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
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Example(8):
Show that if the scalar function , is harmonic at point √3, ,0 .
Solution:
∂
∂θ
H 1 ∂
r ∂r r sin θ e 1
H 1 ∂ ∂H
r r
∂r ∂r 1
r sinθ
sinθ
∂H
∂θ 1
∂ H
r sin θ ∂φ
∂
r sinθ ∂θ sinθ2 sin θ cosθ e
H sin θ
r r e 2r e 2 e
r sinθ sin θ 2sinθ cos θ
H sin θ
r
e r2 2 e
r 2 cos θ sin θ
At point √3, ,0, H0 ⟹ is not harmonic.
1
r sin θ 0
Example (9):
Display whether the field vector is solenoid, conservative or none of them?
Solution:
(1). In order to show that the field is solenoid, it must be satisfy: ∙A 0
∙A 1 r ∂ ∂r r A 1
rsinθ
Therefore the field is not solenoid:
∂
∂θ sin θ A 1 ∂A
rsinθ ∂φ
∙A 1 r ∂ ∂r r 1 r r 3 0
(2). In order to show that the field is conservative, it must be satisfy: A 0
1
1
0 0
1
00 00 00 0
Therefore the field is conservative.
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