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Chapter 25: Problem 66

Sonya is designing a diffraction experiment for her students. She has a laser that emits light of wavelength \(627 \mathrm{nm}\) and a grating with a distance of \(2.40 \times 10^{-3} \mathrm{mm}\) between slits. She hopes to shine the light through the grating and display a total of nine interference maxima on a screen. She finds that no matter how she arranges her setup, she can see only seven maxima. Assuming that the intensity of the light is not the problem, why can't Sonya display the \(m=4\) interference maxima on either side?

Short Answer

Expert verified
Answer: Sonya cannot observe an interference maxima with m=4 on either side because the maximum possible integer value of m that can be observed in this setup is 3. This is because higher-order maxima (m=4) would go beyond the maximum possible angle for interference maxima according to the diffraction grating interference formula.

Step by step solution

01

Understand the diffraction grating interference formula

The formula used to analyze diffraction grating interference is given by: \(d \cdot \sin{\theta} = m \cdot \lambda\) Where \(d\) is the distance between slits, \(\theta\) is the angle of the light's path with respect to the grating's normal, \(m\) is the order of the interference maxima, and \(\lambda\) is the wavelength of the light.
02

Identify the given information

We know the following information: - Wavelength of light (\(\lambda\)) = \(627 \times 10^{-9} \mathrm{m}\) (converting nm to m) - Distance between slits in the grating (\(d\)) = \(2.40 \times 10^{-3} \mathrm{mm} = 2.40 \times 10^{-6} \mathrm{m}\) (converting mm to m) - We are asked to verify if the maximum possible \(m\) is indeed \(3\) and not \(4\).
03

Find the maximum possible value of \(\sin{\theta}\)

The maximum value of \(\sin{\theta}\) is 1, which occurs when \(\theta\) approaches \(90^{\circ}\). Let's use this observation to find the maximum value of \(m\).
04

Calculate the value of \(m_{max}\)

Using the formula, we can substitute the values of \(\lambda\), \(d\), and \(\sin{\theta_{max}}=1\) to find the maximum value of \(m\). Let's call this value \(m_{max}\): \(d \cdot 1 = m_{max} \cdot \lambda\) Now, we can find the value of \(m_{max}\): \(m_{max} = \frac{d}{\lambda} = \frac{2.40 \times 10^{-6} \mathrm{m}}{627 \times 10^{-9} \mathrm{m}} \approx 3.83\)
05

Interpret the result

Since \(m_{max} \approx 3.83\), it means that the maximum possible integer value of \(m\) that Sonya can observe is \(3\) as higher-order maxima (\(m=4\)) would go beyond the maximum possible angle for interference maxima. Therefore, it is not possible for Sonya to observe \(m=4\) interference maxima on either side of her setup.

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