Fields Lec 5&6

1. 1
1
Electrical Boundary Conditions
2

2. 2
3
an2 is a unit vector normal to the interface from region 2 to region1
4
Electric Field Boundary Conditions:

3. 3
5
Magnetic Field Boundary Conditions:
6
K=Js
K=Js

4. 4
7
Dielectric- dielectric boundary conditions
Dielectric materials are dominated by “bound” rather than “free’’
charges (E-fields causes +ve and –ve charges of molecules to separate
and form dipoles throughout the material interior
Therefore, the free charge density sand the surface current density Js
are 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

tt BB

8
Conductor-dielectric boundary conditions
snDn  1

5. 5
9
Conductor-free space boundary conditions
01
1 0  t
t
ED 
snn ED   101
10

6. 6
11
12

7. 7
13
(Not Yet)
14
Electrodynamics
• Electrostatic charges electrostatic fields
• Steady currents (motion of electric charges with uniform velocity
magnetostatic fields
• Time varying currents electromagnetic fields

8. 8
15
Changing Magnetic Field  Current and
Voltage
Current
N S
B, H
16
Faraday’s Law
VLOOP = -E.dl
Fmagnetic = total magnetic “flux” = B.ds
Faraday’s Law : Rate of change of magnetic flux through
a loop = emf (voltage) around the loop
. .mag
loop surface
d d
dt dt
F
    
B
Edl ds
B, H
N S

9. 9
17
Lenz’s Law
B, H
Lenz’s Law emf appears and current flows that creates a
magnetic field that opposes the change – in this case an
increase – hence the negative sign in Faraday’s Law.
B, H
N S
V-, V+
Iinduced
18
Lenz’s Law
B, H
Lenz’s Law emf appears and current flows that creates a
magnetic field that opposes the change – in this case an
decrease – hence the negative sign in Faraday’s Law.
B, H
N S
V+, V-
Iinduced

10. 10
Relative motion between magnet and the coil
galvanometer shows deflection
Relative motion faster more deflection
No of turns of the coil increased greater deflection
19
20
In summary: Faraday’s Law- Integral Form

11. 11
21
Faraday’s Law-Differential Form
-There are different methods to generate
an Electric Field
HW: How? (Use illustrations)
- According to Faraday’s Law: how can you
change the magnetic flux density? (Use
illustrations)
22

12. 12
23
TMS:
24

13. 13
Faraday’s Law in increasing osteogenesis
25
HW
• https://www.youtube.com/watch?v=V7vOgHqE_P4
• http://www.youtube.com/watch?v=XJtNPqCj-iA
• From Basic Introduction to Bioelectromagnetics book:
Read: - Inductive telemetry for communication with
medical implants.
-The difficulty in propagating through skin-fat-
muscle-layers (you could find a question related to
these topics in the exam).
26

14. 14
Telemetry
• It is often useful to monitor body parameters
such as temperature, pressure, oxygen content of
blood, heart rate, etc. The transducers that
measure these parameters must often be inside
the human body. This poses two main problems:
how to get the information of the measured
parameter out of the body and how to power the
circuit.
27
Wires through the skin?
• Not a good idea:
• Nasty wound
• Mechanical stress could pull the wire out
• Danger of electrocution
• Discomfort
Internal batteries?
• Need surgery to replace battery
• Dangerous chemicals could leak
• Telemetry without internal batteries can use
inductive coupling. 28

15. 15
29
The voltage induced in the second winding is given by
V 2= −N2 dφ/dt
If the flux varies as φ = φ0 sinωt
then the induced voltage is V2 = N2 ωφ0 cosωt
Increasing the number of turns increases the induced voltage - but also
the size of the coil!
Ferromagnetization of Target Tissues by
Interstitial Injection of Ferrofluid
• Magnetic Retraction versus graspers.
• Ferromagnetic particles injected into submucosal layer of Porcine
stomach.
• Simple magnetic probe for tissue retraction and resection
30

16. 16
Spatial distribution of the electric field at the bottom of a petri dish
placed at the center between two helmholtz coils (B field is perpendicular
to the dish) –used to expose cells to low frequency EMFs
Arrow plot (direction)of the electric field at the bottom of a petri dish
placed at the center between two helmholtz coils (B field is perpendicular
to the dish)

17. 17
33
Time Harmonic fields and their phasor
representation
34
• 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 jt)
If E = Eocos(t -x)ay, we can write E as: E = Re (Eoe -j x ay e jt )
Es = Eo e -j x ay phasor form
Notice that
 











 
j
E
tE
Ej
t
E
eEjeE
tt
E tj
s
tj
s )Re()Re(

18. 18
35
• Maxwell’s equations in terms of vector field phasors (E, H) and source
phasors (, J) in a simple linear, isotropic and homogeneous medium are:
00 





  
 

dSHH
dvdSEE
dSEjdSJdlHEjJH
dSHjdlEHjE
S
ss
v
vs
S
s
vs
s
L s s
ssssss
L s
ssss






From the table, note that the time factor e jt disappears because it is
associated with every term and therefore factors out, resulting in time
independent equations
36

19. 19
37
Plane Wave Equations
38
Electromagnetic wave equation in free
space (coupling between E and H)

20. 20
39
40
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.

