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 1 cm across, and you estimate that the distance from the window shade to the wall is about 4 m.
Estimate:
a. The average wavelength of the sunlight (in nm)
b. The diameter of the pinhole (in mm).

Answer 1

Given that,

Central maximum = 1 cm

Distance from the window shade to the wall =4 m

We know that,

The visible range of the sun light is 400 nm to 700 nm.

(a). We need to calculate the average wavelength

Using formula of average wavelength

$\lambda_{avg}=\dfrac{\lambda_{1}+\lambda_{2}}{2}$

Put the value into the formula

$\lambda_{avg}=\dfrac{400+700}{2}$

$\lambda_{avg}=550\ nm$

(b). We need to calculate the diameter of the pinhole

Using formula for diameter

$w=\dfrac{2.44\lambda L}{D}$

$D=\dfrac{2.44\lambda L}{w}$

Put the value into the formula

$D=\dfrac{2.44\times550\times10^{-9}\times4}{1\times10^{-2}}$

$D=0.537\ mm$

Hence, (a). The average wavelength 550 nm.

(b). The diameter of the pinhole is 0.537 mm.