522
11 Electrostatics
(iv) q = q; the charge remains constant.
(v) U > U ; energy increases because U =
1
2
qV , with V increasing
and q remaining constant.
11.80 As the plates carry equal but opposite charges, the force of attraction, which
is conservative, is given by
F = −
dU
dx
But U =
1
2
q 2
C
For the parallel plate capacitor,
C =
ε 0 A
x
where x is the distance of separation. Combining the above equations
F = −
d
dx
1
2
q 2 x
ε 0 A
= −
q 2
2ε 0 A
= −
1
2
ε 0 AV 2
d 2
where we have put x = d.
11.81 (a) As the drops are assumed to be incompressible, the volume does not
change.
4
3
π R
3
= n
4
3
πr
3
∴ R = n
1/3 r
(b) C
= 4πε 0 R = 4πε 0 r n
1/3
∴ C
= n
1/3 C
(c) Q = nq (charge conservation)
V
=
Q
4πε 0 R
=
nq
4πε 0 n 1/3 r
=
n 2/3 q
4πε 0 r
= n
2/3 V
(d) σ =
Q
4π R 2 =
nq
4π r 2 n 2/3 = n 1/3 σ
(e) U =
1
2
QV =
1
2
nqn 2/3 V = n 5/3 U
11 Electrostatics
(iv) q = q; the charge remains constant.
(v) U > U ; energy increases because U =
1
2
qV , with V increasing
and q remaining constant.
11.80 As the plates carry equal but opposite charges, the force of attraction, which
is conservative, is given by
F = −
dU
dx
But U =
1
2
q 2
C
For the parallel plate capacitor,
C =
ε 0 A
x
where x is the distance of separation. Combining the above equations
F = −
d
dx
1
2
q 2 x
ε 0 A
= −
q 2
2ε 0 A
= −
1
2
ε 0 AV 2
d 2
where we have put x = d.
11.81 (a) As the drops are assumed to be incompressible, the volume does not
change.
4
3
π R
3
= n
4
3
πr
3
∴ R = n
1/3 r
(b) C
= 4πε 0 R = 4πε 0 r n
1/3
∴ C
= n
1/3 C
(c) Q = nq (charge conservation)
V
=
Q
4πε 0 R
=
nq
4πε 0 n 1/3 r
=
n 2/3 q
4πε 0 r
= n
2/3 V
(d) σ =
Q
4π R 2 =
nq
4π r 2 n 2/3 = n 1/3 σ
(e) U =
1
2
QV =
1
2
nqn 2/3 V = n 5/3 U