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Chapter 25: Problem 40

A grating is made of exactly 8000 slits; the slit spacing is $1.50 \mu \mathrm{m} .\( Light of wavelength \)0.600 \mu \mathrm{m}$ is incident normally on the grating. (a) How many maxima are seen in the pattern on the screen? (b) Sketch the pattern that would appear on a screen \(3.0 \mathrm{m}\) from the grating. Label distances from the central maximum to the other maxima.

Short Answer

Expert verified
Answer: There are a total of 5 maxima seen in the diffraction pattern. The distances of the maxima from the central maximum on the screen are 1.32 m for the maxima at m=1 and 4.0 m for the maxima at m=2.

Step by step solution

01

Calculate the maximum order m

Use the grating equation: \(d \cdot sin(\theta) = m \cdot \lambda\), where d is the slit spacing, \(\theta\) is the diffraction angle, m is the order, and \(\lambda\) is the wavelength of the incident light. To find the maximum order m, set \(\theta\) at \(90^{\circ}\), i.e., \(sin(90^{\circ}) = 1\). Then, the equation becomes \(d = m \cdot \lambda\). Solve for m: \(m = \frac{d}{\lambda}\). Plug in d = \(1.50 \mu\mathrm{m}\) and \(\lambda = 0.600 \mu\mathrm{m}\): \(m = \frac{1.50}{0.600} = 2.5\). As m must be an integer, the maximum order m is 2.
02

Calculate the number of maxima

Including the central maximum (m = 0), we have orders 0, 1, and 2 on both sides of the central maximum. Therefore, the total number of maxima seen in the pattern is \((2 \times 2) + 1 = 5\).
03

Calculate the angular separation of maxima

For each order m, find the angular separation \(\theta\) using the grating equation. For m=1: \(sin(\theta_1) = \frac{1 \cdot 0.600 \mu\mathrm{m}}{1.50 \mu\mathrm{m}}\), so \(\theta_1 = sin^{-1}(0.4) \approx 23.58^{\circ}\). For m=2: \(sin(\theta_2) = \frac{2 \cdot 0.600 \mu\mathrm{m}}{1.50 \mu\mathrm{m}}\), so \(\theta_2 = sin^{-1}(0.8) \approx 53.13^{\circ}\).
04

Calculate distances of maxima from the central maximum on the screen

Using the angular separation, we can calculate the distances of these maxima from the central maximum on a screen placed 3.0 m away from the grating. We do this by using the formula \(y = L \cdot tan(\theta)\), where y is the distance from the central maximum, L is the distance between grating and screen, and \(\theta\) is the angular separation. For the maxima at m=1, the distance is \(y_1 = 3.0 \cdot tan(23.58^{\circ}) \approx 1.32 \mathrm{m}\). For the maxima at m=2, the distance is \(y_2 = 3.0 \cdot tan(53.13^{\circ}) \approx 4.0 \mathrm{m}\).
05

Sketch the pattern and label the distances

Sketch the pattern with the central maximum at the center and two maxima on each side. Label the distances of the maxima at m=1 as \(1.32 \mathrm{m}\), and the maxima at m=2 as \(4.0 \mathrm{m}\).

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