UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 9
Prof. Steven Errede
LECTURE NOTES 9
AC Electromagnetic Fields Associated with a Parallel-Plate Capacitor
Let’s investigate the nature of AC electromagnetic fields associated with a parallel-plate
capacitor, e.g. with circular plates of radius a separated by a small distance d  a as shown in
the figure below – we will neglect edge effects here:
3-D View:
V  V  z  d   V  z  0   Vo
At DC (f = 0 Hz), we know the static solution to this problem, namely that the {free} charge
Qfree on the capacitor is related to the potential difference V across the capacitor’s plates by:
Q free  C V where the capacitance of the capacitor is: C   o A d (Farads) for d  a ;
the area of one plate of the parallel plate capacitor is A   a 2 .
Since there is no free electric charge between the plates of the parallel plate capacitor,

then for d  a , the solution to Laplace’s Equation  2V  r   0 {derived from Gauss’ Law
  
 
 

 E  r    free  r   o  0 , with E  r   V  r  } yields:
V  V  z  d   V  z  0    
z d
z 0
  
E  r  d 
 
But: E  r    Eo zˆ between the plates of the parallel plate capacitor for d  a

V  V  z  d   V  z  0   Vo  0   Vo   Eo d

 
E  r    Vo d  zˆ   free  o  nˆ1
Side View:
where:  free  Q free A
  free  z  d   
Q free
 free  z  0   
Q free
And:
A
A

V
C V  o AVo
  o o   o Eo

d
A
Ad

 AV
V
C V
  o o   o o   o Eo
A
Ad
d
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
1
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 9
Prof. Steven Errede
We ask:
What happens when we slowly raise the frequency from f = 0 Hz (static E-field) to f > 0?
e.g. Apply a sinusoidally time-varying potential difference across the plates of the capacitor of
the form: V  t   Vo eit  Vo  cos t  i sin t   single frequency, f   2 e.g. using a
sine-wave function generator , as shown in the figure below:
Function
Generator

 
V  t 
Vo eit
ˆ
E
r
t


z


zˆ  Eo eit zˆ with: Eo   Vo d
,
For d  a :  
d
d


The potential difference V  t  and electric field E  t  vs. time t: E  t   E  t   Eo eit
Maxwell’s Equations must be obeyed in the gap-region between the parallel plates of






capacitor, where:  free  r , t    bound  r , t   0 and: J free  r , t   J bound  r , t   0 :
  
 E  r , t   0
  
2) No magnetic charges / monopoles :  B  r , t   0
  
 
Maxwell’s Displacement Current:
  E  r , t    B  r , t  t
3) Faraday’s Law:
  
 
 
 
 
4) Ampere’s Law:   B  r , t   o o E  r , t  t  o J D  r , t  where: J D  r , t    o E  r , t  t
1) Coulomb’s Law:
2
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 9
Prof. Steven Errede
Ampere’s Law (with Maxwell’s Displacement Current) in integral form tells us that:
    B  r , t  da  

S
 



 
 B  r , t d    
o S
C




 J  r , t da  n.b. not a closed surface!
S o D
 1 
 
B
r
t
d

 2
,


C
c t
 
 


J D  r , t da  o o  E  r , t  t da
  E  r, t  t da

S
S
Using Stokes’
Theorem
where: c 2  1  o o
Let us consider a contour path of integration C1 enclosing the surface S1 as shown in the figure below:

da  danˆ   dazˆ
da   d  d
(in cylindrical
coordinates)
 
E  r , t   Eo eit zˆ and:


1
 
 B  r , t d   c
2
C1

t
  E  r, t  t da

S1
 
Note that: B  r , t   B   , t  ˆ due to the circular/azimuthal symmetry associated with this problem.



d   d ˆ   dˆ , and da  danˆ   dazˆ by the right-hand rule, da   d  d zˆ , da   d  d
 
 
in cylindrical coordinates, thus B  d  , and E  da , and:

1 
1   2  E  t  
  E  t  
 B   , t  2  2 E  t   2  B   , t   2

 ˆ  2 
 ˆ
c t
c 2    t 
2c  t 
E  t 
 i Eo eit  i E  t 
But: E  t   Eo eit 
t

i
i
 B   , t   2 E  t  ˆ  2 Eo eitˆ  Bo    eitˆ
2c
2c

 Bo   


 i 
  
B   , t   Bo    eitˆ   2 Eo  eitˆ  i  2 Eo  eitˆ
 2c

 2c




n.b. B   , t  also oscillates sinusoidally like E  t  but is 90o out-of-phase with E  t  .
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
3
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 9
Prof. Steven Errede

