288
4 Thermodynamics and Statistical Physics
u ν =
8πν
2
c 3 kT
(Rayleigh-Jeans law)
(b) If hν/kT 1 i.e hc/λkT 1 then we can ignore 1 in the denominator
in comparison with the exponential term in Planck’s formula
u λ dλ = c 1 e
−c2/λkT dλ
(Wien’s distribution law)
where the constants, c 1 = 8π hc and c 2 = hc
4.71 u λ dλ =
8π hc
λ 5 .
1
e hc/λkT − 1
dλ
(Planck’s formula)
The wavelength λ m corresponding to the maximum of the distribution curve
is obtained from the condition
du λ
dλ
λ=λm
= 0
Differentiating and writing hc/kT λ m = β, gives
e
−β
+
β
5
− 1 = 0
This is a transcental equation and has the solution
β = 4.9651, so that
λ m T =
hc
4.9651k
= b = constant.
Thus, the constant
b =
6.626068 × 10
−34
× 2.99792 × 10
8
4.9651 × 1.38065 × 10 −23
= 2.8978 × 10
−3 m-K
a value which is in excellent agreement with the experiment.
4.72 By definition
u =
u ν dν = aT
4
(1)
Inserting Planck’s formula in (1)
u = aT
4
=
8π h
c 3
∞
0
ν
3 dν
e hν/kT − 1
=
8πk
4 T
4
h 3 c 3
∞
0
x
3 dx
e x − 1
where x = hν/kT
a =
8πk
4
h 3 c 3
∞
0
x
3 (e
−x
+ e
−2x
+ . . . e
−r x
+ . . .)
Now,
∞
0 x
3 e
−r x dx =
6
r 4 , and Σ
∞
r =1
1
r 4 =
π
2
90
a =
48πk
4
h 3 c 3 .
π
4
90
=
8
15
π
5 k
4
h 3 c 3
∴ σ =
ac
4
=
2
15
π
5 k
4
h 3 c 2