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, 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 632.8 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 5.22 cm apart.
How thick was this strand of hair???
a= Mm?

Answer 1

Width of hair is "1.96 10⁻⁵ m"

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 = 5.06/12
•      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
• Hence, "the width of the strand of hair is   1.96 10⁻⁵ m" is correct answer