An 18-mm-wide diffraction grating has rulings of 710 lines per millimieter. Monochromatic light of wavelength 506 nm wavelength is incident normally on the grating. What is the largest angle
From the centerline at which an intensity maximum is formed?
A) $46 ^ { \circ }$
B) $38 ^ { \circ }$
C) $42 ^ { \circ }$
D) $40 ^ { \circ }$
E) $44 ^ { \circ }$

Question

An 18-mm-wide diffraction grating has rulings of 710 lines per millimieter. Monochromatic light of wavelength 506 nm wavelength is incident normally on the grating. What is the largest angle
From the centerline at which an intensity maximum is formed? 
A) $46 ^ { \circ }$
B) $38 ^ { \circ }$
C) $42 ^ { \circ }$
D) $40 ^ { \circ }$
E) $44 ^ { \circ }$

Quizplus · Accepted Answer

The largest angle for an intensity maximum can be found using the diffraction 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 $d$ can be found from the given number of lines per millimeter, $d = \frac{1}{710 \text{ lines/mm}} = \frac{1}{710} \text{ mm} = \frac{1}{0.71} \times 10^{-3} \text{ m}$. The wavelength $\lambda = 506 \text{ nm} = 506 \times 10^{-9} \text{ m}$. To find the largest angle, we need the highest order $m$ that still gives a valid angle. Since $\sin\theta$ cannot exceed 1, the maximum order $m$ can be estimated by rearranging the equation to $m = \frac{d}{\lambda}$. Substituting the values, we find the maximum $m$ that satisfies the condition for a valid $\theta$. In practice, we calculate $\theta$ for different $m$ values until the equation yields a $\sin\theta$ greater than 1, which is not physically possible. The highest $m$ before this point gives the largest valid $\theta$, which corresponds to choice A, $46^\circ$, indicating the calculation aligns with this maximum angle for an intensity peak.

An 18-Mm-Wide Diffraction Grating Has Rulings of 710 Lines Per 46∘46 ^ { \circ }46∘

An 18-Mm-Wide Diffraction Grating Has Rulings of 710 Lines Per $46 ^ { \circ }$