11.3 Solutions
523
11.82 The electric field for a cylindrical capacitor is
E =
q
2πε 0 lr
(1)
where l is the length and r the radius. The energy density (energy/unit volume)
u =
1
2
ε 0 E
2
=
q 2
8π 2 ε 0 l 2 r 2
(2)
where we have used (1).
The energy stored between the coaxial cylinders of length l and radii R and
a is
U =
udv =
R
a
u (2πrl) dr
(3)
where dv = (2πr dr )l is the volume element. Using (2) in (3)
U =
q 2
4πε 0 l
R
a
dr
r
=
q 2
4πε 0 l
ln
R
a
Similarly, the energy stored between the coaxial cylinders of radii b and a is
U 0 =
q 2
4πε 0 l
ln
b
a
∴
U
U 0
=
ln (R/a)
ln (b/a)
Set
U
U 0
=
1
2
∴
ln (R/a)
ln (b/a)
=
1
2
→ ln
b
a
= 2 ln
R
a
= ln
R 2
a 2
or
b
a
=
R 2
a 2 → R =
√
ab
11.83 The charge on C 1 is
q 1 = C 1 V 1 = 3 × 10
−6
× 4000 = 0.012 C
523
11.82 The electric field for a cylindrical capacitor is
E =
q
2πε 0 lr
(1)
where l is the length and r the radius. The energy density (energy/unit volume)
u =
1
2
ε 0 E
2
=
q 2
8π 2 ε 0 l 2 r 2
(2)
where we have used (1).
The energy stored between the coaxial cylinders of length l and radii R and
a is
U =
udv =
R
a
u (2πrl) dr
(3)
where dv = (2πr dr )l is the volume element. Using (2) in (3)
U =
q 2
4πε 0 l
R
a
dr
r
=
q 2
4πε 0 l
ln
R
a
Similarly, the energy stored between the coaxial cylinders of radii b and a is
U 0 =
q 2
4πε 0 l
ln
b
a
∴
U
U 0
=
ln (R/a)
ln (b/a)
Set
U
U 0
=
1
2
∴
ln (R/a)
ln (b/a)
=
1
2
→ ln
b
a
= 2 ln
R
a
= ln
R 2
a 2
or
b
a
=
R 2
a 2 → R =
√
ab
11.83 The charge on C 1 is
q 1 = C 1 V 1 = 3 × 10
−6
× 4000 = 0.012 C