Note also that B   , t  is linearly proportional to ρ (the radial distance from the axis of

capacitor) and that B    0   0 at the center of the capacitor:

NOTE: B    a   0

because E    a   0
for d  a in our model
of this capacitor.
Thus, we see that for ω > 0,  (i.e. there exists) an azimuthal, ρ-dependent and time-varying

 i Eo  it
e ˆ in the gap region of the parallel-plate capacitor,
magnetic field B   , t   
2 
 2c 
for d  a . Note also that the azimuthal magnetic field is also linearly proportional to   2 f ,
thus as the frequency increases, this magnetic field also increases in strength.

Note that for ω = 0, B    0 as we obtained for the static limit case!
Furthermore, because the capacitor now has a non-zero magnetic field associated with it, for
ω > 0, the complex, frequency-dependent impedance Z    R    i    (Ohms)
{where R   = AC resistance and    = AC reactance} of the parallel-plate capacitor is no
longer just: Z    i     i 1 C  (Ohms) where     1 C = the AC capacitive
C
C
C
reactance of the capacitor (Ohms), with (complex) AC Ohm’s Law: V    I   Z  
Because of the existence of the magnetic field in gap-region of  -plate
capacitor, EM energy can also be/is stored in the magnetic field of  -plate
capacitor due to the inductance, LC (Henrys) associated with the parallelplate capacitor and hence it has an inductive reactance of  L     L
and hence has an inductive complex impedance associated with it, of
Z L    i  L    i LC (Ohms). Since the inductance associated with
this capacitor is in series with its capacitance, we add the two impedances:
 1 
ZCTOT    ZC    Z L    i  C    i  L    i 
  i LC
 C 
 1

  LC 
ZCTOT    i 
 C

The {complex} form Ohm’s Law {here} is thus: V    I   ZCTOT  
4
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 9
Prof. Steven Errede
Note that at low frequencies   0  for the parallel-plate capacitor with d  a , the capacitive
reactance     1 C       L and thus Z TOT   0   Z   0  . However, at very
C
L
C
C
high frequencies (ω → ∞),  C     L    Z
TOT
c
C
     Z L     , i.e. in the very
high frequency limit, this capacitor instead behaves like a pure inductor!!!
Note also that the electric, magnetic and total EM energy densities in the gap-region of the
parallel plate capacitor, respectively are:
  2
1
1   2





EM
u E  r , t    o E  r , t  , uM  r , t  
B  r , t  and uTOT
 r , t   uE  r , t   uM  r , t 
2
2o

i
Now because the capacitor has a non-zero time-varying magnetic field: B   , t   2 Eo eitˆ
2c
  
 
Faraday’s Law   E  r , t    B  r , t  t tells us that there will be an additional {induced}
 
electric field, because B  r , t  is also varying in time!!!
Faraday’s Law in integral form is:

S

  


  E  r , t  da  
t

 t 
 


m

B
r
t
da


,
S  
t


  t   B  r, t da is the magnetic flux (Webers = Tesla-m2) enclosed by the surface S
where 
m

S
 t 

 
 



m
E
r
t
d


B
r
t
da


,
,







C

S
t
t
where the contour C around a closed path of integration encloses the surface S through which

  t   B  r, t da passes.
magnetic flux 
m

at time t. Applying Stokes’ Theorem, we have:


S

 
Now B  Bˆ (i.e. points in the ̂ {azimuthal} direction) and thus here we need B  da hence

da2  daˆ also, and thus we take the closed contour C2 line-integral path around the surface S2
as shown in the side-view figure below:
Side-View:
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
5
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 9
Prof. Steven Errede
The induced electric field, as created by the time-varying magnetic field is:
 


E
r
t
d
,




ind
C2
t


 t 
 


m
B
r
t
da


,



2
S2
t

  t   B  r, t da = magnetic flux enclosed by contour C2 passing through surface S2
where 
m
2

S2
Then:

C2















Eind  r , t d    Eind  r , t d 1   Eind  r , t d 2   Eind  r , t d 3   Eind  r , t d 4
(1)
(2)
(3)
(4)

d 1  d zˆ

d 2  d ˆ ( ˆ  ŷ here in yˆ  zˆ plane)

d 3  d    zˆ   d zˆ

d 4  d    ˆ   d ˆ
 
 


