Problem 55 - Physics

Question

ssm The same diffraction grating is used with two different wavelengths of light, \(\lambda_{A}\) and \(\lambda_{\mathrm{B}}\). The fourth-order principal maximum of light A exactly overlaps the third-order principal maximum of light \(\mathrm{B}\). Find the ratio \(\lambda_{\mathrm{A}} / \lambda_{\mathrm{B}}\).

Step-by-Step Solution

Answer

The ratio \( \lambda_{A} / \lambda_{B} \) is \( 3/4 \).

Step 1: Understanding the Problem

We have a diffraction grating being used with two different wavelengths \( \lambda_{A} \) and \( \lambda_{B} \). The task is to find the ratio \( \lambda_{A} / \lambda_{B} \) such that the fourth-order maximum of light A overlaps with the third-order maximum of light B.

Step 2: Applying Diffraction Grating Equation

For a diffraction grating, the condition for principal maxima is given by \( d \sin \theta = m \lambda \), where \( d \) is the distance between grating lines, \( \theta \) the angle, \( m \) the order of the maximum, and \( \lambda \) the wavelength. Here, \( m_{A} = 4 \) for light A and \( m_{B} = 3 \) for light B.

Step 3: Equating Conditions for Overlap

Since the maxima overlap, the angle \( \theta \) is the same for both light A and light B. Therefore, we can write the equation as \( m_{A} \lambda_{A} = m_{B} \lambda_{B} \). Plugging in the values, we have \( 4 \lambda_{A} = 3 \lambda_{B} \).

Step 4: Solving for the Wavelength Ratio

To find the ratio \( \lambda_{A} / \lambda_{B} \), rearrange the equation: \( \lambda_{A} / \lambda_{B} = 3 / 4 \).

Step 5: Conclusion

The ratio of the wavelengths \( \lambda_{A} / \lambda_{B} \) is \( 3/4 \), meaning that the wavelength \( \lambda_{A} \) is three-fourths of \( \lambda_{B} \).

Key Concepts

Wavelength

Wavelength is a fundamental concept in physics that describes the distance between consecutive peaks of a wave. This concept is important when studying wave phenomena, such as light waves, sound waves, or water waves.
In the context of a diffraction grating, the wavelength determines how the light waves will spread out and create patterns after passing through the grating.

• A shorter wavelength means that the light will bend less as it passes through the grating.
• Conversely, a longer wavelength means the light will bend more.

Understanding the wavelength's role is crucial for predicting the behavior of light as it interacts with various optical devices, like diffraction gratings used in the discussed problem.

Diffraction Maxima

When light passes through a diffraction grating, it creates a pattern of bright and dark spots. The bright spots, known as diffraction maxima, are points where the light waves constructively interfere.
Each maximum corresponds to a specific integer multiple of the wavelength; this multiple is known as the order of the maximum.

• The first-order maximum is associated with one complete wavelength fitting within the path difference.
• The second-order maximum corresponds to two wavelengths, and so on.

In our problem, the principal maxima that overlap occur at different orders for different wavelengths. Specifically, the fourth-order maximum for wavelength \( \lambda_A \) coincides with the third-order maximum for wavelength \( \lambda_B \). This understanding is pivotal in solving the problem, as it shows how different wavelengths can produce the same angle of diffraction under certain conditions.

Grating Equation

The grating equation is an essential mathematical expression used to describe the conditions under which light diffracts to create maxima. This equation is written as \( d \sin \theta = m \lambda \), where:

• \( d \) is the distance between lines on the diffraction grating.
• \( \theta \) is the angle at which the light is diffracted.
• \( m \) is the order number of the maximum.
• \( \lambda \) is the wavelength of the light.

This equation is crucial because it allows us to calculate the angle and wavelength relationships for diffraction patterns. In our exercise, it helps determine how different wavelengths coincide at certain orders to produce overlapping maxima. By using this equation, we can equate the maxima conditions for different orders, providing insights into how wavelengths interact within a grating.