Diffraction grating: Monochromatic light of wavelength 500 nm is incident normally on a diffraction grating. If the third-order maximum of the diffraction pattern is observed at 32.0° from the central spot, how many TOTAL number of maxima can be seen?
A) 5
B) 7
C) 10
D) 11
E) 13

Question

Quizplus · Accepted Answer

The formula for the maxima in a diffraction grating is given by $d\sin\theta = m\lambda$, where $d$ is the grating spacing, $\theta$ is the angle of the maxima from the central maximum, $m$ is the order of the maximum, and $\lambda$ is the wavelength of the light. Given that the third-order maximum ($m = 3$) is observed at 32.0°, we can use the equation to find the maximum possible value of $m$ that still produces a real value for $\sin\theta$, which must be less than or equal to 1. However, the key insight is that for each order $m$ on one side of the central maximum, there is a corresponding order $-m$ on the other side, plus the central maximum ($m = 0$). If the third-order maximum is visible, it implies that orders $m = -3, -2, -1, 0, 1, 2,$ and $3$ are visible, totaling 7 maxima. But since the question asks for the total number of maxima, we must consider that higher orders could also be visible if they do not exceed the angle at which light can still diffract. Given the angle for $m = 3$, we can infer that orders $m = 4$ and $m = -4$ are also possible before the sine of the angle exceeds 1. This gives us a total of $3 (for m = 1, 2, 3) + 3 (for m = -1, -2, -3) + 1 (for m = 0) + 2 (for m = 4 and m = -4) = 11$ maxima.

Diffraction Grating: Monochromatic Light of Wavelength 500 Nm Is Incident