Now   Eind  r , t    B  r , t  t tells us that if B  Bˆ direction, then in cylindrical coordinates:
0
0 
0

 0




0 

 



 




E
E
E
E z  r , t 
1 

 1 Ez
  Ez 
 


ˆ
ˆ
ˆ

z

     E  
ˆ
  Eind  r , t   

  
z
z 
  
 

 









 





0
0
  
  
Thus, we see that   E  r , t     E  r , t  ˆ only, for all points   ,  , z  in the gap region of
║-plate capacitor and for all times t. However, we see that due to the azimuthal / rotational
 
 
symmetry associated with the cylindrical ║-plate capacitor, neither Eind  r , t  nor B  r , t  can
have any explicit φ-dependence, thus E   0 and E   0 , which in turn respectively

imply that E z  0 and   E

 

must also have Eind  r , t   B  r , t  .

z

  0 . Note further that Faraday’s Law tells us that we
For d  a , the electric field in the gap region of the ║-plate capacitor cannot explicitly
 

depend on z either. Thus, E z  0   the only surviving term in   Eind  r , t  is:

 
E z  r , t 

ˆ .
  Eind  r , t   






 Eind  r , t   Eind  r , t  zˆ i.e. the induced E -field points in the ẑ direction (must be  B  Bˆ )
which is satisfied because ẑ  ˆ .
6
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 9
Prof. Steven Errede



Thus, if the induced electric field Eind  r , t   Eind  r , t  zˆ {only}, then we see that:

C2















Eind  r , t d    Eind  r , t d 1   Eind  r , t d 2   Eind  r , t d 3   Eind  r , t d 4
(1)
  E ind zˆ dzzˆ 
(1)
(2)
(3)
(4)
E ind zˆ d ˆ   E ind zˆ  dzzˆ    E ind zˆ  d ˆ 
(3)
(4)




(2)
0
0
 zˆ  ˆ 
 ẑ  ˆ 
  E ind zˆ dzzˆ   E ind zˆ   dzzˆ 
(1)

z=d
z=0
(3)
z=d
Eind    0  dz   E ind      dz
z=0


But Eind  r , t  has no explicit z-dependence, thus:

C2


z d
z d

Eind  r , t d   E ind    0, t   dzE nd     , t   dz  E ind    0, t   d  E ind     , t   d
Or:
z o

C2
z 0



Eind  r , t d    E ind    0, t   E ind     , t   d

where: Eind   , t   E ind   , t  zˆ


 





E
r
,
t
d
B
r , t da2 where S 2  surface enclosed by contour C2





C2 ind

t S2

and da2  danˆ2  daˆ (i.e. S2 lies in the y-z plane)  daxˆ {here} and da  dydz  d  dz
But:

 i 
Now: B   , t   Bo    eitˆ   2  Eo eitˆ
 2c 



   z  d  i 

it
B
S2   , t da2   0 z 0  2c 2  E0e ˆ d  dzˆ but: ˆ ˆ  1
  z d
 
 i 
 i d 
  2  Eo eit    d  dz   2  E0 eit   d  and:
 0 z 0
 0
 2c 
 2c 
1
 d   2 
2

  i 2 d 
i t

B
,
t
da

S2   2   4c 2  E0e
Then: 
And:
and ˆ   x̂ ( S2 lies in the y-z plane)
 

 i 2 d 


  i 2 d 

it
i t






B
r
,
t
da
E
e
i
E
e










2
0
0
2
2
t S2
t  4c 

 4c 

 i  i    i  i   2  1 2   2
{since i  1 and i   1 }
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
7
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 9
Prof. Steven Errede
 

  2  2d
  B  r , t da2 
E0 eit .

2
S
t 2
4c


 



Then:  Eind  r , t d     B  r , t da2 yields:
C2
S
t 2


2 2 d
d  Eind    0, t   Eind     , t   
E0 eit zˆ


4c 2
Note that the d’s cancel on both sides of the above equation. Note also that because of the

explicit  2 dependence on the RHS of above equation, we see that Eind    0, t   0 .
2

2 2
  
it
it
Eo e zˆ   
Hence: Eind   , t   
 Eo e zˆ
c
4c 2
2



Thus the total E -field in the capacitor gap is:
2



    2 
  
it
it
it
EToT   , t   E  t   Eind   , t   Eo e zˆ  
 Eo e zˆ  1  
  Eo e zˆ
 2c 
  2c  
Thus, we see here that the induced electric field caused by the time-varying magnetic field


points in the direction opposite to the initial/original E -field, reducing the overall E -field for
  0 , as we would expect from Lenz’s Law.

