Question

Although we have discussed single-slit diffraction only for a slit, a similar result holds when light bends around a straight, thin object, such as a strand of hair. In that case, a is the width of the strand. From actual laboratory measurements on a human hair, it was found that when a beam of light of wavelength 633.0 nm was shone on a single strand of hair, and the diffracted light was viewed on a screen 1.25 m away, the first dark fringes on either side of the central bright spot were 5.06 cm apart. How thick was this strand of hair???

Answer 1

The width of the strand of hair is 1.96 10⁻⁵ m

Step-by-step explanation:

For this diffraction problem they tell us that it is equivalent to the diffraction of a single slit, which is explained by the equation

a sin θ =± m λ

Where the different temrs are: “a” the width of the hair, λ the wavelength, θ the angle from the center, m the order of diffraction, which is the number of bright rings (constructive diffraction)

We can see that the diffraction angle is missing, but we can find it by trigonometry, where L is the distance of the strand of hair to the observation screen and "y" is the perpendicular distance to the first minimum of intensity

L = 1.25 m 100 cm/1m = 125 cm

y = 5.06 cm

Tan θ = y/L

Tan θ = 5.06/125

θ = tan⁻¹ ( 0.0405)

θ = 2.32º

With this data we can continue analyzing the problem, they indicate that they measure the distance to the first dark strip, thus m = 1

a = m λ / sin θ

a = 1 633 10⁻⁹ 1.25/sin 2.3

a = 1.96 10⁻⁵ m

a = 0.0196 mm

The width of the strand of hair is 1.96 10⁻⁵ m