Problem 52 - Physics

Question

Review Conceptual Example 2 before attempting this problem. Two slits are \(0.158 \mathrm{~mm}\) apart. A mixture of red light (wavelength \(=665 \mathrm{~nm}\) ) and yellow-green light (wavelength \(=565 \mathrm{~nm}\) ) falls on the slits. A flat observation screen is located \(2.24 \mathrm{~m}\) away. What is the distance on the screen between the third-order red fringe and the thirdorder yellow-green fringe?

Step-by-Step Solution

Answer

The distance is 4.3 mm.

Step 1: Introduction to the problem

We are given two slits separated by a distance (d), through which a mixture of red and yellow-green light is passing. We need to find the distance on a screen located at distance (L) from the slits between the third-order bright fringes of each color.

Step 2: Identify Key Formula

The formula for the position of the m-th order bright fringe in a double-slit interference pattern is given by \[ y_m = \frac{{m \cdot \lambda \cdot L}}{d} \]where \( y_m \) is the position on the screen, \( m \) is the order number, \( \lambda \) is the wavelength of the light, \( L \) is the distance to the screen, and \( d \) is the separation between slits.

Step 3: Calculate the position of the third-order red fringe

For red light, the wavelength \( \lambda = 665 \text{ nm} = 665 \times 10^{-9} \text{ m} \), the order \( m = 3 \), distance \( L = 2.24 \text{ m} \), and slit separation \( d = 0.158 \text{ mm} = 0.158 \times 10^{-3} \text{ m} \). Substituting these values into the formula gives:\[ y_{3, \text{red}} = \frac{{3 \cdot 665 \times 10^{-9} \cdot 2.24}}{{0.158 \times 10^{-3}}} \approx 0.0282 \text{ m} \]

Step 4: Calculate the position of the third-order yellow-green fringe

For yellow-green light, the wavelength \( \lambda = 565 \text{ nm} = 565 \times 10^{-9} \text{ m} \). Substituting this value into the fringe position formula with the same \( m, L, \) and \( d \) values gives:\[ y_{3, \text{yellow-green}} = \frac{{3 \cdot 565 \times 10^{-9} \cdot 2.24}}{{0.158 \times 10^{-3}}} \approx 0.0239 \text{ m} \]

Step 5: Find the distance between the fringes

The distance between the third-order red and yellow-green fringes is the absolute difference between their positions: \[ \Delta y = |y_{3, \text{red}} - y_{3, \text{yellow-green}}| \approx |0.0282 \text{ m} - 0.0239 \text{ m}| = 0.0043 \text{ m} \]

Step 6: Convert answer to millimeters

Since distances on the screen are often expressed in millimeters, convert \(0.0043 \text{ m}\) to millimeters: \[ 0.0043 \text{ m} = 4.3 \text{ mm} \]

Key Concepts

Wavelength Calculation

Understanding how to calculate the wavelength is key when analyzing double-slit interference patterns. The wavelength, denoted as \( \lambda \), is essentially the distance between identical points in successive waves, such as from crest to crest or trough to trough.
Light can have different wavelengths, which are typically measured in nanometers (nm) for visible light, where 1 nm equals \( 1 \times 10^{-9} \) meters. For this exercise, we utilize given wavelengths for red light (665 nm) and yellow-green light (565 nm). It's important to convert these to meters before using them in calculations, ensuring all units are consistent.
In the context of interference, the distinct wavelengths of light dictate how the waves combine, leading to different fringe appearances on the observation screen.

• Ensure proper conversion of units when dealing with wavelengths and other measurements in interference calculations.
• Wavelength changes outcome in fringe positions and impacts the bright and dark stripe pattern formed on the screen.

Fringe Position

Determining the position of fringes involves calculating how light waves interfere after passing through the slits. The formula to find the position \( y_m \) of the m-th order bright fringe on the screen is:
\[ y_m = \frac{{m \cdot \lambda \cdot L}}{d} \]
where \( m \) is the order of the fringe number, \( \lambda \) is the light's wavelength in meters, \( L \) is the distance from the slits to the screen, and \( d \) is the slit separation. This equation is crucial for predicting where colored fringes will appear. In the calculations here, \( L = 2.24 \) meters and \( d = 0.158 \times 10^{-3} \) meters.
Steps to calculate fringe position include:

• Select the appropriate wavelength for the color of light being calculated.
• Insert the correct variable values into the formula, ensuring consistency in units.
• Perform the arithmetic calculations to derive the fringe position in meters.

Once positions for the specific order are determined for each light, differences can highlight where different colored fringes appear on the screen.

Order of Fringes

The order of fringes, represented by \( m \), identifies the sequence in which bright (or dark) bands appear in an interference pattern. Bright fringes occur at integral values of \( m \) (e.g., 1, 2, 3, etc.). In this problem, the focus is on the third-order fringes (\( m = 3 \)) for both red and yellow-green light.
Different orders correspond to varying distances on the screen, as calculated using the fringe position formula. In scenarios with multiple wavelengths present, each order of fringe for each color might not align exactly because of the difference in wavelengths.
Key points about the order of fringes include:

• Higher order fringes (larger \( m \)) appear further from the center of the pattern.
• Colors with different wavelengths have different fringe spacings even if they are of the same order.
• The spacing between similar order fringes across different colors can show the wavelength's impact on pattern spacing.

This understanding allows for the prediction and explanation of fringe occurrences in interference experiments.