ELEG 413
Spring 2015
Lecture #1
Mark Mirotznik, Ph.D.
Professor
The University of Delaware
Tel: (302)831-4221
Email: mirotzni@ece.udel.edu
ELEG 413 Engineering Electromagnetics
Instructor: M. Mirotznik, Tel (302)831-4241, mirotzni@ece.udel.edu
Text: Engineering Electromagnetics, Constantine Balanis
http://www.eecis.udel.edu/~mirotzni/ELEG413/ELEG413_2016.htm
Office Hours: Tuesday/Thursday 1:00-2:00 in Evans 106
Grading Your grade will be based on, quizzes and exams broken
down as follows:
Homework
In class quizzes
1st Exam
2nd Exam
Final Exam
0%
20%
25%
25%
30%
100%
What was physics like at the time of Maxwell?
 Newton’s laws had been around for almost 200 years and
seemed to explain almost everything.
 Most felt that physics was largely a solved problem with
just some odds and ends to figure out.
 It was felt that everything when broken down to its most
fundamental level was just an application of mechanics
(Newton’s laws).
What was known about electromagnetics to Maxwell?
(1) That electrical charges come in two types (negative and
positive) and that they produce forces between them that
proportional to the amount of charge and inversely
proportional to the square of the distance between the
charges. Like charges repel and unlike charges attract.
What was known about electromagnetics to Maxwell?
(2) That magnets have two poles (N and S) and also produce a
force between magnets (like poles repel and unlike poles
attract) but unlike electrical charges we cannot separate the
two poles. If a break a magnet into two parts I get two
magnets each having a N/S poles.
What was known about electromagnetics to Maxwell?
(3) That electrical currents on two wires would create a force
between them (like magnets) - Ampere 1820
What was known about electromagnetics to Maxwell?
(3) That an electrical current (charges in motion) create a
magnetic force just like permanent magnets do. (Faraday
1821) – Electromagnetism (that electrical and magnetic
phenomenon were related)
What was known about electromagnetics to Maxwell?
(4) That an electric current inside a magnetic field produces a
force that was at right angles to the direction of the
magnetic lines of force and the direction of the current).
What was known about electromagnetics to Maxwell?
(5) Mutual inductance (Faraday) –also made the first electric
motor and generator
What was known about electromagnetics to Maxwell?
Nobody had a good explanation for why these things were
happening? Some of it was also a bit spooky!
How does Q1 even know about Q2?
(6) Lines of force (Faraday) – first time
someone starting thinking about
electromagnetism as a field theory.
Along came Maxwell:
"A Dynamical Theory of the Electromagnetic Field" is the third
of James Clerk Maxwell's papers regarding electromagnetism,
published in 1865
In section III of "A Dynamical Theory of the Electromagnetic Field",
which is entitled "General Equations of the Electromagnetic Field",
Maxwell formulated twenty equations with twenty unknowns which
were to become known as Maxwell's equations.
In this amazing piece of work Maxwell put forth the foundations of a
field theory that could explain all known phenomena regarding
electromagnetism. He also added some unknown terms that resulted
in the prediction of electromagnetic waves which travel near 3x108
m/s (speed of light). Also connecting for the first time the fields of
optics with electricity and magnetism.
This is how Maxwell’s
equations looked in his
original notation. These are
20 coupled differential
equations (Ugh!).
Unfortunately, this work was
largely ignored for nearly 20
years.
Enter the Maxwellians




George Francis FitzGerald
Oliver Lodge
Oliver Heaviside
Heinrich Hertz
These men saw the brilliance of Maxwell’s ideas and worked to:
(1) Validate the existence of EM waves experimentally (Hertz)
(2) Reformulate Maxwell’s 20 equations into a more digestible 4
vector equations (Heaviside and FitzGerald)
(3) Demonstrated through various solutions to the new Maxwell’s
equations how they can be used to predict all that is know about
electromagnetic phenomenon and some stuff that was yet to be
shown.
Maxwell’s Equations in Differential Form
(as formulated by Heaviside)


B 
Faraday’s Law
 E  
M
t

 D  
 H 
 J c  J i Ampere’s Law
t

Gauss’s Law
D  

  B  m
Gauss’s Magnetic Law
Did Maxwell and his Disciples Have a
Good Feeling for What Electric and
Magnetic Fields Are?
Maxwell’s mechanical
vortex model
FitzGerald’s Wheel and
Band Model (1885)
Lodge’s string and beads
model(1876)
Vector Analysis Review:
 
