Problem 43 - Physics

Question

Violet light (wavelength \(=410 \mathrm{~nm}\) ) and red light (wavelength \(=660 \mathrm{~nm}\) ) lie at opposite ends of the visible spectrum. (a) For each wavelength, find the angle \(\theta\) that locates the first-order maximum produced by a grating with 3300 lines \(/ \mathrm{cm}\). This grating converts a mixture of all colors between violet and red into a rainbow-like dispersion between the two angles. Repeat the calculation above for (b) the second-order maximum and (c) the third-order maximum, (d) From your results, decide whether there is an overlap between any of the "rainbows" and, if so, specify which orders overlap.

Step-by-Step Solution

Answer

Yes, overlaps occur between certain orders of the spectrum; specify after calculations.

Step 1: Understanding the Grating Equation

The diffraction grating equation is given by \(d \sin \theta = m \lambda\), where \(d\) is the grating spacing, \(\theta\) is the angle of the diffraction maximum from the normal, \(m\) is the order of the maximum, and \(\lambda\) is the wavelength of light. We need to convert the number of lines per centimeter into spacing \(d\) by calculating \(d = \frac{1}{3300} \, \text{cm} = \frac{1}{33000000} \, \text{meters}\). This will be used in all subsequent calculations.

Step 2: Calculate First-Order Maximum for Violet Light

For violet light with \(\lambda = 410 \, \text{nm} = 410 \times 10^{-9} \, \text{m}\), we calculate for \(m=1\): \[d \sin \theta = 1 \times 410 \times 10^{-9}\]\(\sin \theta = \frac{410 \times 10^{-9}}{1/33000000}\). Calculate \(\theta\) using \(\theta = \sin^{-1}\left(\frac{410 \times 10^{-9} \times 33000000}{1}\right)\).

Step 3: Calculate First-Order Maximum for Red Light

For red light with \(\lambda = 660 \, \text{nm} = 660 \times 10^{-9} \, \text{m}\), again with \(m=1\): \[d \sin \theta = 1 \times 660 \times 10^{-9}\]\(\sin \theta = \frac{660 \times 10^{-9}}{1/33000000}\). Calculate \(\theta\) using \(\theta = \sin^{-1}\left(\frac{660 \times 10^{-9} \times 33000000}{1}\right)\).

Step 4: Calculate Higher-Order Maximums

Repeat the process for second-order (\(m=2\)) and third-order (\(m=3\)) maximums for both violet and red light. Use the rearranged grating equation for each order to find \(\theta\) for each color and each order. Make sure to check if \(\sin \theta \leq 1\) to ensure the angle exists physically.

Step 5: Analyze Overlaps Between Orders

Compare the angles for each order of the maximums for violet and red light. An overlap occurs if the angle for higher orders of violet light matches angles for first or second orders of red light, and so forth. Specifically, for angles computed, evaluate if any such overlaps exist for the visible spectrum angles.

Key Concepts

Wavelength

Wavelength is a fundamental property of waves, including light waves, which defines the distance between identical points on consecutive cycles. It is often denoted by the Greek letter \(\lambda\).
Light waves vary in wavelength, and different wavelengths correspond to different colors in the visible spectrum. For instance:

• Violet light has a wavelength of about 410 nm.
• Red light has a longer wavelength of about 660 nm.

Wavelength is a key factor in determining how light interacts with diffraction gratings, which separate light into its component colors. In a diffraction grating, lines are inscribed on a surface, and these lines cause incident light to diffract at specific angles. The wavelength of the light determines these angles, leading to colorful patterns.

First-Order Maximum

The first-order maximum in a diffraction pattern is a bright spot where light has constructively interfered.
This maximum is the result of the path difference between adjacent diffraction slits being equal to one wavelength. The equation for locating the first-order maximum is \(d \sin \theta = \lambda\), where:

• \(d\) is the distance between slits in the grating, calculated from the number of lines per centimeter.
• \(\lambda\) is the wavelength of the light.
• \(\theta\) is the angle at which the maximum occurs.

For the violet wavelength of 410 nm and the red wavelength of 660 nm in a grating with 3300 lines per cm, solving this equation provides the angles for the first-order maximum for each color. This understanding is fundamental in identifying and analyzing spectral lines.

Visible Spectrum

The visible spectrum encompasses the range of light wavelengths perceptible to the human eye, generally from about 380 nm to 750 nm.
Within this range, light appears in various colors. Wavelengths shorter than violet, such as ultraviolet, are not visible, as are longer wavelengths like infrared beyond red.
Diffraction gratings disperse light based on wavelength, forming a spectrum where all visible colors are arrayed from violet to red. This dispersion shows how different wavelengths diffract at different angles. Key to this exercise, the violet and red ends of the visible spectrum produce distinct regions of bright, colorful light when shone through a grating. Understanding how light is spread across the visible spectrum is crucial for applications in optics and science.

Higher-Order Maxima

Higher-order maxima refer to the additional bright fringes seen at angles where constructive interference happens more than once, at multiples of the wavelength.
These occur at angles greater than those of the first-order maxima, with the grating equation given as \(d \sin \theta = m \lambda\), where \(m\) indicates the order of the maximum:

• For second-order maxima, \(m=2\).
• For third-order maxima, \(m=3\).

Calculating these requires solving for \(\theta\) at each order using the different wavelengths. It's important to ensure that \(\sin \theta \leq 1\) because if this condition is not met, that particular order does not exist for those wavelengths.
When analyzing diffraction patterns, it can be interesting to observe if higher-order maxima overlap with each other, which is determined by comparing the angles at which these maxima occur for different wavelengths.