However, note that we now also have an additional contribution to the B -field inside the gap-region


of the parallel plate capacitor, due to the presence of the induced E -field contribution, Eind   , t  .
Before we proceed further on this discussion, it would be best for us change our notation:

 
 
Call our original time-dependent E -field, E  r , t   Eo eit  E1  r , t  .


This E -field in turn creates a time-dependent B -field by Ampere’s Law:
 
 
  
E1  r , t  1 E1  r , t 
  B1  r , t   o o
 2
.
t
c
t
 
However, because B1  r , t  also varies in time, it turn creates another induced time-dependent
electric field by Faraday’s Law:
 
  
B1  r , t 
  E2  r , t   
.
t
 
But E2  r , t  is also time-varying, and so it in turn produces another time-varying contribution to
 
the magnetic field B2  r , t  .
8
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 9
Prof. Steven Errede
 
But because B2  r , t  is also time-varying, it in turn will induce another contribution to the
 
electric field E3  r , t  and so on… i.e.:
  A. L.   F .L.   A. L.   F . L.   A. L.   F . L.   A. L.   F . L.
E1  r , t   B1  r , t   E2  r , t   B2  r , t   E3  r , t   B3  r , t   E4  r , t   B4  r , t   . . . .
 

 
 
 
 
 

 
Then: ETOT  r , t   E1  r , t   E2  r , t   E3  r , t   E4  r , t   E5  r , t   ...   En  r , t 
n 1
And:
 

 
 
 
 
 

 
BTOT  r , t   B1  r , t   B2  r , t   B3  r , t   B4  r , t   B5  r , t   ...   Bn  r , t 
n 1
So thus we see that:
 1 
 
 

C1 B1  r , t d   c 2 t S1 E1  r , t da1

 
 




E
r
,
t
d
B




1  r , t  da2
C2 2

S
t 2
 1 
 
 


,
B
r
t
d
E




2
2  r , t  da1
2
C1

S
c t 1

 
 


C2 E3  r , t d    t S2 B2  r , t da2
 1 
 
 

C1 B3  r , t d   c2 t S1 E3  r , t da1

 
 




E
r
,
t
d
B




3  r , t  da2
C2 4

S
t 2
…. etc.












Algorithmically, this infinite sequence can be written as:
 
n  1,E1  r , t 
Infinite Loop

 1 
 
B
r
t
d

 2
,


n
C1
c t

 

En  r , t da1




En 1  r , t d   
C2
t

 

Bn  r , t da2

S1
S2
 
Bn  r , t 



En 1  r , t 
n  n 1

Where contour C1 enclosing surface S1 and area element da1 are associated with the figure drawn

on page 3 of these lecture notes, and where contour C2 enclosing surface S2 and area element da2
are associated with the figure drawn on page 5 of these lecture notes.
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
9
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 9
Prof. Steven Errede
It can thus be shown for the parallel-plate capacitor with d  a that:
2
4
6



1   
1   
1   
V 
...



ETOT   , t   1 
 Eo eit zˆ with: Eo    o 



2 
2 
2 
d 
 1!  2c   2!  2c   3!  2c 

2
4
6



1   
1   
1   
i



BTOT   , t   1 
...
 Bo eitˆ with: Bo  2 Eo



2 
2 
2 
2c
 1!  2c   2!  2c   3!  2c 

and where: n !  n n  1 n  2  ...21 , 0!  1 , 1!  1 , 2!  2 , 3!= 6, 4! = 24, etc.
2
We also see that: Bn 
i
 i 
 i   i 
  
En and: En 1  
 Bn  
  2  En   
 En
2
2c
 2 
 2   2c 
 2c 
2
 i 
 i   i 
  
and: Bn 1   2  En 1   2  
 Bn   
 Bn
 2c 
 2c   2 
 2c 
Due to the cylindrical geometry / azimuthal symmetry associated with this problem, it should
not come as a surprise that:
x
Defining:
k

c
2
 k  where k 

= wavenumber
c
c

f

f 

2
Then the quantity in square brackets on the previous page becomes:
2
1
4
6
2
4
6
1   
1   
1   
1  x
1 x
1 x
 
 
  ...  1 
 
 
  ...
2 
2 
2 
2 
2 
2 
1!  2c   2!  2c   3!  2c 
1!  2   2!  2   3!  2 
The so-called “ordinary” Bessel function of the first kind, of order zero has a series expansion of
the form:
2
4
6
1 x
1 x
1 x
J0  x   1
 
