A grating with 8000 slits over 2.54 cm is illuminated by light of a wavelength of 546 nm. What is the angle for the third-order maximum?
A) 31.1°
B) 15.1°
C) 26.3°
D) 10.5°

Question

Quizplus · Accepted Answer

The angle for the third-order maximum can be found using the grating equation $d\sin\theta = m\lambda$, where $d$ is the slit spacing, $m$ is the order of the maximum, and $\lambda$ is the wavelength of the light. First, find the slit spacing $d$ by dividing the total length of the grating by the number of slits: $d = \frac{2.54\, \text{cm}}{8000} = 3.175 \times 10^{-4}\, \text{cm} = 3.175 \times 10^{-6}\, \text{m}$. Then, for the third-order maximum ($m = 3$) and $\lambda = 546\, \text{nm} = 546 \times 10^{-9}\, \text{m}$, solve for $\theta$. Rearranging the equation gives $\sin\theta = \frac{m\lambda}{d}$. Substituting the given values, $\sin\theta = \frac{3 \times 546 \times 10^{-9}\, \text{m}}{3.175 \times 10^{-6}\, \text{m}}$, which calculates to a $\sin\theta$ value that corresponds to an angle $\theta \approx 31.1^\circ$.

A Grating with 8000 Slits Over 2