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Electromagnetic Theory
Engr.Mian Shahzad IqbalDepartment of Telecom EngineeringUniversity of Engineering & TechnologyTaxila
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
Field Vector
Cartesian Coordinate System
x
Orthonormal Unit Vectors , ,
| | | | | | 1
x y z
x y z
x y z
e e e
e e e
e e e
Coordinates , ,x y z
= A ( , , ) A ( , , ) A ( , , )
x y z
x y zx y zx y z x y z x y z
A R A R A R A R
e e e
y
z
xxe
yye
zzeR
A R
xe ye
ze
xyz
Limits
Arbitrary Vector Field
Position Vector
( ) ( ) ( )
= ( ) ( ) ( ) ( ) ( )
( )
r z
r zr z
r z
R R R
r z
R R R R R R R
R e R e R e
e e
Cylindrical Coordinate System
Orthonormal Unit Vectors( ), ( ),
( ) ( ) | ( ) | | ( ) | | | 1
r z
r z r z
e e e
e e e e e e
Coordinates , , ; 0 , 0 2 ,r z r z
y
z
x
rr e
zzeR
Field Vector
Spherical Coordinate System
,
= A ( , , ) , ( , , ) , ( , , )
R
R R
t
R A R A R
A R A R A R A R
e e e
Orthonormal Unit Vectors
, , , ,
, ,
| , | | , | | | 1
R
R
R
e e e
e e e
e e e
Coordinates , ,R
y
z
x
R
,RR e
A R
e
,R e
, e
Arbitrary Vector Field
Limits
: Perpendicular / Senkrecht
000 2
R
Cartesian Coordinate System: Coordinate Surfaces, Unit Vectors, Surface Elements and Volume Element
xeye
ze
( , , )P x y z
const.z
const.y
const.x
xzdS
xydS
yzdS
Cylindrical Coordinate System: Coordinate Surfaces, Unit Vectors, Surface Elements and Volume Element
r e
( ) eze
const.z
const.
const.r
rzdS
xydS
zdS
d rdr
( , , )P r z
Spherical Coordinate System: Coordinate Surfaces, Unit Vectors, Surface Elements and Volume Element
const. const.R
, e
( ) e
,R e
rdS
dS
rdS
sin d R
sinR
d R
( , , )P R
Metric Coefficients and Vector Differential Line Elements
Cartesian Coordinate System
1, 1, 1x y zh h h
Cylindrical Coordinate System Spherical Coordinate System
1, , 1r zh h r h 1, , sinRh h R h R
d
d
d
d
d
d
d
d
d
r
rr
r
z
zz
z
R
h r
r
R
h
r
R
h z
z
dR s
e
e
dR s
e
e
dR s
e
e
d
d
d
d
d
d
d
d
sin d
R
RR
R
R
h R
R
R
h
R
R
h
R
dR s
e
e
dR s
e
e
dR s
e
e
d
d
d
d
d
d
d
d
d
x
xx
x
y
yy
y
z
zz
z
R
h x
x
R
h y
y
R
h z
z
dR s
e
e
dR s
e
e
dR n
e
e
Metric Coefficients and Differential Volume and Surface Elements
Cartesian Coordinate System
1, 1, 1x y zh h h
Cylindrical Coordinate System Spherical Coordinate System
1, , 1r zh h r h 1, , sinRh h R h R
d d d d
d d d
d d d
d
( ) d d
d d
d
( ) d d
d d
d
( ) d d
d d
r z
r z
z
zz
r
rz
r zz r
r
rr
z
V h r h h z
h h h r y
r r z
S
h h z
r y z
S
h h r z
r z
S
h h r
r r
dS n
e ×e
e
dS n
e ×e
e
dS n
e ×e
e
2
2
d d d d
d d d
sin d d d
d
( ) d d
sin d d
d
( ) d d
sin d d
d
( ) d d
d d
R
R
R
r
RR
R
RR
V h Rh h
h h h R
R R
S
h h
R
S
h h R
R R
S
h h R
R R
dS n
e ×e
e
dS n
e ×e
e
dS n
e ×e
e
d d d d
d d d
d d d
d
( ) d d
d d
d
( ) d d
d d
d
( ) d d
d d
x y z
x y z
yz
y zy z
x
xz
x zz x
y
xy
x yx y
z
V h xh y h z
h h h x y z
z x z
S
h h y z
y z
S
h h x z
x z
S
h h x y
x y
dS n
e ×e
e
dS n
e ×e
e
dS n
e ×e
e
Spherical CoordinatesCylindrical Coordinates Cartesian Coordinates
x
y
z
cos
sin
r
r
z
sin cos
sin sin
cos
R
R
R
2 2
arctan
x y
y
xz
r
z
sin
cos
R
R
2 2 2
2 2
arctan
arctan
x y z
x y
zy
x
2 2
arctan
r z
r
z
R
Transformation Table
z
y
x
R
Coordinates of Different Coordinate Systems
cos sin cosx r R
1. Formulate x as a function of the cylinder and spherical coordinates.
2. Formulate r as a function of the Cartesian and spherical coordinates.
3. Formulate as a function of the cylinder coordinates. .
2 2 sinr x y R
2 2 2 2 2 2
1
( cos ) ( sin ) cos sinx y r r r r
2 2x y
Examples
Cartesian Coordinates Cylindrical Coordinates Spherical Coordinates
x y zx y zA A A A e e e r zr zA A A A e e e RRA A A A = e e e
x
y
z
A
A
A
cos sin
sin cos
r
r
z
A A
A A
A
sin cos cos cos sin
sin sin cos sin cos
cos sin
R
R
R
A A A
A A A
A A
cos sin
sin cos
x y
x y
z
A A
A A
A
r
z
A
A
A
sin cos
cos sin
R
R
A A
A
A A
sin cos sin sin cos
cos cos cos sin sin
sin cos
x y z
x y z
x y
A A A
A A A
A A
sin cos
cos sin
r z
r z
A A
A A
A
RA
A
A
Transformation Table
Scalar Vector Components in Different Coordinate Systems
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.
