14.3 Solutions
693
∴
V p − V g
V p
=
ωc
V P
dn
dω
(1)
But ω = 2πν = 2π c/λ
(2)
V p = c/n
(3)
dn
dω
=
dn
dλ
dλ
dω
= −
λ 2
2π c
dn
dλ
(4)
Substituting (2), (3) and (4) in (1)
Vp − Vg
Vp
= −nλ
dn
dλ
(5)
n 1 − 1
2.786 × 10 −4
2.781 × 10 −4
n 2 − 1
2.781 × 10 −4
2.777 × 10 −4
n =
n1+n2
2
1 + 2.784 × 10 −4
1 + 2.779 × 10 −4
n
5 × 10 −7
4 × 10 −7
λ 1( A)
4800
5000
λ 2 (A)
5000
5200
λ =
λ1+λ2
2
4900 A
5100 A
λ = λ1 − λ2
−200 A
−200 A
Using formula (5), the first set of data gives (V p − V g )/V p = 1.22 × 10 −5
and the second set 1.02 × 10 −5 .
14.86 ω = ak 2
(a) v p =
ω
k
= ak
(b) v g =
dω
dk
= 2ak = 2v p
14.87 (a) v p =
1
√
εμ
(1)
(b) n =
√ ε r
ε r = n
2
= 1 −
D 2
ω 2
∴ n =
1 −
D 2
ω 2
(2)
υ p =
c
n
=
c
1 −
D 2
ω 2
(3)
Squaring (3) and re-arranging
ω
2
= D
2
+
ω 2 c 2
v 2
p
= D
2
+ k
2 c
2
(4)
∴ v p = ω/k