96
Capacitance and Dielectrics
P26.58
(a)
We use Equation 26.11 to find the potential energy of the capacitor. As we will see, the
potential difference
∆
V
changes as the dielectric is withdrawn. The initial and final
energies are
U
Q
C
i
i
=
F
H
G
I
K
J
1
2
2
and
U
Q
C
f
f
=
F
H
G
I
K
J
1
2
2
.
But the initial capacitance (with the dielectric) is
CC
if
=
κ
. Therefore,
U
Q
C
f
i
=
F
H
G
I
K
J
1
2
2
.
Since the work done by the external force in removing the dielectric equals the change in
potential energy, we have
WU U
Q
C
Q
C
Q
C
fi
iii
=−
=
F
H
G
I
K
J
−
F
H
G
I
K
J
=
F
H
G
I
K
J
−
1
2
1
2
1
2
1
222
κκ
af
.
To express this relation in terms of potential difference
∆
V
i
, we substitute
QC V
ii
=∆
bg
, and
evaluate:
WC
V
=
×
−
=
×
−−
1
2
1
1
2
200 10
100
500 100
400 10
2
9
2
5
∆
a
f
ej
a
f
a
f
..
.
.
F
V
J
.
The positive result confirms that the final energy of the capacitor is greater than the initial
energy. The extra energy comes from the work done
on
the system by the external force that
pulled out the dielectric.

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