Electromagnetic Theory
Engr.Mian Shahzad Iqbal
Department of Telecom Engineering
University of Engineering &
Technology
Taxila
Text Book
Two textbooks will be used
extensively throughout this course
1. Field and Wave Electromagnetic
by David K.Chang
2.
“Engineering Electromagnetic by
William H.Hayt
Yahoo Group
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Group Home Page
http://groups.yahoo.com/group/mianshahzadiqb
al
Group Email
[email protected]
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Field Vector
Cartesian Coordinate System
Coordinates
Limits
z
A  R
R
x, y, z
ez
  x  
  y  
  z  
xe x
x
Orthonormal Unit Vectors
ze z
ey
ex
y
ye y
ex , e y , ez
ex  e y  ez
| e x || e y || e z | 1
Arbitrary Vector Field
A R  Ax R   A y R   Az R 
= A x ( x, y, z) e x  A y ( x, y, z) e y  A z ( x, y, z) e z
Position Vector
Cylindrical Coordinate System
z
R  R r (R)  R (R)  R z (R)
R
= Rr (R) er ( )  R (R) e ( )  Rz (R) e z
 r er ( )  z e z

ze z
r er  
x
Coordinates
Orthonormal Unit Vectors
r,, z;
0  r  , 0    2 ,   z  
er ( ), e ( ), e z
er ( )  e ( )  e z | er ( ) || e ( ) || e z | 1
y
Field Vector
Spherical Coordinate System
Coordinates
Limits
A  R
z
R,,
R

0R
0  
0    2
ReR ,  
eR ,  

x
Orthonormal Unit Vectors
e , 
y
e  
e R  ,   , e  ,   , e  
e R  ,    e  ,    e  
 : Perpendicular / Senkrecht
| e R  ,   || e  ,   || e   | 1
Arbitrary Vector Field
A  R, t   A R  R   A  R   A  R 
= A R ( R, , ) eR  ,   A ( R, , ) e  ,   A ( R, , ) e  
Cartesian Coordinate System: Coordinate Surfaces, Unit Vectors, Surface Elements
and Volume Element
z  const.
dS xy
ez
P ( x, y , z )
ex
dS xz
ey
x  const.
y  const.
dS yz
Cylindrical Coordinate System: Coordinate Surfaces, Unit Vectors, Surface Elements
and Volume Element
dS xy
z  const.
dS rz
e ( )
dS z
ez
P(r ,  , z )
er  
r  const.
dr
  const.
r d
Spherical Coordinate System: Coordinate Surfaces, Unit Vectors, Surface Elements and
Volume Element
R sin  d
dS
eR ,  
P ( R,  ,  )
e ( )
e  ,  
  const.
R  const.
dSr
R d
dS r
R sin 
Metric Coefficients and Vector Differential Line Elements
Cartesian Coordinate System
hx  1, hy  1, hz  1
dR x  s dR
Cylindrical Coordinate System
hr  1, h  r, hz  1
dR r  s dR
 e x hx dx
 e r hr dr
 e x dx
 e r dr
dR y  s dR
dR  s dR
 e y h y dy
 e h d
 e y dy
 e r d
dR z  n dR
 e z hz dz
 e z dz
dR z  s dR
Spherical Coordinate System
hR  1, h  R, h  R sin 
dR R  s dR
 e R hR dR
 e R dR
dR  s dR
 e h d
 e R d
dR  s dR
 e z hz dz
 e h d
 e z dz
 e R sin  d
Metric Coefficients and Differential Volume and Surface Elements
Cartesian Coordinate System
Cylindrical Coordinate System
Spherical Coordinate System
hx  1, hy  1, hz  1
hr  1, h  r, hz  1
dV  hx dx hy dy hz dz
dV  hr dr h d hz dz
dV  hR dR h d h d
 hx hy hz dx dy dz
 hr h hz dr d dy
 hR h h dR d d
 dz dx dz
 r dr d dz
 R 2 sin  dR d d
dS yz  n dS
dS z  n dS
 (e y ×e z ) hy hz dy dz
 (e ×e z ) h hz d dz
 e x dy dz
 er r dy dz
dS xz  n dS
 (e z ×e x ) hx hz dx dz
 e y dx dz
dS xy  n dS
dS rz  n dS
 (e z ×er ) hr hz dr dz
 e dr dz
dS r  n dS
 (e x ×e y ) hx hy dx dy
 (er ×e ) hr h dr d
 e z dx dy
 e z r dr d
hR  1, h  R, h  R sin 
dS  n dS
 (e ×e ) h h d d
 e R R 2 sin  d d
dS r  n dS
 (e ×e R ) hR h dR d
 e R sin  dR d
dS R  n dS
 (e R ×e ) hR h dR d
 e R dR d
z

Coordinates of Different Coordinate
Systems
Transformation Table
R

y
x
Cartesian Coordinates
Cylindrical Coordinates
Spherical Coordinates
R sin  cos 
x
r cos 
y
r sin 
z
z
x2  y2
r
R sin 
y
arctan
x
z

z

R cos 
x2  y 2  z 2
x2  y2
arctan
z
y
arctan
x
r2  z2
r
arctan
z

R sin  sin 
R cos 
R


Examples
1. Formulate x as a function of the cylinder and spherical
coordinates.
x  r cos   R sin  cos 
2. Formulate r as a function of the Cartesian and spherical
coordinates.
r  x 2  y 2  R sin 
3. Formulate
.
x2  y2
as a function of the cylinder coordinates.
x 2  y 2  (r cos  ) 2  (r sin  ) 2  r cos 2   sin 2   r
1
Scalar Vector Components in Different Coordinate Systems
Transformation Table
Cartesian Coordinates
Cylindrical Coordinates
Spherical Coordinates
A  Ax e x  Ay e y  Az e z
A  Ar er  A e  Az e z
A = AR e R  A e  A e
Ax
Ar cos   A sin 
AR sin  cos   A cos  cos   A sin 
Ay
Ar sin   A cos 
AR sin  sin   A cos  sin   A cos 
Az
Az
Ax cos   Ay sin 
Ar
AR sin   A cos 
 Ax sin   Ay cos 
A
A
Az
Az
AR cos   A sin 
Ar sin   Az cos 
AR
Ar cos   Az sin 
A
A
A
Ax sin  cos   Ay sin  sin   Az cos 
Ax cos  cos   Ay cos  sin   Az sin 
 Ax sin   Ay cos 
AR cos   A sin 
Electromagnetic

In EMT, we have to deal with
quantities that depend on both time
and position
Gradient

Gradient of a scalar field is a vector
field which points in the direction of
the greatest rate of increase of the
scalar field, and whose magnitude is
the greatest rate of change.
Gradient
In the above two images, the scalar field
is in black and white, black representing
higher values, and its corresponding
gradient is represented by blue arrows.
Divergence



Divergence is an operator that measures the
magnitude of a vector field's source or sink at
a given point
The divergence of a vector field is a (signed)
scalar
For example, for a vector field that denotes
the velocity of air expanding as it is heated,
the divergence of the velocity field would have
a positive value because the air expands. If
the air cools and contracts, the divergence is
negative. The divergence could be thought of
as a measure of the change in density.
Curl




Curl is a vector operator that shows a vector
field's "rotation";
The direction of the axis of rotation and the
magnitude of the rotation. It can also be
described as the circulation density.
"Rotation" and "circulation" are used here for
properties of a vector function of position,
regardless of their possible change in time.
A vector field which has zero curl everywhere
is called irrotational.