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Electromagnetics Workshop – Week 7
Solutions
Grounding and Divergence Theorem
1
Problem 1:
Concentric spheres are placed in the configuration below.
A solid ball (radius
a
) of
volume charge
3
4
C/m
v
r
a
ρ
π
⎡⎤
=
⎣⎦
is surrounded by a neutral spherical metal shell of
finite thickness (
cb
) and an infinitesimally thin metallic shell of radius
d
which is
grounded
()
abcd
<<<
.
Find the surface charge density at
rb
=
,
rc
=
, and at
,
rdd
−+
=
(just inside of and just outside of the outermost shell, respectively).
Once the
charge distribution is known throughout the geometry solve for the electric field in all
regions of space.
The total charge for
ra
≤
is given by
2
2
4
000
sin
a
a
r
Qr
d
d
d
r
a
ππ
θ
θφ
⎛⎞
=
⎜⎟
⎝⎠
∫∫∫
[]
4
4
0
4
1C
4
a
a
r
Q
a
==
⎢⎥
The field needs to be zero inside the conducting shell
( )
brc
≤
≤
. Hence,
2
2
1
C/m
4
b
s
b
⎡
⎤
=−
⎣
⎦
,
2
2
1
C/m
4
c
s
c
⎡
⎤
=
⎣
⎦
.
The field is also zero outside of
rd
=
due to the grounding of the outer shell.
2
2
1
C/m
4
d
s
d
−
, and
0
d
s
+
=
.
The fields can be solved for using Gauss’ law:
2
4
0
ˆ
V/m
4
r
a
πε
=
Er
,
≤
2
0
1
ˆ
V/m
4
r
=
,
arb
≤<
0
=
E
,
<<
2
0
1
ˆ
V/m
4
r
=
,
crd
0
=
E
,
>
a
b
c
d
v
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View Full DocumentElectromagnetics Workshop – Week 7
Solutions
Grounding and Divergence Theorem
2
Problem 2:
A vector field
3
ˆ
ρ
=
D
ρ
exists in the region between two concentric cylindrical surfaces
defined by
1
=
and
2
=
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 Spring '08
 Staff
 Electromagnet

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