 
  ...
2 
2 
2 
1!  2   2!  2   3!  2 
2k
2k
2k


1
1  x 
1  x 



x
J0  x   
  
  

2 
k  0 k !  k  1  2 
k  0 k ! k !  2 
k  0  k !  2 
k !  k  k  1 k  2  ...321 where:   k  1  k ! (for k = integer)

k
k
k
In general, the series expansion of the ordinary Bessel functions of the first kind, of order n are:
n2k
1

 x
Jn  x  
 
k  0 k !  n  k  1  2 

10
k
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 9
Prof. Steven Errede
Thus, for the cylindrical ║-plate capacitor with d  a the electric and magnetic fields in the gap
region are of the form:

V

E   , t   J 0 c Eo eit zˆ  J 0  k   Eo eit zˆ with k 
and Eo   o
d
c


B is 90o out-of-phase with E {here}:
 

B  , t   J0
  B e

c
it
o
ˆ  J 0  k   Bo    eitˆ with Bo    
i
  
Eo  i  2  Eo
2
2c
 2c 


Note that for   0 that: E    0, t   Eo eit but: B    0, t   0 .
 
x  k    
c
The {Radial} Zeroes of J0(x):
n.b. the zeroes of Jn(x) are not integer related!!


Since E   , t   J 0  k   Eo eit zˆ and B   , t   J 0  k   Bo    eitˆ we see that the zeroes xn

of J 0  x  are physically where the electric and magnetic fields vanish (!!!), i.e. E   , t   0 and

B   , t   0 when  n  xn k  cxn  with x1  2.4048 , x2  5.5201 , x3  8.6537 , etc.!!!


So let’s now examine the frequency-dependence of the E and B fields of the ║-plate capacitor:

E  , t   J0
 E e

B  , t   J0
  B e

c
o

c
it
zˆ  J 0  k   Eo eit zˆ
it
o
a.) When: ω = 0, f = 0 then: k 
with: k 

c
and: Eo  
ˆ  J 0  k   Bo    eitˆ with: Bo    

c
Vo
d
i
ik 
Eo 
Eo
2
2c
2c
 0     (static case). Then: x  k   0 and J 0  0   1


V
V
E   , t   Eo zˆ   o zˆ with: Eo   o and: B   , t   0 ←
d
d
n.b. Same result as original
static calculation
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
11
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 9
Prof. Steven Errede
b.) When:   0 , e.g. f = 60 Hz → ω = 2πf = 120π rad/sec
2  120 rad / sec
Then: k 
 
 1.257 106 rad /meter
8
c
3 10 m / sec

Suppose the radius of the capacitor is a = 1 cm = 102 m (reasonable/typical diameter)
Then:
ka  1.257 108  x (dimensionless)
And:
J 0  ka   J 0 1.257 108   1.0 (n.b. see/refer to above graph of J0(x) vs. x)


Thus, we see that at f = 60 Hz, the E -field is ≈ that of the DC E -field, and e.g. if Vo = 10 V and
V
10 V
d = 0.1 mm  a = 1 cm, i.e. d = 104 m  a = 102 m, then: Eo   o   4  105 Volts /m
10 m
d
8
a
ka
1.257 10
E0 
and: Bo    a   2 E0 
105  2.11012 Tesla {i.e. is very small}.
8
2c
2c
2  3 10
Another way to see this: c Bo    a   6.3 104 Volts /m  Eo  105 Volts /m
c.) Now suppose: f = 1 MHz = 106 Hz and   2 f  2 106 rads / sec
Then: k 
Then:

2 106
 2.1102  0.021 radians /m and if a = 1 cm = 0.01 m
8
c
3 10

(ka) = 0.021 x 0.01 = 2.1 x 104 and J0(ka) = J0 (2.1 x 104) ≈ 1 (still).