21. 21
41
The possible solution in free space is of the form:
)cos()cos(
]Re[
)Re(),(
)()(
ztEztE
eEeE
eEtzE
omom
ztj
m
ztj
m
tj
xx
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 traveling
or negative going wave)
42

22. 22
43
A plane wave traveling in the positive z
direction
)cos(
]Re[
])(Re[),(
0
)(0
ztEor
eEor
ezEtzE
ox
ztj
x
tj
xx
o








44
What do Faraday and Ampere mean?
. .
t
 
   



dlE s
B
d
. .C
t


 
   
 
sH
D
dl 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 and
magnetic fields that, once created, sustain one another?

23. 23
45
Cross-breed Ampere and Faraday!
C
t
t t
t



 
    
 
    




... all in terms of E and H
... all in terms of
D
E
E
H J
a
B H
E
E nd H
 
 
dt d t
d
t
d
t

 
 
   
 
 
 
   



... differentiate both sides
... curl of both
E
H E
H
E sides
 
2
2
d
d
d d
dt dt
d
t
d
t



   



H E
H
E
E
 
2
2
d
dt
d
dt
     
E
E
E
46
Cross-breed Ampere and Faraday!
C
t
t t
t



 
    
 
    




... all in terms of E and H
... all in terms of
D
E
E
H J
a
B H
E
E nd H
   
 
t
t
t
   

  
        
 





  
... curl of b
H
E
oth sid s
E
E E e
E
H
 
2
2
t t
 
 
    
 
H H
H

24. 24
47
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)
48
Plane Wave
We will also look for a plane wave solution – where the field EY
is the same (at an instant in time) across the entire zy plane.
Here is an animation to see
what this means - looking at the
yz plane, down the direction of
travel
Look
down
here
E = (0,EY,0) only
x
y
z
EY = EY0sin(ωt-βx)

25. 25
49
Cross-breed Ampere and Faraday!
,0 ,
0 0
y y
y
dE dEd 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 Ed d d
dx dy dz dxdy dzdydz dx
dE dE
dz dx
 
      
 

i j k
E
And, as we have simplified down to E=(0,Ey,0), with |EY| constant
in the zy plane, this reduces to …
 
2
2
y
y
d E
dx
     E
50
Cross-breed Ampere and Faraday!
• 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 yd E dE d E
dtdx dt
     
Becomes the 1D equation
 
2
2
d d
dt dt
      
E E
ESo (in 3D)

26. 26
51
What have we here?
2 2
2 2
y y yd E dE d E
dtdx dt
   
Variation of Ey in space
(x=direction of propagation)
Variation of Ey with time
Magnetic permeability
(4px107 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)
52
Start with an insulator to make life easy (=0)
2 2
2 2
y yd E d E
dx dt

( )
0
j t x
y yE 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 yd E dE d E
dtdx dt
    becomes
2 2 2
2 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 d
h
an= f v f
p

p
p
 p 

27. 27
53
Still don’t know what it means …
• Travelling wave of
the form
 ( )
0 0 cosj t x
y y yE E e E t x 
 
  
2
12
It travels with a velocity fv
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, =r0=5x8.85x10-12
8
0 0
1
1.43 10 / ... light slows down in glass
r
v m s
  
  
54
This is why lenses work …
V=3x108m/s V=1.43x108m/s V=3x108m/s

28. 28
55
What is H up to?
( )
(0, ,0)j t x
yE e  
E
 ( )
00,0 , 0,0, , j t x
y
ydE
d
j
x
E
t
e  
  
 
 

   


H
Faraday says E E
   
 0 0(0,0, ) , 0,0,
H
H j t jx
z z z
t x
zH H H e
t
j H e   
  
   

So and if
  ( )
0 0
j t x j t x
z yH e E e   
  

H E time-phaand are in in a non-conduse ctor
0 0 0 0
1 1
Also, z y y yH E E E
 

   
 
   
 
(0,0, ) (0, ,0)So and are at 90 to one another ... andz yH EH E
iZ


, the intrinsic impedance ( )of t realhe medium, is for an insulator
56
Summary so far : Insulator
• H and E both obey ej(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

