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HW10
1.
(a) Two metal objects are embedded in weakly conducting material of conductivity
.
Show
that the resistance between them is related to the capacitance of the arrangement by
C
R
0
(b) Suppose you connected a battery between the two conductors and charge them to a
potential difference V0. If you then disconnect the battery, the charge will gradually leak off.
Show that
/
0
)
(
t
e
V
t
V
, and find the time constant,
, in terms of
0
and
.
Solution:
(a)
From the Ohm’s law,
E
J
, we have the relation between the charge of the
conductor, Q, and the current out of the conductor, I, by performing a surface
integration over the conductor surface.
0
Q
S
d
E
S
d
E
S
d
J
I
This is true for both conductors, i.e., all current flowing out of one conductor
flows into another conductor. Then if V is the potential difference between the
two conductors, we have
C
V
Q
Q
V
I
V
R
0
0
0
/
/
,
where
V
Q
C
is the capacitance.
(b)
After disconnecting the battery, the change of the Q is the leak current.
V
CR
V
C
I
dt
dQ
C
dt
dV
0
1
The solution with the initial condition of V(t=0)=V
0
is an exponential function.
t
V
t
V
0
0
exp
)
(
The time constant is
0
, which is roughly the time needed to leak out the
charge.
So the higher the conductivity is, the shorter time to leak off the charge.
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This note was uploaded on 02/06/2010 for the course PHYSICS 11 taught by Professor Qiu during the Fall '09 term at University of California, Berkeley.
 Fall '09
 Qiu
 Magnetism

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