Physics 101 Calculating Diameter of Hair

Question

You stretch a strand of your hair across a laser beam and observe the diffraction pattern this produces on a sheet of paper 1.25 m from your hair strand. You mark the center of the dark regions on either side of the central bright spot and measure the distance between these marks to be 21.6 mm. You are given that the wavelength of the laser light is 633 nm. What is the diameter of your hair strand (in mm)? Make use of the small angle approximation.

Answer 1

Babinet's principle is that the diffraction pattern of an opaque body is the same as that from a hole of the same size and shape. Thus, this problem is very nearly the same as that of single slit diffraction. For the single slit of width w, the angular deviation for the two dark spots adjacent to the central bright region are

θ = ± λ/w where λ is the wavelength of the light. The spatial separation of these two dark spots will thus be

D = 2 R λ/ w where R is the distance to the screen

Solving for w : w = 2 R λ / D plugging in D = 21.6 E-3 R = 1.25 λ = 633E-9 gives

w = 73.3E-6 (meters) or .0733 mm