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http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
1
Revision of Electrostatics (week 1)
Vector Calculus (week 2-4)
Maxwell’s equations in integral and differential form (week 5)
Electromagnetic wave behaviour from Maxwell’s equations (week 6-7.5)–Heuristic description of the generation of EM waves
–Mathematical derivation of EM waves in free space
–Relationship between E and H
–EM waves in materials
–Boundary conditions for EM waves
Fourier description of EM waves (week 7.5-8)
Reflection and transmission of EM waves from materials (week 9-10)
22.3MB1 ElectromagnetismDr Andy Harvey
e-mail a.r.harvey@hw.ac.uk
lecture notes are at:http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
2
Maxwell’s equations in differential form
continuity of Equationt
law s Faradaydt
law s Amperedt
tics magnetosta for law Gauss
tics electrosta for law Gauss
J
BE
DJ H
B
D
.
'
'
' 0 .
' .
HB
ED
• Varying E and H fields are coupled
NB Equations brought from elsewhere, or to be carried on to next page, highlighted in this colour
NB Important results highlighted in this colour
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
3
Fields at large distances from charges and current sources
• For a straight conductor the magnetic field is given by Ampere’s law
dt
HE
dt
EJH
I H
• dH/dt produces E
• dE/dt term produces H
• etc.
• How do the mixed-up E and H fields spread out from a modulated current ?– eg current loop, antenna etc
• At large distances or high frequencies H(t,d) lags I(t,d=0) due to propagation time– Transmission of field is not instantaneous
– Actually H(t,d) is due to I(t-d/c,d=0)
• Modulation of I(t) produces a dH/dt term
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
4
A moving point charge:• A static charge produces radial
field lines
• Constant velocity, acceleration and finite propagation speed distorts the field line
• Propagation of kinks in E field lines which produces kinks of and
• Changes in E couple into H & v.v– Fields due to are short range
– Fields due to propagate
• Accelerating charges produce travelling waves consisting coupled modulations of E and H
tE22 t E
tE22 t E
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
5
Depiction of fields propagating from an accelerating point charge
• Fields propagate at c=1/xm/sec
E in plane of page
H in and out of page
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
6
Examples of EM waves due to accelerating charges
• Bremstrahlung - “breaking radiation”
• Synchrotron emission
• Magnetron
• Modulated current in antennas– sinusoidal v.important
• Blackbody radiation
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
7
2.2 Electromagnetic waves in lossless media - Maxwell’s equations
SI Units• J Amp/ metre2 • D Coulomb/metre2
• H Amps/metre• B Tesla
Weber/metre2
Volt-Second/metre2
• E Volt/metre• Farad/metre• Henry/metre• Siemen/metre
dt
BE
dt
DJH
D.
0. B
EJ
HHB
EED
or
or
t
J.
Maxwell
Equation of continuity
Constitutive relations
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
8
2.6 Wave equations in free space• In free space
– =0 J=0
– Hence:
dt
BE
dtdt
DDJH
2
2
t
tt
ttt
o
o
o
EE
D
HBB
E
– Taking curl of both sides of latter equation:
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
9
Wave equations in free space cont.
• It has been shown (last week) that for any vector A
where is the Laplacian operator
Thus:
2
2
to
E
E
AAA 2.
2
2
2
2
2
22
zyx
2
22.
to
E
EE
2
22
to
E
E
• There are no free charges in free space so .E==0 and we get
A three dimensional wave equation
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
10
• Both E and H obey second order partial differential wave equations:
2
22
to
E
E
Wave equations in free space cont.
2
22
to
H
H
22 seconds
eVolts/metr
metre
eVolts/metr o
• What does this mean – dimensional analysis ?
– has units of velocity-2
– Why is this a wave with velocity 1/ ?
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
11
The wave equation
2
22
to
A
A
)( vtzfAA o
vtzfvAvtzfA ooo 2
ov 1cv oo m/sec109979.21 8
v
Substitute back into the above EM 3D wave equation
This is a travelling wave if
• In free space
• Why is this a travelling wave ?