29. 29
57
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!
58
Now a conductor …
• In general: the electric field in a conductor may be
expressed in the form:
)cos()cos(
))(Re(),(



 aa

xteExteE
exEtxE
x
m
x
m
tj
yy
Where 
mm EandE were replaced in terms of their mag. and phases

30. 30
59
Now a conductor …  >0 ( > )
2 2
2 2
y y yd E dE d E
dtdx dt
    
( )
0
j t x
y y
x
E E e e  a 
Look for a solution of the form
 
0
j xj t
y yE E e e a   

 2 2 2
0 0 0 0y y y yj E E j E Ea        
.j  a  For tidiness, write is called the propagation constant
   2
,j j j j          
0
x j t
y yE E e e 

60
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 yE E e e j j 
   
  
   3 3 5
790 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<< speed of light
 a = 1.54x105 >>1 … rapid attenuation via e-ax
Let’s have a look at e-ax …

31. 31
61
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 6m i.e. the wave doesn’t get far in
copper!
Skin Depth : the depth of penetration into a good conductor (the wave will
be attenuated by a factor
pa

f
e
11
368.01


62
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 e
t
H
H
 
 00,0 , 0 ,, ,0
H
Faraday says E E y j t x
y
d
e
E
E
dxt
 
 
   


 
 
 
 
0 0 0y z i z
i
j
E H Z H
j j j
Z
jj j


  
    
 
  
 
  


32. 32
63
Example : Good Conductor,
E=Zi H…. Intrinsic Impedance
4
0 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 E
y
z
j
E
H
e
p


So relates the magnitudes of and
0 0
4
y zE H
p
and leads by
Recall
Plane wave propagation:
Reduction of plane wave amplitude in tissue (propagation
along z)


22
22
0for0

 EkE
(Derive?)/11
2
2
mNpo

















a



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33. 33
HW
• For a frequency of 433 MHz, the muscle has r= 64.21, 
= 0.9695 S/m
• If we use 2/3 muscle to represent the human body, then *
r ,  by 2/3 to give r = 42.81 and σ = 0.6463 S/m.
• Calculate α, β, λ = 2p/ β, velocity of propagation vp
• If the electric field just inside the body Eo= 1 V/m, the
field at z = 10 cm inside the 2/3 muscle material will be
E0 e–αz =?. The power will be reduced by e–2αz to ? %
of its original value.
65
Electromagnetic waves and the human body
66

34. 34
Can you generate both figures (calculate a,  and
substitute in E = Eo exp(- a z)
67
68
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 are
uniform in the plane perpendicular to the direction
of propagation.
• The electric and magnetic fields are perpendicular to
each other, and to the direction of propagation i.e.no
component of either the electric or the magnetic
field is in the direction of propagation.

35. 35
69
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
70
Polarization of plane waves
• For a wave propagating along the z axis, the electric field may be expressed as
having two components in the x and y direction:
E = (A ax + B ay) e -jz
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
jbja
eBBeAA  ,
)cos()(
)( )(
aztaBaAE
eaBaAE
yx
azj
yx

 


The tip of the E vector follows a line Linear polarization

36. 36
71
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(
zbtBE
zatAE
y
x




72
- 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.

37. 37
73
74

38. 38
75
Circular Polarization
76

39. 39
77
Reflection at a Boundary (reflection of EM waves
from body tissues)
78
ET
HTHI
EI
HR
ER

40. 40
79
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R
IR
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TRITRI
HW
• The electric field associated with a uniform plane
wave propagating in air is given by
• If this wave is normally incident on a glass
medium of the following electrical properties (g
= 5 o , µg = µo and g =0, determine the following
1.  in air and  in glass.
2. Reflection Γ and transmission Τ coefficients
3. Amplitudes of reflected and transmitted electric
fields 80
mVztE xo
i
/a)10cos(1000 8
p 

41. 41
81
HW: Report
1. Cell phones and health ( Tumors and cell phone use ). You need
to support your claims by summarizing recent published
scientific research. You also need to state what type of
experiments/models were used to reach conclusions.
2. Exposure to electromagnetic radiation emitted by cell tower base
stations. You need to include an answer to the question we asked
at the beginning of the EM class. You also need to support your
claims by summarizing recent publications.
3. Choose at least 3 medical (therapeutic or diagnostic) applications
of EMFs and summarize the most recent publications related to
the design, implementation, modeling and was it FDA approved
(it does not have to be). You need to also show how did the
inventors or users of the idea benefit from studying the material
we covered in class so far. If you need ideas, contact me. (Hint:
TMS, but do not use it in your report)
82