Question

you stretch a strand of your hair across a laser beam and observe the diffraction pattern this produces on a sheet of paper 1.30 1.30 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 24.6 24.6 mm. you are given that the wavelength of the laser light is 633 633 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 find the diameter of your hair strand, you can use Babinet's principle and the small angle approximation. Using the distance between the marks on either side of the central bright spot, you can calculate the order of the fringe. Then, by doubling the distance between the marks, you can estimate the diameter of your hair strand.

Step-by-step explanation:

To find the diameter of your hair strand, we can use Babinet's principle which equates the spot pattern created by light going around the hair to the pattern when light goes through a slit. According to Babinet's principle, the distance between dark regions on either side of the central bright spot can be used to determine the size of the hair strand. We have the distance between the marks as 24.6 mm.

Using the small angle approximation, we can assume the diameter of the hair strand is much smaller than the distance to the screen. Now, the diffraction pattern created by a single slit can be given by the formula sin(θ) = (mλ)/D, where θ is the angle between the center and the dark region, m is the order of the fringe, λ is the wavelength, and D is the distance between the slit and the screen.

In this case, the angle θ can be approximated as the angle θ ≈ tan(θ) = (Δx)/D, where Δx is the distance between the center and the dark region, and D is the distance between the hair strand and the screen. Since sin(θ) ≈ tan(θ), we can equate the equations sin(θ) = (mλ)/D and tan(θ) = (Δx)/D to get (Δx)/D = (mλ)/D.

Cancelling out D on both sides, we get Δx = mλ. Rearranging the equation, we have m = Δx/λ. Since we are given the distance between the marks as 24.6 mm and the wavelength as 633 nm, we can substitute these values into the equation to find the order of the fringe. m = (24.6 mm)/(633 nm) = (24.6×10-3 m)/(633×10-9 m) = 38.85.

The diameter of the hair strand can be approximated as the distance between dark regions on either side of the central bright spot, which is twice the distance between the marks. So, the diameter of the hair strand is approximately 2 × 24.6 mm = 49.2 mm.