Light from a HeNe laser (632.8 nm wavelength) is directed upon a straightened strand of human hair, $7.5 \mathrm{~m}$ from the wall. By examining the diffraction pattern produced on the wall, the diameter of the hair can be determined (the diffraction pattern thus produced has minima in the same locations as if the hair were a slit of the same width in an otherwise opaque barrier). If the central diffraction maximum is measured to be $14.6 \mathrm{~cm}$ wide, what is the diameter of the hair?
A) $65 \mu \mathrm{m}$
B) $86.6 \mu \mathrm{m}$
C) $43.3 \mu \mathrm{m}$
D) $32.5 \mu \mathrm{m}$

Question

Quizplus · Accepted Answer

The width of the central maximum in a single-slit diffraction pattern is given by:$$w=\frac{2\lambda L}{d}$$where $w$ is the width of the central maximum, $\lambda$ is the wavelength of the light, $L$ is the distance from the slit to the screen, and $d$ is the width of the slit. In this case, we have:$$\lambda = 632.8 \mathrm{~nm} = 6.328 \times 10^{-7} \mathrm{~m}$$$$L = 7.5 \mathrm{~m}$$$$w = 14.6 \mathrm{~cm} = 0.146 \mathrm{~m}$$Solving for $d$, we get:$$d = \frac{2\lambda L}{w} = \frac{2(6.328 \times 10^{-7} \mathrm{~m})(7.5 \mathrm{~m})}{0.146 \mathrm{~m}} = 65 \mu \mathrm{m}$$Therefore, the diameter of the hair is $65 \mu \mathrm{m}$.

Light from a HeNe Laser (632 7.5 m7.5 \mathrm{~m}7.5 m From the Wall 14.6 cm14.6 \mathrm{~cm}14.6 cm

Light from a HeNe Laser (632 $7.5 \mathrm{~m}$ From the Wall $14.6 \mathrm{~cm}$