2

Vo
ka
(constant), and: Bo    a  
Eo  3.5 108 Tesla  35 nT (still very small)
d
2c
5
for Eo  10 Volts/meter and {still} c Bo    a   10.5 Volts /m  Eo  105 V /m
→ Eo  
for Vo = 10 Volts, d = 0.1 mm and
a = 1 cm = 102 m.
d.) Now suppose: f = 100 GHz = 1011 Hz and ω = 2πf = 6.3 x 1011 rads/sec
 2 6.3 1011

 2.1103 radians/m
Then: k  
8
c
3 10

Then:
(ka) = 2.1 x 103 x 102 = 21 → J0 (kρ) has 5 zeroes in it !!!

EEK!! → the E -field points in the reverse direction depending on 0 ≤ ρ ≤ a value !!!
(see above graph of J 0  x  vs. x on page 11 of these lecture notes)
Suppose instead that we pick: ka = 2.4048 = x1 = 1st zero of J0(x) = J0(kρ) (a = 102 m = 1 cm)
Then: k = 240.48 radians/m =

→ f = 1.15 x 1010 Hz = 11.5 GHz (in the microwave region)
c


2
ik 
 2.61 cm , B   , t   J 0  k   Bo eitˆ , Bo 
Eo
Then: E   , t   J 0  k   Eo eit zˆ and  
k
2c
12
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 9
Prof. Steven Errede

E 
Eo
Electric field
= 0 at   a
when
ka  2.4048
0
 a

c B
f = 11.5 GHz
a = 0.01 m

cBo  12 kaEo
Magnetic field
= 0 at   a
when

B 0  0
ka  2.4048
0
 a
f = 11.5 GHz
a = 0.01 m

The Inductance of a Parallel-Plate Capacitor
Equate: Wm 
1 2
1
LI 
2
2 o
Capacitance: C 
o A
d
v
  , I  V Z
V  IZ
ToT
ToT
V  Vo eit
 1

 L 
ZToT  ZC  Z L  i 
 C

 for d  a 
I 
Thus: Wm 

2
B d
Vo eit
Vo e it
Vo2
2
* 
, I  
then: I  II
2
 1

 1

 1

i
 L 
i 
 L 
 L 

 C

 C

 C

Vo2
1
1
1
2
LI 
L

2
2
2 o  1
2 o

 L 

 C


v
2
k 
B d   J 02  k   2 Eo2  d  ddz
4c
2
2
2
2
2
1 d  Vo d  2  d k 2 2 a 2

Eo  J 0  kp   3 d 
= L
2
2
0
2 
2 o 4 c
1

2


 C   L 
= L
Eo2 d
 1

 L 


C


2

 o o k 2 Eo2
2 o

a
o
J 02  k   3 d  with
1
  o o ,   ck , k   c
c2
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
13
UIUC Physics 436 EM Fields & Sources II

  o 2

 2
2
 1

 2c d

L



 C

L

a
o
2
Fall Semester, 2015
Lect. Notes 9
Prof. Steven Errede

J 02  kp   3 d    A

2
2
L
 1

 1 
 1 
L
2 2
 L   A 
 A 2 L2  A 
 L  A
  2A
  2A    A L
C
 C

 C 
 C 
C 
2
 2A 
 1 
 1 L  A 
 L 
or: A
 0

C

C






a
2
2
b
c
b  b 2  4ac
.
2a
 Quadratic equation of the form: aL2  bL  c  0 , solve for L: L 

1  2A
L
C
 
1  2A

C

2
2
2
 4A 
2 C2
2A 2

L
1  2A


2
2
 1  4A  4A 2  4A 2
C
C
C
C
2A

1  2A
C
 
1  2A

C
 2A
C

2
2A 2
 1  2A C  
1  4A
2A
2
 
2
C
2
Physically, we want L  0 when   0  must choose – (negative) sign in above formula!
 L
1  2A C  
1  4A C
2A
Now:
2
2
 o 2
1  4A 


A
8 C 
2c 2 d


L 
2A 2
 2C 2

a
0
2
1
1
4A
1   1    2 ... for  
 1 , thus:
2
8
C
J 02  k    3 d 
 A
2
o
2

d
2c 2 o A2

a
0
J 02  k    3 d  
o d
2A
2

a
0
J 02  k    3 d 
d2
Where the capacitance and inductance of the parallel-plate capacitor, for d  a are:
C
o A
d
and L 
Note that for
14
o d
ka =
2A
2

a
0
4A   2 2
J 02  k    3 d  for   

2
 C   Ac
2.4048 – 1st
5.5201 – 2nd
8.6537 – 3rd
11.7915 – 4th
14.9309 – 5th
18.0711 – 6th
.
.
.

a
o

J 02  kp   3 d    1 .

