Problem 42 - College Physics

Question

White light of wavelength ranging from \(400 \mathrm{nm}\) to \(700 \mathrm{nm}\) is used for a diffraction grating with 6500 lines per centimeter. (a) In a particular order of maximum, red color will have (1) a larger, (2) the same, or (3) a smaller angle than blue color. Explain. (b) Calculate the angles for \(400 \mathrm{nm}\) and \(700 \mathrm{nm}\) in the second-order maximum. (c) What is the angular width of the whole spectrum in the second order?

Step-by-Step Solution

Answer

(a) Red has a larger angle. (b) Calculate angles using \(\sin \theta = \frac{m \lambda}{d}\). (c) Angular width is the difference in these angles.

Step 1: Understanding the Diffraction Grating Equation

A diffraction grating separates light into its component wavelengths. The diffraction condition is given by the equation \(d \sin \theta = m \lambda\), where \(d\) is the grating spacing, \(\theta\) is the diffraction angle, \(m\) is the order number, and \(\lambda\) is the wavelength.

Step 2: Determine the Grating Spacing

Given the number of lines per centimeter is 6500, we convert this to the grating spacing \(d\) by taking the reciprocal of 6500 lines/cm, i.e., \(d = \frac{1}{6500} \text{ cm/line} = \frac{1}{6500 \times 10^2} \text{ m} = \frac{1}{6.5 \times 10^4} \text{ m}\).

Step 3: Predict Angle for Red and Blue Colors

The angle \(\theta\) is directly proportional to the wavelength \(\lambda\) (\(\theta \propto \lambda\)). Red has a larger wavelength (700 nm) than blue (400 nm), so the angle for red light will be larger than for blue light in any given order. Therefore, the correct answer is (1): a larger angle for red.

Step 4: Calculate Second-Order Maximum Angle for 400 nm

The second-order maximum \(m = 2\). For \(\lambda = 400\, \text{nm} = 400 \times 10^{-9} \text{ m}\), use the equation \[ \sin \theta = \frac{m \lambda}{d} = \frac{2 \times 400 \times 10^{-9}}{1/6.5 \times 10^4} \]. This needs to be simplified to find \(\theta\).

Step 5: Calculate Second-Order Maximum Angle for 700 nm

Using the same method as Step 4 for \(\lambda = 700 \text{ nm} = 700 \times 10^{-9} \text{ m}\), calculate \( \sin \theta = \frac{2 \times 700 \times 10^{-9}}{1/6.5 \times 10^4} \). Simplify to find \(\theta\).

Step 6: Solve for Angles and Calculate Angular Width

For 400 nm, solve for \(\theta\) from Step 4. For 700 nm, solve for \(\theta\) from Step 5. The angular width of the spectrum is the difference in angles for the 400 nm and 700 nm wavelengths in the second-order maximum.

Key Concepts

Wavelength

The concept of wavelength is crucial in understanding how waves, including light, behave. Wavelength is the distance between consecutive crests or troughs in a wave. It is generally denoted by the Greek letter \( \lambda \) and is measured in units like nanometers (nm) or meters (m). The spectrum of visible light includes wavelengths from about 400 nm (violet) to 700 nm (red).
Changing the wavelength affects the color of light. For instance, shorter wavelengths are seen as blue or violet, while longer wavelengths appear red. When it comes to diffraction gratings, the separation of light into different wavelengths creates the spectrum. The variation in angles for different wavelengths helps us see different colors.
In practical applications, understanding the wavelength is key to designing devices like spectrometers, which use diffraction gratings to analyze light. It helps us learn more about the physical properties of stars, chemicals, and other light-emitting or absorbing objects.

Diffraction Angle Calculation

Calculating the diffraction angle is an essential step in studying how light interacts with diffraction gratings. The angles where specific wavelengths of light are visible are determined using the diffraction grating equation, \( d \sin \theta = m \lambda \). In this equation, \( d \) represents the grating spacing, \( \theta \) is the diffraction angle, \( m \) is the order number, and \( \lambda \) the wavelength of the light.
First, you need to find the grating spacing \( d \). For example, if a grating has 6500 lines per centimeter, the grating spacing is the inverse of that number: 1/6500 cm per line, or \( \frac{1}{6.5 \times 10^4} \text{ m} \).
Once you have \( d \), you can select an order number, \( m \), such as 1, 2, or higher depending on how many times the spectrum repeats. Using these values, plug them into the equation to find \( \theta \). This equation helps predict how light is distributed when it passes through a diffraction grating, allowing scientists to analyze the composition of light sources.

Optical Physics

Optical physics is the field of science dedicated to the study of light and its interactions with matter. It encompasses concepts like refraction, reflection, diffraction, and interference. Each principle helps explain how light behaves under different circumstances.
Diffraction, a core phenomenon in optical physics, refers to the bending and spreading of waves around obstacles. When light passes through a diffraction grating, it splits into various components based on their wavelengths. This separation enables the study of different light properties and phenomena, such as analyzing the light from stars to understand their composition.
Understanding optical physics also has practical applications, from the design of optical instruments like microscopes and telescopes to telecommunications and the development of laser technology. By exploring how light interacts with different materials and devices, optical physics continues to be a pivotal area in both scientific research and technological advancements.