• A 1D travelling wave has a solution of the form:
Constant for a travelling wave
vtzfvAt
vtzfAz oo
2
2
2
2
2 AA
z
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
12
• Substitute this 1D expression into 3D ‘wave equation’ (Ey=Ez=0):
Wave equations in free space cont.
vtzEx 2
sin
o
o
v
vtxvvtx
vtzvvtzt
vvtzt
vtzvtzz
vtzz
1
sinsin
sincossin
sincossin
222
222
22
2
22
to
E
E
• Sinusoidal variation in E or H is a solution ton the wave equation
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
13
• Maxwells equations lead to the three-dimensional wave equation to describe the propagation of E and H fields
• Plane sinusoidal waves are a solution to the 3D wave equation
• Wave equations are linear
– All temporal field variations can be decomposed into Fourier components• wave equation describes the propagation of an arbitrary disturbance
– All waves can be written as a superposition of plane waves (again by Fourier analysis)
• wave equation describes the propagation of arbitrary wave fronts in free space.
Summary of wave equations in free space cont.
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
14
Summary of the generation of travelling waves
24 r
qE
o
022 tE• We see that travelling waves are set up when – accelerating charges
– but there is also a field due to Coulomb’s law:
– For a spherical travelling wave, the power carried by the travelling wave obeys an inverse square law (conservation of energy)
• P E2 1/r2
– to be discussed later in the course
• E/r
– Coulomb field decays more rapidly than travelling field– At large distances the field due to the travelling wave is much larger than the
‘near-field’
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
15
Heuristic summary of the generation of travelling waves
E due to stationary charge (1/r2)
due to charge constant velocity (1/ r2)
tE
22 t E kinks due to charge
acceleration (1/r)
22 t H due to E (1/r)
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
16
• Consider a uniform plane wave, propagating in the z direction. E is independent of x and y
2.8 Uniform plane waves - transverse relation of E and H
0.
zyx
zyx EEEE
00
yx
EE
)waveanotisconst(000,0
zzzyx EE
z
E
y
E
x
E
In a source free region, .D= =0 (Gauss’ law) :
E is independent of x and y, so
• So for a plane wave, E has no component in the direction of propagation. Similarly for H.
• Plane waves have only transverse E and H components.
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
17
Orthogonal relationship between E and H:• For a plane z-directed wave there are no variations along x and y:
y
A
x
A
x
A
z
A
z
A
y
AA
xyz
zxy
yzx
a
a
a
z
H
z
Hx
yy
x
aaH
t
E
z
H
t
E
z
H
yx
xy
t
H
z
E
t
H
z
E
yo
x
xo
y
to HE • Equating terms:dt
DJH
• and likewise for :
t
E
t
E
t
E
t
zy
yy
xx aaa
D
• Spatial rate of change of H is proportionate to the temporal rate of change of the orthogonal component of E & v.v. at the same point in space
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
18
t
E
z
H
t
E
z
H
yx
xy
• Consider a linearly polarised wave that has a transverse component in (say) the y direction only:
vtzfvEt
E
vtzfEE
oy
oy
t
H
z
E
t
H
z
E
yo
x
xo
y
xo
y EH
• Similarly
yo
x
y
oox
EH
vE
vtzfvEconstzvtzfvEH
d
z
H x
Orthogonal and phase relationship between E and H:
• H and E are in phase and orthogonal
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
19
• The ratio of the magnetic to electric fields strengths is:
which has units of impedance
yo
x EH
xo
y EH
o
yx
yx
H
E
HH
EE
22
22
metreamps
metreVolts
/
/
37712010
36
1104
9
7
o
o
cH
E
B
E
ooo
1
Note:
• and the impedance of free space is:
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
20
Orientation of E and H• For any medium the intrinsic impedance is denoted by
and taking the scalar product
so E and H are mutually orthogonal
y
x
x
y
H
E
H
E
0
.
