Homework #7 Solutions
Question 1)
K+K, Chapter 4, Problem 8. See Figure (1) for the setup.
Figure 1: The situation before (top) and after (bottom) a temperaturefloating
shield is placed between a hot surface (at temperature
T
u
) and a cold surface
(at temperature
T
l
).
We expect
J
1
=
J
2
in equilibrium, where
J
1
=
σ
B
(
T
4
u

T
4
m
) and
J
2
=
σ
B
(
T
4
m

T
4
l
). Solve for
T
m
:
T
m
=
T
4
u
+
T
4
l
2
1
/
4
(1)
Now the heat current is given by
J
=
σ
B
(
T
4
u

T
4
m
) =
σ
B
T
4
u

T
4
l
2
=
J
u
2
(2)
This is half the original heat current. Note that we would get the same answer
if we calculated the flux between the inserted plate and the low temperature
plate (up to a possible sign difference). This has to be so in equilibrum for a
black body due to energy conservation.
The general case of N floating heat shields reduces the heat current by a
factor of (N+1). This is the principle behind superinsulation, in which many
layers of Alcoated mylar are wrapped around a cold object to insulate it from
radiative heating from nearby warm objects.
1
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Question 2)
K+K, Chapter 4, Problem 11 (First part only). Starting with
Eq. (41) in K+K we can write the energy of phonons in a solid as
U
=
9
Nτ
4
(
k
B
θ
D
)
3
Z
k
B
θ
D
/τ
0
d
x
x
3
e
x

1
(3)
We are working in the limit where the upper limit of the integral is small:
k
B
θ
D
/τ
1. Hence
x
remains small in the integrand, and we can expand the
denominator as
e
x

1 = 1 +
x
+
· · · 
1
≈
x
. So,
U
≈
9
Nτ
4
(
k
B
θ
D
)
3
Z
k
B
θ
D
/τ
0
d
xx
2
(4)
The integration is now simple and gives
k
3
B
θ
3
D
/
3
τ
3
. Simplifying the expression
we get
U
= 3
Nτ
(5)
Differentiating this with respect to temperature we get the DulongPetit law for
the heat capacity of a solid at high temperature (
θ
D
T
):
C
V
=
∂U
∂T
U
= 3
Nk
B
for
θ
D
T
(6)
Question 3)
K+K, Chapter 4, Problem 12. The phonon specific heat at 1
K is
C
phonon
V
=
12
π
4
Nk
B
5
T
θ
D
3
since
T
θ
D
.
(7)
For photons, we know from Eqs. (2526) in K+K
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 Fall '11
 Anlage
 Thermodynamics, mechanics, Energy, Work, Statistical Mechanics, chemical potential, K+K

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