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3
The Quantum Theory of Light
31
(a)
E
2
π
r
=
r
2
dB
dt
⎛
⎝
⎜
⎞
⎠
⎟
E
=
r
2
⎛
⎝
⎜
⎞
⎠
⎟
dB
dt
⎛
⎝
⎜
⎞
⎠
⎟
(b)
If
r
remains constant, then:
E
=
Eq
=
r
2
⎛
⎝
⎜
⎞
⎠
⎟
dB
dt
⎛
⎝
⎜
⎞
⎠
⎟
e
so that
Fdt
=
r
2
⎛
⎝
⎜
⎞
⎠
⎟
dB
dt
⎛
⎝
⎜
⎞
⎠
⎟
dt
=
m
e
dv
, or
dv
=
re
2
m
e
⎛
⎝
⎜
⎞
⎠
⎟
dB
dv
v
v
+∆
v
∫
=
er
2
m
e
⎛
⎝
⎜
⎞
⎠
⎟
dB
0
B
∫
∆
v
=
erB
2
m
e
B
E
E
r
(c)
∆
ω
=
∆
v
r
=
eB
2
m
e
=
1.6
×
10
−
19
C
()
1 T
2
9.1
×
10
−
31
kg
=
8.8
×
10
10
rad sec
=
2
f
=
2
c
λ
=
2
3.0
×
10
8
ms
(
)
500
×
10
−
9
m
=
3.8
×
10
15
rad sec
;
∆
=
2.3
×
10
−
5
(d)
For the
0
line the electrons’ plane is parallel to
B
, therefore, the magnetic flux,
Φ
B
is
always zero. This means that
F
and
E
are zero and as a consequence, there is no force
on the electrons and there will be no
∆
v
for the electrons. The
0
is the case
calculated in parts (a)–(c). The
0
− ∆
will have the same magnitude for
F
,
B
, and
∆
v
as in (a)–(c) but the direction will be opposite.
e
B
32
Assume that your skin can be considered a blackbody. One can then use Wien’s displacement
law,
max
T
=
0.289 8
×
10
−
2
m
⋅
K
with
T
=
35
0
C
=
308 K
to find
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max
=
0.289 8
×
10
−
2
m
⋅
K
308 K
=
9.41
×
10
−
6
m
=
9 410 nm
.
34
(a)
From Stefan’s law, one has
P
A
=
σ
T
4
. Therefore,
P
A
=
5.7
×
10
−
8
Wm
2
K
4
()
3000 K
4
=
4.62
×
10
6
2
.
(b)
A
=
P
4.62
×
10
6
2
=
75 W
4.62
×
10
6
2
=
16.2 mm
2
.
35
(a)
Planck’s radiation energy density law as a function of wavelength and temperature is
given by
u
,
T
=
8
π
hc
5
e
hc
B
T
−
1
. Using
∂
u
∂
=
0
and setting
x
=
hc
max
k
B
T
, yields an
extremum in
u
T
with respect to
. The result is
0
=−
5
+
hc
max
k
B
T
⎛
⎝
⎜
⎞
⎠
⎟
e
hc
max
k
B
T
e
hc
max
k
B
T
−
1
−
1
or
x
=
51
−
e
−
x
(
)
.
(b)
Solving for
x
by successive approximations, gives
x
≅
4.965
or
max
T
=
hc
k
B
⎛
⎝
⎜
⎞
⎠
⎟
4.965
=
2.90
×
10
−
3
m
⋅
K
.
310
The energy per photon,
E
=
hf
and the total energy
E
transmitted in a time
t
is
Pt
where
power
P
=
100 kW
. Since
E
=
nhf
where
n
is the total number of photons transmitted in the
time
t
, and
f
=
94 MHz
, there results
nhf
=
100 kW
(
)
t
=
10
5
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 Fall '08
 Hirsch
 Physics, Light

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