Question

A red laser pointer shines light with a wavelength of \(635 \mathrm{nm}\) on a diffraction grating with 300 slits \(/ \mathrm{mm}\). A screen is placed a distance of \(2.0 \mathrm{~m}\) behind the diffraction grating to observe the diffraction pattern. How far away from the central maximum will the next bright spot be on the screen? a) \(39 \mathrm{~cm}\) c) \(94 \mathrm{~cm}\) e) \(9.5 \mathrm{~m}\) b) \(76 \mathrm{~cm}\) d) \(4.2 \mathrm{~m}\)

Short answer

Answer: The next bright spot is approximately 94 cm away from the central maximum.

Step-by-step solution

List down the given information

We are given the following information: - Wavelength of light, λ = \(635 \mathrm{nm}\) - Number of slits per unit length in the diffraction grating, N = \(300 \spaceslits\space \mathrm{mm}^{-1}\) - Distance between the diffraction grating and the screen, D = \(2.0 \spaces\mathrm{m}\)

Find the slit separation

We need to find the slit separation, denoted as d. Slit separation is the inverse of the number of slits per unit length. d = \(\frac{1}{N}\) = \(\frac{1}{300 \spaceslits\space \mathrm{mm}^{-1}} = \frac{1}{300} \space\mathrm{mm}\) Now convert d to meters: d= \(\frac{1}{300} \space\mathrm{mm} \times \frac{1 \spaces \mathrm{m}}{1000\spaces\mathrm{mm}} = \frac{1}{300000} \spaces\mathrm{m}\)

Calculate the angular position of the first bright spot

To find the angular position of the first bright spot, we will use the formula for the angular position of maxima in a diffraction grating: \(mλ = d\sin\theta\) Here, m is the order number of the bright spot. For first bright spot, m=1 and λ = \(635 \times 10^{-9}\spaces\mathrm{m}\). We will solve for the angle θ. \(1(635 \times 10^{-9}\spaces\mathrm{m})= \frac{1}{300000} \spaces\mathrm{m} \sin\theta\) \(\sin\theta = \frac{635 \times 10^{-9}\spaces\mathrm{m}}{\frac{1}{300000}\spaces\mathrm{m}}\) \(\theta = \sin^{-1}\left(\frac{635 \times 10^{-9}\spaces\mathrm{m}}{\frac{1}{300000}\spaces\mathrm{m}}\right)\)

Convert the angular position to the distance on the screen

Now that we have found the angle θ, we can use this angle to find the distance from the central maximum to the first bright spot on the screen. We will use the formula for finding the distance from the central maximum on the screen: \(y=D\tan{\theta}\) \(y=2.0 \spaces\mathrm{m} \times\tan\left(\sin^{-1}\left(\frac{635 \times 10^{-9}\spaces\mathrm{m}}{\frac{1}{300000}\spaces\mathrm{m}}\right)\right)\)

Calculate the value of y and find the answer

Now, calculate the value of y using the above equation which will give us the distance of the first bright spot from the central maximum. After evaluating the expression, we get: \(y ≈ 0.94\spaces \mathrm{m}\) or \(94\spaces\mathrm{cm}\) Therefore, the next bright spot will be approximately \(94\spaces\mathrm{cm}\) away from the central maximum on the screen, which corresponds to option c) \(94 \spaces\mathrm{cm}\).