Numerade Educator: Donald A.

Calculus: Early Transcendentals

Light enters the eye through the pupil and strikes the retina, where photoreceptor cells sense light and color. W. Stanley Stiles and B. H. Crawford studied the phenomenon in which measured brightness decreases as light enters farther from the center of the pupil. (see the figure.)

They detailed their findings of this phenomenon, known as the Stiles-Crawford effect of the first kind, in an important paper published in 1933. In particular, they observed that the amount of luminance sensed was not proportional to the area of the pupil as they expected. The percentage $ P $ of the total luminance entering a pupil of radius $ r mm $ that is sensed at the retina can be described by
$$ P = \frac{1 - 10^{-pr^2}}{pr^2 \ln 10} $$
where $ p $ is an experimentally determined constant, typically about $ 0.05 $.
(a) What is the percentage of luminance sensed by a pupil of radius $ 3 mm $? Use $ p = 0.05 $.
(b) Compute the percentage of luminance sensed by a pupil of radius $ 2 mm $. Does it make sense that it is larger than the answer to part $ (a) $?
(c) Compute $ \displaystyle \lim_{r\to 0^+} P $. Is the result what you would expect? Is this physically possible?

Source: Adapted from W. Stiles and B. Crawford, "The Luminous Efficiency of Ray Entering the Eye Pupil at Different Points." Proceedings of the Royal Society of London, Series B: Biological Sciences 112(1933): 428-50.

Chapter 4: Applications of Differentiation

Section 4: Indeterminate Forms and l'Hospital's Rule