1) P m ℓ (cos q ) = - 3/ 2 sin q (5 cos 2 q - 1 )
If m = 1 and ℓ = 3, show that the function is accurate.
Soln.: We may use:
Soln.: We may use:
P ℓm (q ) = (-1 ) m sin m q d (m) [P ℓ (cos q )] / d (cos q)m
For ℓ =3 and m = 1 , so:
P ℓm (q ) = (-1 ) 1 sin 1 q d (1) [P 3 (cos q )] / d (cos q)1
And: P 3 (cos q ) = 1/2 (5 cos 2 q - 3 cos q)
Hence take: z = cos q and substitute, viz :
d/ dz [P 3 (z)] = d/ dz (5 z 2/2 - 3z /2 )
è
P ℓm (q ) = (-1 ) sin q (5 cos 2 q - 3/2)
= - 3 sin q/ 2 (5 cos 2 q - 1) = - 3/ 2 sin q (5 cos 2 q - 1 )
d/ dz [P 3 (z)] = d/ dz (5 z 2/2 - 3z /2 )
è
P ℓm (q ) = (-1 ) sin q (5 cos 2 q - 3/2)
= - 3 sin q/ 2 (5 cos 2 q - 1) = - 3/ 2 sin q (5 cos 2 q - 1 )
2)Write the full Laplace equation in spherical coordinates for a homogeneous medium with magnetic permeability m , permittivity e, , conductivity, s and frequency w.
For case in vacuo: Ñ 2 V = 0
For a homogeneous medium with magnetic permeability m , permittivity e, , conductivity, s and frequency w.
Ñ 2 V = = g 2 V = [ j w m (s + j w e)] V
Where g = Ö [ j w m (s + j w e)] is the propagation constant
So that:
3) We have:
d2 Q / d 2 q + cot q (d Q / d q ) + (a2 - m2/ sin2 q ) Q = 0
For case in vacuo: Ñ 2 V = 0
Which in spherical coordinates becomes:
1/ r 2 ¶ / ¶ r (r 2 ¶ V/¶ r )
+ 1/( r 2 sin q) ¶ / ¶ q (sin q ¶ V/¶ q) + 1/( r 2 sin q) ¶ 2 V/¶ j2 = 0
For a homogeneous medium with magnetic permeability m , permittivity e, , conductivity, s and frequency w.
Ñ 2 V = = g 2 V = [ j w m (s + j w e)] V
Where g = Ö [ j w m (s + j w e)] is the propagation constant
So that:
1/ r 2 ¶ / ¶ r (r 2 ¶ V/¶ r ) + 1/( r 2 sin q) ¶ / ¶ q (sin q ¶ V/¶ q)
+ 1/( r 2 sin q) ¶ 2 V/¶ j2 = [ j w m (s + j w e)] V
+
3) We have:
d2 Q / d 2 q + cot q (d Q / d q ) + (a2 - m2/ sin2 q ) Q = 0
Where: a2 = n(n + 1)
Then let:
x = cos q, sin 2 q = 1 – x 2 , d/ dq = - sin q d/ dx
Then we rewrite the Legendre equation after subst. as:
( 1 – x 2 ) d2 Q / d 2 x - 2x d Q / d x + [n(n + 1) - m2/ 1 – x 2] Q = 0
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