Problem 11 - University Physics with Modern Physics

Question

Two slits spaced 0.450 \(\mathrm{mm}\) apart are placed 75.0 \(\mathrm{cm}\) from a screen. What is the distance between the second and third dark lines of the interference pattern on the screen when the slits are illuminated with coherent light with a wavelength of 500 \(\mathrm{nm} ?\)

Step-by-Step Solution

Answer

The distance between the second and third dark lines is approximately 0.417 mm.

Step 1: Understand the Problem

We have two slits producing an interference pattern on a screen. We are asked to find the distance between the second and third dark fringes when coherent light of 500 nm wavelength is used. The key is to determine the position of these dark fringes using the formula for dark fringe positions in a double-slit experiment.

Step 2: Identify the Formula

For dark fringes in a double-slit interference pattern, the condition is given by \( d \sin \theta = (m + 0.5) \lambda \), where \(d\) is the distance between the slits, \(\lambda\) is the wavelength, and \(m\) is the order of the dark fringe.

Step 3: Calculate Positions of Dark Fringes

To find the positions, consider small angles where \( \sin \theta \approx \tan \theta \approx \frac{y}{L} \). Hence, \( y = (m + 0.5) \frac{\lambda L}{d} \). Substitute \( d = 0.450 \, \mathrm{mm} = 0.00045 \, \mathrm{m} \), \( \lambda = 500 \, \mathrm{nm} = 500 \times 10^{-9} \, \mathrm{m} \), and \( L = 0.75 \, \mathrm{m} \).

Step 4: Calculate Position of Second Dark Line

Substitute \( m = 1 \) (second dark fringe), we get \( y_2 = (1 + 0.5) \frac{500 \times 10^{-9} \times 0.75}{0.00045} \). Calculate \( y_2 \).

Step 5: Calculate Position of Third Dark Line

Substitute \( m = 2 \) (third dark fringe), we get \( y_3 = (2 + 0.5) \frac{500 \times 10^{-9} \times 0.75}{0.00045} \). Calculate \( y_3 \).

Step 6: Find the Distance Between Dark Lines

The distance \( \Delta y \) between the second and third dark lines is \( y_3 - y_2 \). Substitute the calculated values of \( y_2 \) and \( y_3 \) to find \( \Delta y \).

Key Concepts

Dark Fringes in Double-Slit Interference

Dark fringes are key elements in understanding interference patterns created by light. In a double-slit experiment, light waves from each slit overlap, creating regions of both constructive and destructive interference. Dark fringes occur where destructive interference happens - spots where the waves from the two slits cancel each other out.

In simpler terms, a dark fringe is like a shadow on the bright background of light. These fringes occur at specific positions along the interference pattern on a screen.

The condition for dark fringes is given by the formula: \[ d \sin \theta = (m + 0.5) \lambda \]

• \(d\) is the slit separation (0.450 mm in our problem).
• \(\lambda\) is the wavelength of the light used (500 nm here).
• \(m\) represents the order of the dark fringe (like second, third, etc.).

To find the locations of these fringes on a distant screen, we assume small angles where \(\sin \theta \approx \tan \theta \approx \frac{y}{L}\), letting us calculate the position \(y\) of dark fringes.

Wavelength and Its Role in Interference

Wavelength is a fundamental aspect of wave physics, representing the distance between successive peaks of a wave. It plays a crucial role in interference patterns, determining how waves interact to form bright and dark fringes.

The wavelength, \(\lambda\), is especially important in a double-slit experiment. It dictates the distance between interference fringes.

Longer wavelengths result in fringes that are farther apart, while shorter wavelengths create fringes that are closer together.

• For the exercise, the wavelength of coherent light is given as 500 nm (nanometers).
• This value tells us the light is visible and helps to find accurate positions of the dark lines.

Understanding wavelength helps in predicting and calculating the exact fringe pattern observed on the screen, enhancing insights into how light behaves when passing through slits.

Understanding the Interference Pattern

Interference patterns result from the interaction of overlapping waves, leading to regions of alternating light and dark stripes or bands on a screen. In the case of a double-slit experiment, the pattern stems from two sources of light waves interfering with each other.

Depending on where you look on the screen, you will either see a bright spot (where constructive interference occurs) or a dark fringe (where destructive interference occurs).

• Constructive interference happens when light waves align to amplify the light, producing bright areas.
• Destructive interference occurs when waves cancel each other out, leading to dark fringes.

Insight into this pattern aids in understanding fundamental wave properties, offering a window into the wave nature of light. The positions and spacing of these fringes are governed by the light’s wavelength, the distance between the slits, and the distance to the screen.

Coherent Light in Interference Experiments

Coherent light is essential in performing double-slit interference experiments effectively. It refers to light waves that are in phase and have a constant amplitude. Such light is characterized by having a fixed phase relationship, which means that the peaks and troughs of the waves match up over time.

In our exercise, the coherent light ensures the waves emerging from the slits maintain a consistent relationship. This consistency is key to forming clear interference patterns on the screen.

• Lasers are a common source of coherent light due to their monochromatic and phase-aligned nature.
• Other light sources can also be made coherent using special filters or apparatus.

With coherent light, you can accurately predict and measure the positions of dark and bright fringes, such as determining the distance between specific dark fringes like the second and third ones in this case.