Problem 11 - College Physics

Question

(a) In a double-slit experiment, if the distance from the double slits to the screen is increased, the separation between the adiacent maxima will (1) increase, (2) decrease, (3) remain the same. Explain. (b) Yellow-green light \((\lambda=550 \mathrm{nm})\) illuminates a double-slit separated by \(1.75 \times 10^{-4} \mathrm{~m} .\) If the screen is located \(2.00 \mathrm{~m}\) from the slits, determine the separation between the adjacent maxima. (c) What if the screen is located \(3.00 \mathrm{~m}\) from the slits?

Step-by-Step Solution

Answer

(a) Increase. (b) 6.29 mm. (c) 9.43 mm.

Step 1: Analyze the change in separation

In a double-slit experiment, the fringe separation or the distance between adjacent maxima is given by the equation \( x = \frac{\lambda L}{d} \), where \( \lambda \) is the wavelength of the light used, \( L \) is the distance to the screen, and \( d \) is the distance between the slits. If \( L \) increases, \( x \) will also increase. Therefore, the separation between adjacent maxima will increase.

Step 2: Calculate the separation for given distance (2.00 m)

Using the formula \( x = \frac{\lambda L}{d} \), substitute \( \lambda = 550 \times 10^{-9} \) m, \( L = 2.00 \) m, and \( d = 1.75 \times 10^{-4} \) m. \[ x = \frac{550 \times 10^{-9} \times 2.00}{1.75 \times 10^{-4}} \]Perform the calculation:\[ x \approx \frac{1.1 \times 10^{-6}}{1.75 \times 10^{-4}} \approx 6.29 \times 10^{-3} \text{ m} \]So, the separation is approximately 6.29 mm.

Step 3: Calculate the separation for increased distance (3.00 m)

Again, use the formula \( x = \frac{\lambda L}{d} \), now with \( L = 3.00 \) m:\[ x = \frac{550 \times 10^{-9} \times 3.00}{1.75 \times 10^{-4}} \]Perform the calculation:\[ x \approx \frac{1.65 \times 10^{-6}}{1.75 \times 10^{-4}} \approx 9.43 \times 10^{-3} \text{ m} \]Therefore, the separation is approximately 9.43 mm.

Key Concepts

Double-Slit Experiment

The double-slit experiment is a fundamental and highly intriguing demonstration in physics that illustrates the wave nature of light. When light passes through two closely spaced slits, it creates an interference pattern on a screen placed at some distance. This pattern consists of alternating bright and dark bands, known as fringes, on the screen.
The essential idea behind this experiment is that as light waves pass through the slits, they spread out and overlap, resulting in constructive and destructive interference. Constructive interference happens when waves align in phase, creating brighter regions, whereas destructive interference occurs when they are out of phase, resulting in darker regions. This forms a pattern of alternating light and dark bands.

• The phenomenon reveals the wave-like behavior of light.
• The primary formula used is: \( x = \frac{\lambda L}{d} \), where \(x\) is the fringe separation, \(\lambda\) is the wavelength, \(L\) is the distance from the slits to the screen, and \(d\) is the separation between the slits.
• From this formula, as the distance \(L\) increases, the separation \(x\) between adjacent maxima also increases.

This experiment is significant in reinforcing the concept of wave-particle duality, displaying the remarkable properties of light acting both as a wave and a particle.

Diffraction Patterns

Diffraction patterns are a hallmark of wave behavior observed when waves encounter an obstacle or opening. In the context of the double-slit experiment, these patterns appear as a series of alternating light and dark bands on a screen.
When waves such as light pass through narrow slits, bending occurs at the edges. This bending causes the waves to overlap and interfere with each other, creating the observed patterns.

• The bright bands in a diffraction pattern correspond to regions where constructive interference takes place; the waves reinforce each other to create intensity peaks.
• In contrast, the dark bands are areas of destructive interference, where waves cancel each other out, leading to intensity minima.
• The spacing and prominence of diffraction patterns depend on factors such as the wavelength of light and the geometry of the slit arrangement.

Understanding how diffraction patterns form helps illuminate the nature of wave interference and provides insights into the behavior of waves in various media.

Wavelength of Light

The wavelength of light is a fundamental property that defines many of its behaviors, including color and interference patterns. Wavelength is typically denoted by the Greek letter \(\lambda\), and it represents the distance between two consecutive peaks of a wave.
In the double-slit experiment, the wavelength of the light source significantly affects the resulting interference pattern. A key insight is that the wavelength determines the position and spacing of the maxima and minima on the screen.

• Longer wavelengths result in broader interference fringes, as the waves spread out more extensively as they propagate past the slits.
• Shorter wavelengths result in narrower fringes since the waves remain more closely packed.
• In visible light, different colors correspond to different wavelengths, with red having the longest wavelength and violet the shortest.

Understanding wavelength is crucial in fields ranging from optics to spectroscopy, where it is essential for analyzing light behavior and applications.