zeroes of J0(ka)
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 9
Prof. Steven Errede

2
The electric field E    a   0 for these values of ka, corresponding to wavelengths  
k
c 
ck
and frequencies f  

 2 2
λ1 = 2.61 cm ↔ f1 = 1.15 x 1010 Hz = 11.5 GHz
λ2 = 1.14 cm ↔ f2 = 2.64 x 1010 Hz = 26.4 GHz
λ3 = 0.73 cm ↔ f3 = 4.13 x 1010 Hz = 41.3 GHz
λ4 = 0.53 cm ↔ f4 = 5.63 x 1010 Hz = 56.3 GHz
λ5 = 0.42 cm ↔ f5 = 7.13 x 1010 Hz = 71.3 GHz
λ6 = 0.35 cm ↔ f6 = 8.63 x 1010 Hz = 86.3 GHz
.
.
.
.
.
.

Note that because the electric field E    a   0 for these specific frequencies
Compare with radius
a = 1.0 cm and
diameter D = 2a = 2.0 cm
of ║-plate cylindrical
capacitor, as well as
the gap dimension of
d = 0.1 mm = 0.01 cm
(corresponding to the zeroes of J0(x)=J0(ka)), this means that physically, we could actually short
out the capacitor at ρ = a and it wouldn’t make any difference to the behavior / physics of this
“capacitor” at these specific frequencies f1, f2, f3, . .!!!
For ka = zero of J0(ka) {i.e. J0(ka) = 0), we can short out the capacitor by wrapping it e.g. with
sheet metal at ρ = a , thus turning it into a cylindrical, fully-enclosed can with d  a !
→ No change in physics for frequencies f1, f2, f3, . .

because E    a   0 for these frequencies!
Thus, at these frequencies f1, f2, f3, . . fn corresponding
to the zeroes of the Bessel Function J0(ka) (i.e. J0(ka) = 0),
a cylindrical conducting metal can of radius a and height
d  a is actually a resonant cavity with electric field:

E   , t   J 0  kn   Eo eint zˆ and magnetic field:

B   , t   J 0  kn   B0    eitˆ
subject to the boundary conditions that:
E    a, t   E z    a, t   0 and also that:
B    a, t   B    a, t   0


n
, n  2 f n , n = 1, 2, 3, . . .
c
ik 
V
Eo and: Eo   o
with: Bo 
d
2c
for:
kn 
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.
15
UIUC Physics 436 EM Fields & Sources II
Fall Semester, 2015
Lect. Notes 9
Prof. Steven Errede
We will see shortly in the next set of P436 Lecture Notes (# 10) that the resonant frequencies
of a resonant cavity and the allowed modes of EM wave propagation in wave guides can be
derived directly from the wave equation for EM waves in these structures, as determined by the
boundary conditions imposed on the EM waves by the conducting walls of these devices and also
the allowed polarization states of these EM waves.


Here in these lecture notes, we obtained harmonic EM wave solutions for E and B in the gap
region of a parallel plate capacitor (and cylindrical can capacitor, subject to boundary condition
E = 0 at ρ = a) via a perturbative technique, analogous to what we did last semester in P435 for


the E -field associated with a dielectric sphere immersed in an initially uniform external E -field

and the B -field associated with a magnetizable sphere immersed in an initially uniform external

B -field. (See/work Griffiths Problems 4.23 and 6.18).
“Homework” Exercises:
1.) Calculate the electric, magnetic and total energy densities u E   ,  , z , t  , um   ,  , z , t  and
uTot   ,  , z , t  and their time averages; make e.g. plots of these vs.  . Investigate/plot their
behavior for low frequencies   0  and at higher frequencies, when n  ckn  c  xn a 
where xn  kn a = zeroes of J 0  xn   0 .


1 
2.) Calculate Poynting’s vector S   ,  , z , t  
E   ,  , z, t   B   ,  , z, t  and its time average;
o

make plots of S    vs.  , investigate/plot its behavior for low frequencies   0  and
when n  ckn  c  xn a  .


3.) Calculate the linear EM momentum density, 
EM   ,  , z , t    o  o S   ,  , z , t  and angular

 
momentum density,  EM   ,  , z , t   r 
EM   ,  , z , t  , and their time averages; make plots


of 
EM    and  EM    vs.  ; investigate/plot their behavior for low frequencies
  0  and when n  ckn  c  xn a  .
16
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2015. All Rights Reserved.