In this problem, you will explore the plausibility of the responsibility of photon shot noise for the ``night-time reflection effect'' and then use that effect to make an arm-chair estimate of the value of Planck's constant h.
We will use the following assumptions about the human eye:
for the human eye is 
,
somewhere between the imperceptible cycle time of fluorescent lighting
and the noticeable flicker of movies at 15 frames per second.
(the angle of a single pixel on a 
CRT at about 100 cm), corresponding to a solid angle of
.
In (a)-(c) it is convenient to express your answers in
terms of numerical constants and an unknown
parameter 
, the energy/photon expressed in units of
.
a) If we take the average intensity of indoor lighting to be that of
the uniform illumination of the 100 Watt light-bulb of problem
3, at a distance of 3 meters, how many photons
in the visible
region of the spectrum illuminate each 
in the room. (For
simplicity, use your result from the previous problem for the total energy flux
and make the approximation that all the photons in the visible range
carry the same energy, which we will take for now to be 
. We expect but do not yet assume that 
is on the
order of unity.) In terms of 
, assuming that objects reflect
the light impinging on them uniformly in all directions, how many
photons
are received by each receptor in the eye? (See Figure
4 as an aid in your thinking.) Again, in terms of 
,
how many photons are impinging during a single response time,
?
b) Now, suppose that ten percent (0.1) of the light-energy impinging
on a glass window is reflected back. How many photons (on average)
per sampling time are received by each receptor from the reflection,
, if the room is illuminated under the conditions (and
approximations) described above? What size shot noise fluctuations,
, do you expect about this average?
c) If we take the average outdoors night-time lighting condition to
correspond to the illumination of the same 100 Watt bulb but at an
average distance of 30 meters, what number of photons is expected to
arrive at each receptor in a sampling time from objects outside the
window, 
?
d) Suppose that under these conditions, you had observed that you
could just barely make out images from the outside, what relationship
must exist between your final answers to (b) and (c)? Using this
relationship, determine the value of the unknown parameter
and give absolute numbers for your responses to (b)
and (c). Using the fact that the wavelength of visible light is on
the order of 5100Å, what order of magnitude would you then assign to
Planck's constant h? How close is this to the accepted value?