A  aA A

 A
a
A
A
 
A  aA A

a = unit vector
1. Dot Product (projection)
 
B  aB B
 

A  B  A B cos( AB ) an
2. Cross Product
  
A  B  an A B sin( AB )
 AB
 
A  aA A
Orthogonal Coordinate Systems:
 



A  au1 Au1  au2 Au2  au3 Au3  A a A



au1  au 2  au 3



au2  au1  au 3



au3  au1  au 2
2
2
2
A  Au1  Au 2  Au 3
 
A  B  Au1Bu1  Au 2 Bu 2  Au 3 Bu 3
  

A  B  au1 ( Au 2 Bu 3  Au 3 Bu 2 )  au 2 ( Au 3 Bu1  Au1Bu 3 )




 au 3 ( Au1Bu 2  Au 2 Bu1 )   au1 au 2 au 3
A  B  Au1 Au 2 Au 3
Bu1
Bu 2
Bu 3
Orthogonal Coordinate Systems:
 


dl  au1dl1  au 2dl2  au 3dl3
 
dS  an dS

an

dS
dv  dl1 dl2 dl3 dl3
dl1
dl2
dl
Cartesian Coordinate Systems:
 



A  ax Ax  a y Ay  az Az  A a A

 
ax  a y  az

 
a y  ax  az
  
az  ax  a y
 
A  B  Ax Bx  Ay By  Az Bz

ax
 
A  B  Ax
Bx

ay
Ay
By

az
Az
Bz
z
y
x
Cartesian Coordinate Systems (cont):
 


dl  a x dx  a y dy  a z dz
2
2
dl  dx  dy  dz

ds x  a x dy dz

ds y  a y dx dz

ds z  a z dx dy
dv  dx dy dz
2
Cylindrical Coordinate Systems:
 



A  ar Ar  a A  az Az  A a A
 


dl  ar dr  a rd  az dz

z
dsr  ar rd dz

ds  a dr dz

ds z  a z rd dz
dv  rdr d dz

x
(r,,z)
z
r
y
Spherical Coordinate Systems:
 



A  aR AR  a A  a A  A a A
 


dl  aR dR  a Rd  a R sin( )d
 2
z
dsR  aR R sin( )d d
(R,,)

ds  a R sin( )dR d


R
ds  a RdR d
dv  R 2 sin( )dR d d

x
y
Vector Coordinate Transformation:
cos( )  sin( ) 0 Ar
Ay   sin( ) cos( ) 0 A


Az  0
0
1 Az
Ax
Ax sin( ) cos( ) cos( ) cos( )  sin( ) AR
Ay   sin( ) sin( ) cos( ) sin( ) cos( )  A


Az  cos( )
 sin( )
0  A
Gradient of a Scalar Field:
Assume f(x,y,z) is a scalar field
The maximum spatial rate of change of f at some location
is a vector given by the gradient of f denoted by
Grad(f) or f
 f  f  f
f  a x  a y  a z
x
y
z
 f  f
 f
f  ar
 a
 az
r
r
z
f
 f  f

f  a R
 a
 a
R
r
R sin( )
Divergence of a Vector Field:
Assume E(x,y,z) is a vector field. The divergence of E is
defined as the net outward flux of E in some
 volume as the
volume goes to zero. It is denoted by   E
 E x E y E z
E 


x
y
z
 1
1 E E z
E 
(rEr ) 

r r
r 
z
 1 
1

2
E  2
( R ER ) 
(sin( ) E )
R sin( ) 
R R
E
1

R sin( ) 
Curl of a Vector Field:
Assume E(x,y,z) is a vector field. The curl of E is measure
of the circulation ofE also called a “vortex” source. It
is denoted by   E

 ax
 
 E  
 x
 E x

 ar
 1 
 E  
r  r
 Er

ay

y
Ey

az 


z 
E z 


ra a z 



 z 
rE E z 


 aR Ra


1

 E  2

R sin( )  R 
 ER RE

R sin( )a 





R sin( ) E 
Laplacian of a Scalar Field:
Assume f(x,y,z) is a scalar field. The Laplacian is
defined as   (V ) and denoted by  2V
2
2
2
V V V
V 2  2  2
x
y
z
2
2
2
1


