Diffraction grating: An 18-mm wide diffraction grating has rulings of 880 lines per mm. Monochromatic light of 590 nm wavelength is incident normally on the grating. What is the largest angle, measured from the central bright spot, at which an intensity maximum is formed?
A) 31°
B) 29°
C) 27°
D) 25°
E) 23°

Question

Quizplus · Accepted Answer

The largest angle for an intensity maximum can be found using the grating equation $d\sin\theta = m\lambda$, where $d$ is the grating spacing, $\theta$ is the diffraction angle, $m$ is the order of the maximum, and $\lambda$ is the wavelength of the light. The grating spacing $d$ is the inverse of the number of lines per mm, so $d = \frac{1}{880 \text{ lines/mm}} = \frac{1}{880 \times 10^3 \text{ lines/m}}$. Given $\lambda = 590 \text{ nm} = 590 \times 10^{-9} \text{ m}$, we find the maximum order $m$ that satisfies the equation before $\sin\theta$ exceeds 1. For the highest order maximum that can be observed, $\sin\theta$ approaches 1, which corresponds to $\theta = 90^\circ$, but practically, we look for the highest $\theta < 90^\circ$. Rearranging the equation for $\theta$, and incrementing $m$ from 1 upwards, we find the maximum $m$ that yields a valid $\theta$. For the largest angle less than 90°, we calculate $\theta$ for the highest possible $m$. Given the constraints, the calculation will show that the angle for the highest observable order is closest to 31°, making option A correct.

Diffraction Grating: an 18-Mm Wide Diffraction Grating Has Rulings of 880