yxxy
yyxx
HHHH
HEHE
HE
2
22
H
HH
HEHE
z
xyz
xyyxz
aHE
a
aHE
xyyxz
zxxzy
yzzyx
BABA
BABA
BABA
a
a
aBA
• Taking the cross product of E and H we get the direction of wave propagation
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
21
A ‘horizontally’ polarised wave
Hy Ex
xo
y EH
HE
• Sinusoidal variation of E and H
• E and H in phase and orthogonal
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
22
A block of space containing an EM plane wave• Every point in 3D space is characterised by
– Ex, Ey, Ez
– Which determine• Hx, Hy, Hz
• and vice versa– 3 degrees of freedom
HEHy
Ex
yo
x
xo
y
EH
EH
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
23
An example application of Maxwell’s equations:
The Magnetron
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
24
Poynting vector S
Displacement current, D
Current, J
The magnetron
Lorentz force F=qvB
• Locate features:
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
25
Power flow of EM radiation• Intuitively power flows in the direction of
propagation– in the direction of EH
2
2
E
H
HEHE
z
z
xyyxz
a
a
aHE
• What are the units of EH?– H2=.(Amps/metre)2 =Watts/metre2 (cf I2R)
– E2/=(Volts/metre)2 / =Watts/metre2 (cf I2R)
• Is this really power flow ?
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
26
• Energy stored in the EM field in the thin box is:
Power flow of EM radiation cont.
HEHy
Ex
dx
Area A
xAuuUUU HEHE dddd
2
22
2
Hu
Eu
oH
E
xAE
xAHE
U o
d
d22
d
2
22
xo
y EH
• Power transmitted through the box is dU/dt=dU/(dx/c)....
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
27
• This is the instantaneous power flow– Half is contained in the electric component
– Half is contained in the magnetic component
• E varies sinusoidal, so the average value of S is obtained as:
2
22
W/mddd
d
E
xAcxA
E
tA
US
o
xAEU dd 2
Power flow of EM radiation cont.
2sinRMS
sin
2sinE
222
2
22
oo
o
o
o
EvtzE
ES
vtzES
vtzE
• S is the Poynting vector and indicates the direction and magnitude of power flow in the EM field.
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
28
Example problem• The door of a microwave oven is left open
– estimate the peak E and H strengths in the aperture of the door.
– Which plane contains both E and H vectors ?
– What parameters and equations are required?
222
W/mHE
S
• Power-750 W
• Area of aperture - 0.3 x 0.2 m
• impedance of free space - 377 • Poynting vector:
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
29
Solution
μTesla2.775.5104B
A/m75.5377
2170H
V/m171,22.0.3.0
750377
WattsA
7
22
H
E
A
PowerE
HAE
SAPower
o
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
30
• Suppose the microwave source is omnidirectional and displaced horizontally at a displacement of 100 km. Neglecting the effect of the ground:
• Is the E-fielda) vertical
b) horizontal
c) radially outwards
d) radially inwards
e) either a) or b)
• Does the Poynting vector pointa) radially outwards
b) radially inwards
c) at right angles to a vector from the observer to the source
• To calculate the strength of the E-field should onea) Apply the inverse square law to the power generated
b) Apply a 1/r law to the E field generated
c) Employ Coulomb’s 1/r2 law
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
31
Field due to a 1 kW omnidirectional generator (cont.)
