10.3 Solutions
439
10.32 Radiant intensity at the sun’s surface is the power emitted by 1 m 2 of sun’s
surface.
σ T
4
= 63 × 10
6
T =
63 × 10 6
5.67 × 10 −8
1/4
= 5773 K
10.33 −
dE
dt
= σ A
T
4
1 − T
4
2
(Stefan–Boltzmann formula)
= σ π r
2
T
4
1 − T
4
2
= π(5.67 × 10
−8
)(0.05)
2
(500
2
− 300
2
)
= 7.12 × 10
−5 W
10.34 λ m T = b = 3 × 10
−3 m K
(Wien’s law)
(1)
dE
dt
= σ T
4
(Boltzmann law)
(2)
If dE/dt goes down to 1/16 of its original value then by (2) the temperature
T → T /2. Therefore λ m → 2λ m by (1). Thus the wavelength under new
conditions λ
m = 2λ m = 2 × 480 = 960 nm.
10.3.4 Specific Heat and Latent Heat
10.35 Q =
80
20
mC p dT + m L +
200
80
mC p dT
=
200
20
mC p dT + m L
(∵ C p relation for solid and
liquid phase is identical)
=
200
20
1 × (30.6 + 0.0103 T )dT + 1 × 6000
= 11710 J
10.36 C =
CdT
dT
=
T
0 (A + BT 2 )dT
T
=
1
T
AT +
BT 3
3
= A +
BT 2
3
C (midpoint) = A + B(T /2) 2 = A +
BT 2
4
∴
C − C (midpoint) = A +
BT 2
3
−
A +
BT 2
4
=
BT 2
12