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
aˆ
aˆ
aˆ
aˆ
aˆ
aˆ
r
sin
cos
0
aˆ
aˆ
aˆ
aˆ
aˆ
aˆ
r
0
0
1
aˆ
aˆ
aˆ
z
z
z
aˆ
aˆ
aˆ
r
cos
sin
0
Therefore, when we have a vector in cylindrical coordinate given by:
A A aˆ
A aˆ
A ˆ
z
a z
Then this vector is transformed to spherical coordinate as follows:
A ˆ ˆ ˆ ˆ ˆ ˆ ˆ
r
A ar
A
( a
ar
) A
( a
ar
) A
z
( az
ar
)
A ˆ ( ˆ ˆ ) ( ˆ ˆ ) ( ˆ ˆ
A a
A
a
a
A
a
a
A
z
az
a
)
A A aˆ
A ( aˆ
aˆ
) A ( aˆ
aˆ
) A ( aˆ
aˆ
)
In matrix notation, we can write the transformation of vector (A) from A , A
, A ) to
( , A , A
) as:
A r
z
z
(
z
A
A
A
r
aˆ
a
ˆ
a
ˆ
aˆ
r
aˆ
aˆ
aˆ
aˆ
aˆ
aˆ
r
aˆ
aˆ
aˆ
aˆ
z
z
z
r
aˆ
aˆ
aˆ
aˆ
A
A
A
z
A
A
A
r
sin
cos
0
0
0
1
cos
sin
0
A
A
A
z
While when we have a vector ( A ) in spherical coordinate given by:
A( r,
,
)
A aˆ
A aˆ
A aˆ
, then this vector can be transformed to cylindrical
r
r
coordinate as:
A ˆ ˆ ˆ ( ˆ ˆ ˆ ˆ
A a
A
r
( ar
a
) A
a
a
) A
( a
a
)
A aˆ
( ˆ ˆ ) ( ˆ ˆ ) ( ˆ ˆ
A
A
r
ar
a
A
a
a
A
a
a
)
A A aˆ
A ( aˆ
aˆ
) A ( aˆ
aˆ
) A ( aˆ
aˆ
)
z
z
r
r
z
These equations in matrix notation can be written as:
z
z
A
A
A
z
aˆ
r
aˆ
a
ˆr
aˆ
a
ˆr
aˆ
z
aˆ
aˆ
aˆ
aˆ
aˆ
aˆ
z
aˆ
aˆ
aˆ
aˆ
aˆ
aˆ
z
A
A
A
r
A
A
A
z
sin
0
cos
cos
o
sin
0
1
0
A
A
A
r
Example(1):
Let A cos
aˆ
z
2
sin
aˆ
z
, then :
a. Transform A into Cartesian coordinate and calculate its magnitude at point (3,‐4,0).
b. Transform A into spherical coordinate and calculate its magnitude at point (3,‐4,0).
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