Solution
Physics 214 Problem 4 Week 3
Photon Detection
In order for rod cells in your eye to detect light and transmit a signal to the visual center of your
brain, the retina needs to receive a minimum of about 6 photons in a time span of ~100 ms. (In
reality, this threshold varies somewhat from person to person.)
a) For orange light of wavelength 600 nm incident on the retina, what is the energy per photon
(give your answer in electon-volts and in joules)? What is the minimum optical power (in
watts) of this orange light falling on the retina which is detectable by the human eye?
Use Ephoton = hc/
λ
To get the answer in eV, it is convenient to use hc = 1240 eV-nm:
E
photon = 1240/600 = 2.07 eV = 3.3 × 10-19 J
The power is given by P = Ephoton × # photons/ time.
Use the values above to get P = 2.0 × 10-17 Watts (pretty sensitive detector!)
b) If instead, the eye received violet light of wavelength 400 nm at the same optical power
level, at what rate would it be receiving photons? Would the brain be able to detect this
light? (Here we assume – somewhat incorrectly – that the detection efficiency of the rods is
completely independent of frequency.)
Since the power is kept the same, for the two different wavelengths,
R1/
λ
1 = R2/
λ
2 , where R is the photon rate
(you could alternatively repeat the calculation of part a) here as well).
This gives the rate of 4 photons in 100ms, below the detectable threshold.
Note: No calculation is actually needed here. Since the power from a) was already the
threshold amount to give 6 photons in 100ms, if we partition this energy into fewer, more
energetic photons, there clearly won’t be enough.
c) Now consider a radio-wave detector receiving signals at 97.1 MHz (FM) at the same power
level. What is the wavelength of the radio signal? What is the rate of photons hitting the
detector? How does this compare to the rates obtained above for visual light? (Yes, radio
waves are light too! But here you see why the discrete nature of photons at radio frequencies
is rarely an issue.)
λ
= c/f = 3.1 m
Using the formula in part b), the rate is 31 million photons in 100 ms, or 3.1 x 108 s-1.
Note: Although this has units of s-1, it should not be confused with the frequency of
the radiation!
