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
â r
â
â
r sin
r sin cos
r sin sin
x r sin
cos
y r sin
sin
z r cos
r
x
tan
2
1
cos
y
1
y
x
z
r
2
z
2
aˆ
aˆ
aˆ
x
x
x
aˆ
r
aˆ
aˆ
sin
cos
cos
cos
sin
aˆ
aˆ
aˆ
y
y
y
aˆ
aˆ
aˆ
r
sin
sin
cos
sin
cos
aˆ
aˆ
aˆ
z
z
z
aˆ
aˆ
aˆ
r
cos
sin
0
Therefore, when we have a vector in Cartesian coordinate given by:
A A aˆ
A aˆ
A aˆ
x
x
y
y
z
z
Then this vector is transformed to spherical coordinate as follows:
A
A
A
r
A aˆ
A aˆ
A aˆ
r
A
A
A
x
x
x
( aˆ
x
( aˆ
( aˆ
x
x
aˆ
( ˆ
r
) A
y
a
y
aˆ
) ( ˆ
A
y
a
aˆ
) A ( aˆ
y
y
y
aˆ
( ˆ ˆ
r
) A
z
az
ar
)
aˆ
) ( ˆ ˆ
A
z
az
a
)
aˆ
) A ( aˆ
aˆ
)
In matrix notation, we can write the transformation of vector (A) from A , A , A ) to
( A r
, A , A
) as:
z
z
(
x y z
Ar
aˆ
x
aˆ
A
a
ˆx
aˆ
A
a
ˆx
aˆ
r
aˆ
aˆ
y
y
aˆ
y
aˆ
r
aˆ
aˆ
aˆ
aˆ
z
z
z
r
aˆ
aˆ
aˆ
aˆ
A
A
A
x
y
z
A
A
A
r
sin
cos
cos
cos
sin
sin
sin
cos
sin
cos
cos
sin
0
A
A
A
x
y
z
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