Dr. Gangopadhyaya
Test III
Physics 351
December 4, 2002
As always, to receive full credit you must show your work in a neat, organized and a legible form.
Some Relevant Formuae are given below:
V
sph
(
r, θ, φ
) =
∞
X
l
=0
A
l
r
l
+
B
l
r
l
+1
¶
P
l
(cos
θ
)
V
cyl
(
ρ, φ
) = (
a
0
+
b
0
log
ρ
) (
c
0
+
d
0
φ
) +
∞
X
n
=1
(
a
n
ρ
n
+
b
n
ρ

n
)
(
c
n
cos(
nφ
) +
d
n
sin(
nφ
))
(
E
2

E
1
)
·
ˆ
n
1
→
2
=
σ
²
0
;
(
E
2

E
1
)
×
ˆ
n
1
→
2
= 0 ;
cos(
a
+
b
) = cos
a
cos
b

sin
a
sin
b
,
P
0
(
x
) = 1
,
P
1
(
x
) =
x ,
P
2
(
x
) =
1
2
(3
x
2

1)
,
P
3
(
x
) =
1
2
(5
x
3

3
x
) ;
Z
1

1
dx P
m
(
x
)
P
n
(
x
) =
2
2
m
+ 1
δ
mn
(1 +
x
)
ν
= 1 +
νx
+
ν
(
ν

1)
2!
x
2
+
ν
(
ν

1)(
ν

2)
3!
x
3
+
ν
(
ν

1)(
ν

2)(
ν

3)
4!
x
4
+
· · ·
;
converges for

x

<
1
.
ln (1 +
x
) =
x

1
2
x
2
+
1
3
x
3
+
· · ·
+ (

1)
n
+1
1
n
x
n
+
· · ·
1
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1. A spherical shell of radius
R
and made of an insulating material has a surface charge density
σ
=
σ
0
cos
θ
sin
2
θ
glued on it.
a) Express this surface charge density as a linear sum of Legendre polynomials in cos
θ
.
b) Determine the electric potential inside and out side the sphere.
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 Spring '09
 Gangopashyaya
 Electrostatics, Magnetism, Work, Electric charge, charge density, surface charge density

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