t-x)

• H and E are in time-phase• |E|=Zi|H| is the characteristic impedance

– Zi is real in an insulator – Zi = 377Ω in free space (air!) – Zi ≈ 150Ω in glass

• Wave travels at a velocity v =1/√ – 3x108 m/s in free space

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Now a conductor …

• Fields lead to currents

• Currents cause “Joule heating” (I2R)• Leads to loss of energy

• Fields still oscillate, but they decay

• Multiply the solution we have already by a term e-ax?

e-ax e-ax sin(ωt-βx)HEAT!

HEAT!

HEAT!

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Now a conductor …

• In general: the electric field in a conductor may beexpressed in the form:

)cos()cos(

))(Re(),(

a a

xt e E xt e E

e x E t x E

x

m

x

m

t j

y y

Wheremm E and E were replaced in terms of their mag. and phases

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Now a conductor … >0

2 2

2 2

y y y d E dE d E

dt dx dt

( )0

j t x y y

x E E e e a Look for a solution of the form

0

j x j t y y E E e e

a

2 2 2

0 0 0 0y y y y j E E j E E a

. j a For tidiness, write is called the propagation constant

2 , j j j j

0 x j t

y y E E e e

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Example : Good Conductor

f a v

6x107 (S/m) 100MHz 6.28x108 8.85x10-12 1.26x10-6 1.54x105 1.54x105 4x103m/s

0 , x j t

y y E E e e j j

3 3 5790 6 10 0.006 790 6 10 1.54 10 (1 ) j x j x j x j

Comments : a= , so E and H are 45° out of (time) phase v>1 … rapid attenuation via e-ax

Let’s have a look at e-ax

…

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Example : Good Conductor

e-ax

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0μm 10μm 20μm 30μm

0.36=1/e

Amplitude falls by 0.36=1/e in 6m i.e. the wave doesn’t get far incopper!

Skin Depth : the depth of penetration into a good conductor (the wave will

be attenuated by a factor

p a

f

e

11

368.01


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Example : Good Conductor,

E=ZiH …. Intrinsic Impedance

0 0(0,0, ) , 0,0,

So and if j t j t x

z x

z z z j H eH H H et

HH

00,0 , 0 ,, ,0H

Faraday says E E y j t x

y

d e

E E

dx t

0 0 0y z i z

i

j E H Z H

j j j Z

j j j

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Example : Good Conductor,

E=Zi H…. Intrinsic Impedance

40 0 0 0 0

j

y i z z z z

j j E Z H H H e H j

p

0

0

4

H Ey

z j

E H

e

p

So relates the magnitudes of and

0 04

y z E H p

and leads by

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Similarities and differences between the propagation of

uniform plane waves in free space and conductive

medium

Similarities:

• In both cases, the electric and magnetic fields areuniform in the plane perpendicular to the direction

of propagation.

• The electric and magnetic fields are perpendicular toeach other, and to the direction of propagation i.e.nocomponent of either the electric or the magnetic

field is in the direction of propagation.

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Differences:

Free Space Conductive Medium

• E, H vectors are in phase, the E, H vectors are not in phase, the

intrinsic wave impedance ois a real intrinsic wave impedance is a com-

number. plex number.

• The phase velocity = c (speed of The phase velocity is less than the

light. speed of light.

• For a plane wave of a given freq., o The =2p/ is shorter than o

is longer than in the material medium.• Does not attenuate in magnitude as it It exponentially attenuates, with

propagates. the skin depth by = 1/a

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Polarization of plane waves

• For a wave propagating along the z axis, the electric field may be expressed ashaving two components in the x and y direction:

E = (A ax + B ay) e-jz

where the amplitudes A and B may be complex.

1. If A and B have the same phase angle (a = b). In this case, the x and y

components of the electric field will be in phase

jb ja e B Be A A ,

)cos()(

)( )(

a z t a Ba A E

ea Ba A E

y x

a z j

y x

The tip of the E vector follows a line Linear polarization

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2. If A and B have different phase angles. In this case, E will no

longer remain in one plane:

The locus of the end point of the electric field vector will trace out

an ellipse once each cycle Elliptical polarization

3. If A and B are equal in magnitude and differ in phase angle by

p/2, the ellipse becomes a circle Circular Polarization

)cos(

)cos(

z bt B E

z at A E

y

x


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- If one takes a snapshot of a circularly polarized wave at any

instant then he will see the picture below.

- The E-field vector does not change in magnitude but its

direction “twists” in space.

- An observer sitting in the path of the wave will see the E-

field vector rotate in a circular trajectory at his location as the

wave passes by.

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