mV/m5.1104
750377
4
nW/m97.5104
10
4
252
225
3
2
r
Power
A
PowerE
r
PSPower
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
32
4.1 Polarisation
• In treating Maxwells equations we referred to components of E and H along the x,y,z directions– Ex, Ey, Ez and Hx, Hy, Hz
• For a plane (single frequency) EM wave – Ez=Hz =0
– the wave can be fully described by either its E components or its H components
yo
x
xo
y
EH
EH
– It is more usual to describe a wave in terms of its E components
• It is more easily measured
– A wave that has the E-vector in the x-directiononly is said to be polarised in the x direction
• similarly for the y direction
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
33
Polarisation cont..• Normally the cardinal axes are Earth-referenced
– Refer to horizontally or vertically polarised
– The field oscillates in one plane only and is referred to as linear polarisation• Generated by simple antennas, some lasers, reflections off dielectrics
– Polarised receivers must be correctly aligned to receive a specific polarisation
A horizontal polarised wave generated by a horizontal dipole and incident upon horizontal and vertical dipoles
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
34
Horizontal and vertical linear polarisation
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
35
Linear polarisation• If both Ex and Ey are present and in phase then components
add linearly to form a wave that is linearly polarised signal at angle
x
y
E
E1tan
Horizontal polarisation
Verticalpolarisation
Co-phased vertical+horizontal=slant Polarisation
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
36
Slant linear polarisation
Slant polarised waves generated by co-phased horizontal and vertical dipoles and incident upon horizontal and vertical dipoles
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
37
Circular polarisation
vtzEE ohh sin
vtzE
vtzEE
ov
ovv
cos
2sin
vtzEE ohh sin
vtzE
vtzEEv
ov
ov
cos
2sin
RHC polarisation
LHC polarisation NB viewed as approaching wave
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
38
Circular polarisation
LHC & RHC polarised waves generated by quadrature -phased horizontal and vertical dipoles and incident upon horizontal and vertical dipoles
LHC
RHC
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
39
Elliptical polarisation - an example
3sin5.1
vtzEv
vtzEh sin0.1
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
40
Constitutive relations• permittivity of free space =8.85 x 10-12 F/m
• permeability of free space o=4x10-7 H/m
• Normally r(dielectric constant) andr
– vary with material
– are frequency dependant• For non-magnetic materials r ~1 and for Fe is ~200,000
• ris normally a few ~2.25 for glass at optical frequencies
– are normally simple scalars (i.e. for isotropic materials) so that D and E are parallel and B and H are parallel
• For ferroelectrics and ferromagnetics r and r depend on the relative orientation of the material and the applied field:
EJ
HHB
EED
or
or
z
y
x
zzzyzx
yzyyyx
xzxyxx
z
y
x
H
H
H
B
B
B
o
ij
00
0j
0jAt microwavefrequencies:
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
41
Constitutive relations cont...• What is the relationship between and refractive index for non
magnetic materials ?– v=c/n is the speed of light in a material of refractive index n
– For glass and many plastics at optical frequencies• n~1.5
• r~2.25
• Impedance is lower within a dielectric
What happens at the boundary between materials of different n,r,r
?
r
roo
n
n
cv
1
ro
ro
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
42
Why are boundary conditions important ?• When a free-space electromagnetic wave is incident upon a medium secondary waves are
– transmitted wave
– reflected wave
• The transmitted wave is due to the E and H fields at the boundary as seen from the incident side
• The reflected wave is due to the E and H fields at the boundary as seen from the transmitted side
• To calculate the transmitted and reflected fields we need to know the fields at the boundary– These are determined by the boundary conditions
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
43
Boundary Conditions cont.
• At a boundary between two media, r,r are different on either side.
• An abrupt change in these values changes the characteristic impedance experienced by propagating waves
• Discontinuities results in partial reflection and transmission of EM waves
• The characteristics of the reflected and transmitted waves can be determined from a solution of Maxwells equations along the boundary
2,22
1,11
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
44
2.3 Boundary conditions• The tangential component of E is continuous
at a surface of discontinuity
– E1t,= E2t
• Except for a perfect conductor, the tangential component of H is continuous at a surface of discontinuity
– H1t,= H2t
E1t, H1t
2,22
1,11
E2t, H2t
2,22
1,11D1n, B1n
D2n, B2n
• The normal component of D is continuous at the surface of a discontinuity if there is no surface charge density. If there is surface charge density D is discontinuous by an amount equal to the surface charge density.
– D1n,= D2n+s
• The normal component of B is continuous at the surface of discontinuity
– B1n,= B2n
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
45
Proof of boundary conditions - continuity of Et
• Integral form of Faraday’s law:A
BsE d
td
A..
yxt
BxE
yE
yExE
yE
yE z
xyyxyy
243112 2222
21
21 0xx
xxEE
xExE
00 00
0That is, the tangential
component of E is continuous
0,0As yxtBy z
2,22
1,11y
x1yE
2yE
2xE
1xE 3yE
4yE
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
46
• Ampere’s law
Proof of boundary conditions - continuity of Ht
AJD
sH dt
dA
..
