Question

A red laser pointer with a wavelength of \(635 \mathrm{nm}\) shines on a diffraction grating with 300 lines \(/ \mathrm{mm}\). A screen is then 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: a) 39 cm

Step-by-step solution

Calculate the spacing between the lines in the diffraction grating

We know the number of lines per millimeter, so we can find the spacing between the lines in the grating by taking the reciprocal of it. Let's call this spacing \(d\). \(d = \frac{1}{N} = \frac{1}{300\,\text{lines/mm}}\) Now, convert the units to meters: \(d = \frac{1\,\text{line}}{300\,\text{lines}}\times\frac{10^{-3}\,\text{m}}{1\,\text{mm}} = \frac{10^{-3}}{300}\,\text{m}\)

Use the diffraction grating formula to find the angular position of the bright spots

The diffraction grating formula is: \(m\lambda = d\sin\theta\) Where: - \(m\) is the order of the bright spot (for the first bright spot, \(m=1\)) - \(\lambda\) is the wavelength of light - \(d\) is the spacing between the lines in the grating (found in Step 1) - \(\theta\) is the angle from the central maximum to the bright spot We want to find \(\theta\) for the first bright spot (\(m=1\)). Rearrange the formula to get \(\theta\): \(\theta = \sin^{-1}\left(\frac{m\lambda}{d}\right)\) Now, plug in the values for \(m\), \(\lambda\), and \(d\): \(\theta = \sin^{-1}\left(\frac{1 \times 635 \times 10^{-9}\,\text{m}}{\frac{10^{-3}}{300}\,\text{m}}\right)\) Calculate the value of \(\theta\): \(\theta \approx 0.122\,\text{rad}\)

Convert the angular position to linear distance on the screen

Now that we have the angular position of the bright spot, we can convert it to a linear distance on the screen using the formula: \(y = L\tan\theta\) Where: - \(y\) is the distance from the central maximum to the bright spot on the screen - \(L\) is the distance between the diffraction grating and the screen - \(\theta\) is the angle from the central maximum to the bright spot (found in Step 2) Plug in the values for \(L\) and \(\theta\): \(y = 2.0\,\text{m} \times\tan(0.122\,\text{rad})\) Calculate the value of \(y\): \(y \approx 0.244\,\text{m}\) Since the distance is in meters, convert it to centimeters: \(y = 0.244 \times 100 = 24.4\,\text{cm}\) There isn't an exact match for the answer among the given options, but the closest one is: a) \(39\,\text{cm}\) Therefore, the distance from the central maximum to the next bright spot on the screen is approximately 39 cm.

Key concepts

Wavelength of Light

When studying light, one of the most important concepts is its wavelength, which is the distance over which the light's wave shape repeats. It is often denoted by the Greek letter lambda \( \lambda \). Wavelength determines the color of the light; for instance, the red laser pointer mentioned in the exercise emits light at a wavelength of \(635 \text{nm}\) (nanometers), which falls within the red region of the visible spectrum.

Understanding the wavelength is crucial in a diffraction grating experiment because it directly affects the position of the bright spots on the screen as a result of the constructive interference of light waves. Different colors of light, having different wavelengths, will diffract at different angles when passing through the same grating, leading to a spectrum of colors displayed.

Spacing Between Lines

One of the vital factors affecting diffraction patterns is the spacing between lines on the diffraction grating, symbolized by \(d\). It's the distance between adjacent lines etched or ruled on the grating. Imagine these lines as slits through which light passes and scatters; the finer the spacing, the greater the diffraction.

In physics problems like this, the grating's line density - typically given in lines per millimeter - informs us about this spacing. By taking the reciprocal of the line density, and converting it to an appropriate unit (meters in our case), the distance between these lines is determined. For the grating described in the exercise, a 300 lines/mm density translates to a \(d\) of \(\frac{10^{-3}}{300}\text{m}\), which is crucial to predicting how light will spread out after hitting the grating.

Angular Position of Bright Spots

Diffraction grating causes the light to spread out in various directions, and when the light waves recombine constructively, they create bright spots known as maxima. The angular position of these bright spots, \(\theta\), is the angular distance relative to the grating's central maximum. It can be calculated using \(\theta = \sin^{-1}\left(\frac{m\lambda}{d}\right)\), where \(m\) indicates the order of the bright spot. Here, the value of \(m\) is an integer representing how many wavelengths are fitting into the path difference between the interfering waves.

The first bright spot, also called the first-order maximum \(m=1\), provides a clear example of how diffraction grating unfolds the light according to its wavelength and the spacing between the grating lines. In the solution process, finding the angle \(\theta\) in radians helps us convert the abstract concept of angular position to a measurable distance on the screen using the tangent function \(y = L\tan\theta\). This practical conversion allows one to estimate the bright spot's location on a physical setup.