Chapter One: Vector Analysis The use of vectors and vector analysis ...

Electromagnetic Theorem
(Dr. Omed Ghareb Abdullah) University of Sulaimani –College of Science – Physics Department
v
(
E
) dv
first term
2
4 2
0 0 0
2
4 2
0 0 0
second
term
third term
(20 e
20 e
2
4 2
0 0 0
2
4 2
0 0 0
10 e
10
e
2
dd
dz 80
e
3 d
d
dz
d
d
dz 160
e
d
2
d
2
3 d
d
dz 24
d
24
2
2
0
2
0
2
0
by ( u dv)
by ( u dv)
2
0
48
Example(5):
Verify Stoke’s theorem for a vector field
, in the segment of cylindrical surface
defined by 2 , 60 90 , and 0 3 .
The mathematical representation of Stokes’s theorem is given by:
L
A dl
L
A ds
The line integral around a closed path defined by these bounded region is as follows:
L
b
a
c
b
d
c
a
d
B dl
B dl
B dl
B dl
B dl
b
a
B dl
B aˆ
z
z
z
z
z
B aˆ
z
B aˆ
z
B aˆ
z
c
b
d
aˆ
dz aˆ
B dl
z
0
d
( aˆ
) 0
3
0
dz ( aˆ
)
z
d
c
B dl
cos
cos90
dz z
2
3
0
a
d
B dl
0
cos
cos60
dz z
2
3
0
3
0
3
4
ρ
The left hand side of the Stokes’s theorem is :
A
s
ds
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