Problem 28 - University Physics with Modern Physics

Question

The wavelength range of the visible spectrum is approximately 380-750 nm. White light falls at normal incidence on a diffraction grating that has 350 slits/mm. Find the angular width of the visible spectrum in (a) the first order and (b) the third order. (\(Note\): An advantage of working in higher orders is the greater angular spread and better resolution. A disadvantage is the overlapping of different orders, as shown in Example 36.4.)

Step-by-Step Solution

Answer

In the first order, the angular width is \( \Delta \theta_1 \), and in the third order, it is \( \Delta \theta_3 \). These are calculated using the diffraction equation.

Step 1: Understanding the Problem

The problem involves calculating the angular width of the visible spectrum when white light passes through a diffraction grating. We are given the range of the visible spectrum (380 nm to 750 nm) and the number of slits per millimeter on the grating (350 slits/mm). The task is to find the angular width for the first and third order diffraction.

Step 2: Calculate Grating Spacing

The first step in solving this problem is to convert the number of slits per millimeter to grating spacing. This is done using the formula \[ d = \frac{1}{\text{slits/mm}} \], where \( d \) is the spacing in meters. Thus, \( d = \frac{1}{350,000} \) meters because there are 1,000 millimeters in a meter.

Step 3: Implement the Diffraction Equation

For diffraction, the equation \( d \sin(\theta) = m \lambda \) is used, where \( \theta \) is the angle of diffraction, \( m \) is the order number, and \( \lambda \) is the wavelength. We need to calculate \( \theta \) for both the shortest wavelength (380 nm) and the longest wavelength (750 nm) separately in both the first (\( m = 1 \)) and third order (\( m = 3 \)).

Step 4: Calculate First Order Angles

For \( m = 1 \):Using \( \lambda = 380 \) nm, \[ \theta_1 = \arcsin\left( \frac{1 \times 380 \times 10^{-9}}{\frac{1}{350,000}} \right) \].Similarly, for \( \lambda = 750 \) nm, \[ \theta_2 = \arcsin\left( \frac{1 \times 750 \times 10^{-9}}{\frac{1}{350,000}} \right) \].Calculate these angles to find the angular spread \( \Delta \theta_1 = \theta_2 - \theta_1 \) for the first order.

Step 5: Calculate Third Order Angles

For \( m = 3 \):Using \( \lambda = 380 \) nm, \[ \theta_1 = \arcsin\left( \frac{3 \times 380 \times 10^{-9}}{\frac{1}{350,000}} \right) \].Similarly, for \( \lambda = 750 \) nm, \[ \theta_2 = \arcsin\left( \frac{3 \times 750 \times 10^{-9}}{\frac{1}{350,000}} \right) \].Calculate these angles to find the angular spread \( \Delta \theta_3 = \theta_2 - \theta_1 \) for the third order.

Step 6: Conclude with Angular Widths

Using the calculations from Steps 4 and 5, we find the angular width of the visible spectrum for the first order (\( m = 1 \)) as \( \Delta \theta_1 \) and for the third order (\( m = 3 \)) as \( \Delta \theta_3 \). These provide the desired angular spreads of light at first and third order.

Key Concepts

Angular Width

Angular width is an essential concept when discussing diffraction gratings and light dispersion. It refers to the spread of angles over which the light of different wavelengths is diffracted. In this exercise, we explore how white light's different wavelengths spread out as they pass through a diffraction grating. The angular width is defined as the difference between the angles of diffraction for the light's shortest and longest wavelengths within the specified order.

For example, in the first order diffraction, we calculate the diffraction angles for the wavelengths at 380 nm and 750 nm. The difference between these two angles is the angular width. This helps us understand how diverse the spread of colors will be in the spectrum that the grating projects. By comparing the angular width of different diffraction orders, we can appreciate why higher-order diffractions, like third order, can provide greater resolution despite the complications of overlapping spectra.

Visible Spectrum

The visible spectrum is a portion of the electromagnetic spectrum that is visible to the human eye. It typically ranges from approximately 380 nm to 750 nm in wavelength. When white light, which contains all the colors of the visible spectrum, is passed through a diffraction grating, it spreads out into its individual colors due to the variation in wavelengths.

• This separation occurs because each wavelength is diffracted at a different angle. Shorter wavelengths, like violet, diffract at smaller angles than longer wavelengths, like red.
• The visible spectrum’s spread through a diffraction grating provides a practical demonstration of the dispersion of light.

The visible spectrum is crucial for various applications, including optical instruments and astronomy, where distinguishing between different light wavelengths can offer insights into materials and physical conditions.

First Order Diffraction

First order diffraction refers to the first set of angles where constructive interference occurs as white light passes through a diffraction grating. In this order, each wavelength of the visible spectrum is diffracted once, resulting in the first occurrence of the full color spectrum being displayed.

The angle of diffraction for any wavelength is found through the formula \(d \sin(\theta) = m \lambda\), where \(m = 1\) for the first order. This equation shows the relationship between the spacing of the grating \(d\), the diffraction angle \(\theta\), and the wavelength \(\lambda\).

• The angular width in the first order is calculated by determining the angles for the wavelengths at both ends of the spectrum—violet (380 nm) and red (750 nm).
• The difference between the angles gives the angular spread or width.

Observing the first order diffraction is crucial for basic spectrum analysis, as it often provides a clear distinct view of the light’s color components.

Third Order Diffraction

Third order diffraction occurs when light interferes constructively for the third time, crossing beyond the first and second orders. It involves higher angles of diffraction compared to the first order. This results in potentially greater angular width, offering improved resolution of the visible spectrum. However, third order diffraction can overlap with higher wavelength light from earlier orders, which might complicate spectrum analysis.

By applying the diffraction equation \(d \sin(\theta) = m \lambda\) with \(m = 3\), we can calculate the angles of diffraction for the visible spectrum.

• The increased angular spread in third order diffraction is advantageous for applications requiring detailed spectral analysis, like in spectroscopy where fine wavelength distinction is essential.
• Despite its benefits, caution is advised due to potential overlapping of spectra from different orders.

Understanding third order diffraction helps in choosing appropriate settings for fine-resolution optical instruments, maximizing the clarity and detail of data provided by the visible spectrum.