A red laser pointer with a wavelength of 635 nm is shined on a 200-lines/mm diffraction grating.A screen is then placed a distance of 1.0 m behind the diffraction grating to observe the diffraction pattern.How far away from the central maximum will the next bright spot be on the screen?

Question

Quizplus · Accepted Answer

The distance to the next bright spot from the central maximum can be calculated using the formula for diffraction grating: $d \sin \theta = m\lambda$, where $d$ is the distance between grating lines (1/mm converted to meters for calculation), $\lambda$ is the wavelength of the light (in meters), and $m$ is the order of the maximum (m=1 for the first order maximum). The distance between grating lines, $d = \frac{1}{200 \text{ lines/mm}} = \frac{1}{200,000 \text{ lines/m}} = 5 \times 10^{-6} \text{ m}$. The wavelength, $\lambda = 635 \text{ nm} = 635 \times 10^{-9} \text{ m}$. Solving for $\sin \theta$ gives $\sin \theta = \frac{m\lambda}{d}$. For the first order maximum ($m=1$), $\sin \theta = \frac{635 \times 10^{-9}}{5 \times 10^{-6}} = 0.127$. The angle $\theta$ can be found using the inverse sine function, but for small angles, $\sin \theta \approx \tan \theta = \frac{y}{L}$, where $y$ is the distance from the central maximum to the first order maximum, and $L$ is the distance from the grating to the screen (1.0 m). Therefore, $y = L \times \sin \theta = 1.0 \times 0.127 = 0.127 \text{ m} = 12.7 \text{ cm}$, which rounds to 13 cm.

A Red Laser Pointer with a Wavelength of 635 Nm