yxJt
DxH
yH
yHxH
yH
yH z
zxyyxyy
243112 2222
0,0As yxJtDy zz
21
21 0xx
xxHH
xHxH
That is, the tangential component of H is
continuous
00 00
0
2,22
1,11y
x1yH
2yH
2xH
1xH 3yH
4yH
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
47
• The integral form of Gauss’ law for electrostatics is:
Proof of boundary conditions - Dn
V
dVAD d.
applied to the box gives
yxyxDyxD snn edge21
0,0dAs edge z hence
snn DD 21 The change in the normal component of D at a boundary is equal to the surface charge density
y
2,22
1,11
1nD
2nD
z
x
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
48
• For an insulator with no static electric charge s=0snn DD 21
Proof of boundary conditions - Dn cont.
21 nn DD
• For a conductor all charge flows to the surface and for an infinite, plane surface is uniformly distributed with area charge density s
In a good conductor, is large, D=E0 hence if medium 2 is a good conductor
snD 1
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
49
Proof of boundary conditions - Bn
• Proof follows same argument as for Dn on page 47,
• The integral form of Gauss’ law for magnetostatics is
– there are no isolated magnetic poles 0d. AB
21
edge21 0
nn
nn
BB
yxByxB
The normal component of B at a boundary is always continuous at a boundary
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
50
2.6 Conditions at a perfect conductor• In a perfect conductor is infinite• Practical conductors (copper, aluminium silver) have
very large and field solutions assuming infinite can be accurate enough for many applications– Finite values of conductivity are important in calculating
Ohmic loss
• For a conducting medium– J=E
• infinite infinite J
• More practically, is very large, E is very small (0) and J is finite
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
51
• It will be shown that at high frequencies J is confined to a surface layer with a depth known as the skin depth
• With increasing frequency and conductivity the skin depth, x becomes thinner
2.6 Conditions at a perfect conductor
Lower frequencies, smaller
Higher frequencies, larger
xx
Current sheet
• It becomes more appropriate to consider the current density in terms of current per unit with:
0A/mlim
xx s
JJ
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
52
• Ampere’s law:AJ
DsH d
td
A..
yxJt
DxH
yH
yHxH
yH
yH z
zxyyxyy
243112 2222
szzz xJyxJyxtDy ,0,0As
szxx JHH 21
That is, the tangential component of H is discontinuous by an amount equal to the
surface current density
Jszx0 00
0
2,22
1,11y
x1yH
2yH
2xH
1xH 3yH
4yH
Conditions at a perfect conductor cont. (page 47 revisited)
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
53
• From Maxwell’s equations:– If in a conductor E=0 then dE/dT=0– Since
Conditions at a perfect conductor cont. (page 47 revisited)cont.
dt
HE
szx JH 1
Hx2=0 (it has no time-varying component and also cannot be established from zero)
The current per unit width, Js, along the surface of a perfect conductor is equal to the magnetic field just outside the surface:
• H and J and the surface normal, n, are mutually perpendicular:
HnJ s
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
54
Summary of Boundary conditions
21
21
21
21
nn
nn
tt
tt
BB
DD
HH
EE
0.
0.
0
0
21
21
21
21
BB
DD
HH
EE
n
n
n
n
At a boundary between non-conducting media
0.
.
0
0
21
21
21
21
BB
DD
HH
EE
n
n
n
n
s
At a metallic boundary (large )
At a perfectly conducting boundary
0.
.
0
1
1
1
1
B
D
JH
E
n
n
n
n
s
s
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
55
2.6.1 The wave equation for a conducting medium• Revisit page 8 derivation of the wave equation with J0
dt
BE
dt
DJH
As on page 8:HB
BE
ttt o
2
2
2
2
tt
tt
tt
oo
oo
o
EE
DJ
DJE
ED
EJ
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
56
In the absence of sources hence:
2
22
2
2
.tt
EEE
tt
E
oo
oo
E
EE
2.6.1 The wave equation for a conducting mediumcont.
0. E
2
22
tt oo
EE
E
• This is the wave equation for a decaying wave– to be continued...
