Question

You stretch a strand of your hair across a laser beam and observe the diffraction pattern this produces on a sheet of paper 1.15 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 21.4 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. (Hint: recall Babinet's principle which equates the spot pattern created by light going around the hair to the pattern when light goes through a slit.)

Answer 1

To determine the diameter of the hair strand, you can use Babinet's principle and the small angle approximation. By measuring the distance between the marks on the paper and the distance between the hair strand and the paper, you can calculate the diameter of the hair strand using the formula d = 2 * (L * tan(theta)). The diameter of the hair strand is approximately 0.0251 mm.

Step-by-step explanation:

To determine the diameter of the hair strand, we can make use of Babinet's principle, which states that the spot pattern created by light going around the hair is equivalent to the pattern when light goes through a slit. Using the small angle approximation, we can relate the distance between the marks on the paper to the diameter of the hair strand.

The small angle approximation allows us to approximate the angle of diffraction as the opposite over the hypotenuse of a right triangle, where the opposite side is half the distance between the marks and the hypotenuse is the distance between the hair strand and the paper. Since we are given the wavelength of the laser light and can assume the distance between the hair and the paper to be much larger than the distance between the marks, we can solve for the diameter of the hair strand.

Using the formula: d = 2 * (L * tan(theta)), where d is the diameter of the hair strand, L is the distance between the hair and the paper, and theta is the angle of diffraction (tan(theta) = opposite/hypotenuse).

Substituting the given values, we have: d = 2 * (1.15 m * tan(theta)). Converting the wavelength to meters (633 nm = 633 * 10^-9 m), we can solve for the angle of diffraction using the small angle approximation: theta = tan(theta) = (21.4 mm / 2) / 1.15 m = 0.0093 radians.

Now, substituting the values into the formula for the diameter of the hair strand, we have: d = 2 * (1.15 m * 0.0093) = 0.0251 mm. Therefore, the diameter of the hair strand is approximately 0.0251 mm.