Electromagnetics Workshop – Week 11
Solutions
Dielectric Breakdown & Stored Energy
1
Problem 1:
Dielectric breakdown occurs in a material whenever the magnitude of the field
E
exceeds the
dielectric strength anywhere in the material.
a)
In a coaxial capacitor filled with insulating material of permittivity
ε
, at what value of
ρ
is
E
a maximum?
b)
What is the breakdown voltage if
1
a
=
[cm],
2
b
=
[cm], and the dielectric material is
mica with
6
r
=
and
[ ]
200
MV/m
BR
=
E
?
c)
If
24
V
=
[V] for the material
6
r
=
, what is the energy stored per unit length in this
coaxial capacitor?
Problem 1 Solution:
a) From Gauss’s Law, we can easily show that
2
E
περ
=−
A
Hence,
E
is a maximum when
a
=
.
b)
()
0
ln
ln 2
22
1
2
b
a
b
Vd
a
ρρ
πε
⎛⎞
===
⎜⎟
⎝⎠
∫
AA
A
0
12
2
BR
BR
EE
a
a
=⇒
=
A
A
0
62
0
12
ln 2
ln 2
200 10 1 10
ln 2
1.386 MV
12
BR
BR
BR
Ea
VE
a
−
==
=
×
×
≈
c)
0
0
288
ln 2
24
12
ln 2
V
⇒
=
A
A
,
24
ln 2
E
2
2
2
00
6
3456
24
l
n
2
l
n
2
Lb
e
Va
L
Wd
V
d
d
d
z
π
ρρφ
−
=
∫∫∫
∫ ∫ ∫
E
[]
0
3456
/
138.7 nJ/m
ln 2
e
WL
=≈
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Solutions
Dielectric Breakdown & Stored Energy
2
Problem 2:
A typical thundercloud can be modeled as a capacitor with horizontal plates with 10 [km
2
] area
separated by a vertical distance of 5 [km].
Just before a large lightning discharge, the upper plate
may have a total positive charge of up to 300 [C], with the lower plate having an equal amount of
negative charge.
a)
What is the average electric field within the cloud?
How does this compare to the
dielectric breakdown field of dry air (3 [MV/m])?
b)
What is the potential difference between the top and bottom plates?
c)
Find the electrostatic energy stored in the cloud just before a discharge.
Problem 2 Solution:
Choosing
z
as our vertical axis, we can easily show using Gauss’s Law that
0
s
z
E
ρ
ε
=−
,
[ ]
62
2
300 C
30 10
C/m
10 km
s
−
⎡
⎤
==
×
⎣
⎦
⎡⎤
⎣⎦
a)
[]
6
0
30 10
3.39 MV/m
avg
−
×
=≈
E
, This is pretty close to the dielectric breakdown field of dry
air.
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 Spring '08
 Staff
 Electrostatics, Electromagnet, Electric charge, dielectric breakdown

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