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
57
Reflection and refraction of plane waves
• At a discontinuity the change in and results in partial reflection and transmission of a wave
• For example, consider normal incidence: ztj
ieE waveIncident ztj
reE waveReflected
• Where Er is a complex number determined by the boundary conditions
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
58
Reflection at a perfect conductor• Tangential E is continuous across the
boundary – see page 45
• For a perfect conductor E just inside the surface is zero– E just outside the conductor must be zero
ri
ri
EE
EE
0
• Amplitude of reflected wave is equal to amplitude of incident wave, but reversed in phase
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
59
Standing waves• Resultant wave at a distance -z from the interface
is the sum of the incident and reflected waves
tj
i
tjzjzji
ztjr
ztji
T
ezjE
eeeE
eEeE
tzE
sin2
wavereflectedwaveincident,
and if Ei is chosen to be real
tzE
tjtzjEtzE
i
iT
sinsin2
sincossin2Re,
j
ee jj
2sin
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
60
Standing waves cont...
• Incident and reflected wave combine to produce a standing wave whose amplitude varies as a function (sin z) of displacement from the interface
• Maximum amplitude is twice that of incident fields
tzEtzE iT sinsin2,
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
61
Reflection from a perfect conductor
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
62
Reflection from a perfect conductor• Direction of propagation is given by EH
If the incident wave is polarised along the y axis:
xixi
yiyi
HH
EE
a
a
xiyiz
xiyixy
HE
HE
a
aaHE
then
From page 18
That is, a z-directed wave.
xiyiz HEaHΕ For the reflected wave and yiyr EE aSo and the magnetic field is reflected without change in phase
ixixr HHH a
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
63
• Given that
derive (using a similar method that used for ET(z,t) on p59) the form for HT(z,t)
Reflection from a perfect conductor
2cos
jj ee
tj
i
tjzjzji
ztjr
ztjiT
ezH
eeeH
eHeHtzH
cos2
,
As for Ei, Hi is real (they are in phase), therefore
tzHtjtzHtzH iiT coscos2sincoscos2Re,
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
64
• Resultant magnetic field strength also has a standing-wave distribution
• In contrast to E, H has a maximum at the surface and zeros at (2n+1)/4 from the surface:
tzHtzH iT coscos2,
Reflection from a perfect conductor
free space silver
resultant wave
z = 0
z [m]
E [V/m]
free space silver
resultant wave
z = 0
z [m]
H [A/m]
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
65
Reflection from a perfect conductor
• ET and HT are /2 out of phase( )
• No net power flow as expected– power flow in +z direction is equal to power flow in - z direction
tzHtzH iT coscos2,
tzEtzE iT sinsin2,
2/cossin tt
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
66
Reflection by a perfect dielectric
• Reflection by a perfect dielectric (J=E=0)– no loss
• Wave is incident normally– E and H parallel to surface
• There are incident, reflected (in medium 1)and transmitted waves (in medium 2):
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
67
Reflection from a lossless dielectric
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
68
Reflection by a lossless dielectric
• Continuity of E and H at boundary requires:
tri
tri
HHH
EEE
tt
rr
ii
HE
HE
HE
2
1
1
Which can be combined to give
rittriri EEEHEEHH 221
111
1212
12
21
11
ri
riri
riri
EE
EEEE
EEEE
12
12
i
rE E
E
The reflection coefficient
roj
j
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
69
• Similarly
Reflection by a lossless dielectric
tri
tri
HHH
EEE
12
2
12
12
12
12 21
i
r
i
ir
i
tE E
E
E
EE
E
E
12
22
E
The transmission coefficient
http://www.cee.hw.ac.uk/~arharvey/22.3MB1electromagnetism.zip Dr Andy Harvey
70
• Furthermore:Reflection by a lossless dielectric
Hi
t
i
t
Hi
r
i
r
E
E
H
H
E
E
H
H
12
1
12
2
2
1
2
1 22
And because =o for all low-loss dielectrics
21
2
21
2
21
1
21
1
21
21
21
21
22
22
nn
n
nn
n
E
E
nn
nn
E
E
H
i
rE
Hi
rE
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