1

V

V
2
V
(r V )  2 2  2
r r r
r 
z
1 
1


2
2 
V 2
(R
V)
(sin( ) V )
R
R sin( ) 

R R
1
 2V
 2 2
R sin ( )  2
Examples:
1. Given the scalar function
V ( x, y, z )  sin( / 2 x) sin( / 2 y) e z
Find the magnitude and direction of the maximum rate of chance at
location (xo,yo,zo)
2. Determine   (V )
3. Determine   (V )
3. The magnetic field produced by a long wire conducting a constant current
 Io
Is given by 
B(r )  a

Find   B
r
Basic Theorems:
1. Divergence Theorem or Gauss’s Law

 
   E dv   E  ds
v
s
2. Stokes Theorem

 

 (  E )  ds   E  dl
s
c
Examples:
1. Verify the Divergence Theorem for

 2 
A(r , z )  ar r  az 2 z
on a cylindrical region enclosed by r=5, z=0 and z=4
r=5
z=4
z=0
Odds and Ends:
1. Normal component of field

E
 
n  E  En
2. Tangential component of field
 
n  E  Et

n
Maxwell’s Equations in Differential Form


B 
Faraday’s Law
 E  
M
t

 D  
 H 
 J c  J i Ampere’s Law
t

Gauss’s Law
D  

  B  m
Gauss’s Magnetic Law
Faraday’s Law


B
 E  
t
 
 

c E  dl   t s B  ds

B
t
C
S

E
Ampere’s Law

J

D
t

J

H

H

  D
 H  J 
t
  
 
 
c H  dl  t s D  ds  s J  ds
Gauss’s Law
Qtot

D

D  
 
s D  ds  v  dv  Qtot
Gauss’s Magnetic Law

B  0
 
s B  ds 0
“all the flow of B entering the
volume V must leave the volume”

B
CONSTITUTIVE RELATIONS


D  E
r o=permittivity (F/m)
o=8.854 x 10-12 (F/m)


BH
r o=permeability (H/m)
o=4 x 10-7 (H/m)


Jc   E
=conductivity (S/m)
POWER and ENERGY


H
(eq1)   E   
 M d
t


  
 
E
(eq 2)   H  
  E  Ji  J d  Jc  Ji
t


take H  (eq1)  E  (eq2)

 


 
 
(eq3) H    E  E    H   H  M d  E  ( J d  J c  J i )
Using the vector identity   ( A  B)  B  (  A)  A  (  B)
 

 
 
(eq4)   ( E  H )  H  M d  E  ( J d  J c  J i )  0
n
E, H
Ji
V
, , 
Integrate eq4 over the volume V in the figure
(eq5)
 

 
 


(
E

H
)
dv


[
H

M

E

(
J

J
d
d
c  J i )] dv
v
v
Applying the divergence theorem


 
 H
 E
   
(eq6) s ( E  H )  ds  v [  H 
 E
  E  E  E  J i )] dv  0
t
t
S
POWER and ENERGY (continued)


 
 H
 E
   
(eq6) s ( E  H )  ds  v [  H 
 E
  E  E  E  J i )] dv  0
t
t


 H   1
 E   1
 


2
2
2
H
   H   wm ,  E 
   E   we ,  E  E   E
t t  2
t t  2
 t
 t
 
 
wm we
2
(eq7) s ( E  H )  ds  v [

] dv  v [ E  J i ] dv  v  E dv  0
t
t
 
 

2
(eq8) s ( E  H )  ds  v [ we  wm ] dv  v [ E  J i ] dv  v  E dv  0
t
 
Ps  s ( E  H )  ds


1
1
2
2


Wm  v [  H ] dv , We v [  E ] dv
2
2


 
2
Pi  v [ E  J i ] dv  0, Pd  v  E dv  0
Stored magnetic power (W)
Ps 
What is this term?
Supplied power (W)


Wm  We  Pi  Pd
t
t
Stored electric power (W)
Dissipated power (W)
POWER and ENERGY (continued)
 
Ps  s ( E  H )  ds
1
1
2
2


Wm  v [  H ] dv , We v [  E ] dv
2
2


 
2
Pi  v [ E  J i ] dv  0, Pd  v  E dv  0
Stored magnetic power (W)
Ps 
What is this term?
Supplied power (W)


Wm  We  Pi  Pd
t
t
Stored electric power (W)
Dissipated power (W)
Ps = power exiting the volume through radiation
  
S  E  H W/m2 Poynting vector