One day, after pulling down your window shade, you notice that sunlight is passi

Question

One day, after pulling down your window shade, you notice that sunlight is passing through a pinhole in the shade and making a small patch of light on the far wall. Having recently studied optics in your physics class, you're not too surprised to see that the patch of light seems to be a circular diffraction pattern. It appears that the central maximum is about 6 cm across, and you estimate that the distance from the window shade to the wall is about 3 m . Knowing that the average wavelength of sunlight is about 500 nm, estimate the diameter of the pinhole.

Explanation / Answer

This would be related to the diffraction pattern for a circular aperture. If illuminated by a single wavelength, the angle for the first minimum (which would define the ending of the central maximum) is

Angle = Arcsin(1.22*WL/D)

where:

WL = wavelength
D = aperture diameter

Given data

WL = wavelength = 500nm

D = aperture diameter =3 m

So:

sin(Angle) = 1.22*WL/D

Since
sin(Angle) = 6 cm/(3 m) = 6/300
WL = D*sin(Angle)/1.22 *6
= D/(300*1.22) *6
= D/366

WL for average wavelength of sunlight:
According to the graph in the Wiki article, it looks like a little over 500 nm. On the other hand, another article on visible light gives the wavelength of yellow light as being 570 nm. I'll take 570 nm.

b) Diameter of pinhole:
D = 366 * WL
= 366 * 570 nm
= 2.09e5*e-9 m
= 2.09e-4 m
= 0.209 mm

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