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
(c).
V (
r
aˆ
r
1
r
aˆ
r
1
sin
aˆ
)( r
2
cos
cos)
2 r cos
cos
aˆ
r
r sin
cos
aˆ
r cot
sin
aˆ
Example(4):
For the vector field
E 10 e aˆ
3 z aˆ
, verify the divergence theorem for the cylindrical
z
region enclosed by : 2 m , z 0 and z 4 m ?
The divergence theorem
mathematically states that:
E ds ( E)
dv
S
Z=4 m
Z
=2 m
There are three closed
surfaces bounded by the
defined region, they are:
Z=0
Y
Therefore the right hand side of the equation of divergence theorem is:
E ds E ds E ds E ds
S
S1
S 2
S 2
S 2
S
E ds
E ds
E ds
S1
2
2
0 0
2
2
0 0
2
2
0 0
(10
(10
(10
E ds
43.3
S 2
aˆ
E ds 0
48 43.3
4.7
e
e
e
aˆ
aˆ
S 3
3 z aˆ
)
d
d
( aˆ
)
z
3 z aˆ
) d
d
aˆ
z
3 z aˆ
) dz d
aˆ
z
z
z
2
2
0 0
2
2
0 0
2
2
0 0
1 2
3 z d
d
3 z (2
) 3
0 2 2
0
2
1 2
3 z d
d
3 z (2
) 3
4
2
2
48
2
10
e
2
dz d
10
e
2
z (2
) 10
e
2
4
4 (2
)
While the Left hand side is:
1 (
E
) 1 E
E
z
E
z
1
2
E (20 e 10e
) 3
2
1 (
10e
) 1 (0) (
3z)
z
57