Capacitance and Geometry Great mysteries of the universe elucidated by your good friend Gauss…
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Transcript of Capacitance and Geometry Great mysteries of the universe elucidated by your good friend Gauss…
Capacitance and Geometry
Great mysteries of the universe elucidated by your good friend Gauss…
Capacitance
• Capacitors are charge-storage devices– Capacitance, C, is a
measure of the ability to store charge
• You have to do work to store charge on a capacitor – Energy is stored in the e-
field between the cap’s electrodes
V
QC
1 C/volt = 1 farad [F]
221 CVUE
Parallel-plate capacitors
Consider two plates of opposite charge separated by a gap d and carrying a charge density
The field between the plates is
and the field beyond them is zero. ooo
EEE
22
Parallel-plate capacitors
Recall that
Since and
the potential difference across this capacitor is
Given the definition of capacitance
the capacitance of this geometry is
q
dqE
q
WV
)(
o
E
A
q
V
QC
d
AC o
oA
qdV
Cylindrical capacitors
Inner cylinder carries a charge density of + while the outer carries -
At any arbitrary location
so a
b
b
a
baab EdrqEdr
WVV
1
a
bV ln
2 0
rE
o
2
therefore
Cylindrical capacitors
Since
for this geometry,
LQ
abL
Cln
2 0
Spherical shellsConsider an inner shell
charged at +Q and an outer shell charged at -Q
At any arbitrary location
and a
b
ab EdrVV
ba
QV
11
4 0
24 r
QE
o
enclosed
and for this geometry
therefore
ab
abC
04
Isolated conductors as capacitors
Think of the 2nd electrode as infinitely far away
• At the surface,
• At the 2nd electrode V=0, so
for this geometryaC o4
a
QV
o4