Problem 10 - University Physics with Modern Physics

Question

Coherent light with wavelength 450 nm falls on a double slit. On a screen 1.80 \(\mathrm{m}\) away, the distance between dark fringes is 4.20 \(\mathrm{mm} .\) What is the separation of the slits?

Step-by-Step Solution

Answer

The separation of the slits is approximately 193 µm.

Step 1: Introduction to the Problem

We have coherent light with wavelength \( \lambda = 450 \text{ nm} \) (nanometers), which falls on a double slit, producing an interference pattern on a screen. The distance between consecutive dark fringes on the screen is given as 4.20 mm, and the distance from the slits to the screen is 1.80 m. We need to find the separation between the slits, denoted as \( d \).

Step 2: Understanding Fringe Spacing

The distance between dark fringes in a double slit experiment is given by the formula: \[ \Delta y = \frac{\lambda L}{d} \] where \( \Delta y \) is the distance between two consecutive dark fringes, \( \lambda \) is the wavelength of the light, \( L \) is the distance from the slits to the screen, and \( d \) is the separation of the slits.

Step 3: Solving for Slit Separation

We can rearrange the formula to solve for \( d \): \[ d = \frac{\lambda L}{\Delta y} \] Substitute the given values into this formula: \( \lambda = 450 \text{ nm} = 450 \times 10^{-9} \text{ m} \), \( L = 1.80 \text{ m} \), and \( \Delta y = 4.20 \text{ mm} = 4.20 \times 10^{-3} \text{ m} \).

Step 4: Calculation of Slit Separation

Plug the values into the equation: \[ d = \frac{450 \times 10^{-9} \times 1.80}{4.20 \times 10^{-3}} \]First, calculate the product in the numerator:\[ 450 \times 10^{-9} \times 1.80 = 810 \times 10^{-9} \]Then divide by \( 4.20 \times 10^{-3} \):\[ d = \frac{810 \times 10^{-9}}{4.20 \times 10^{-3}} \approx 1.93 \times 10^{-4} \text{ m} \].

Step 5: Converting to Micrometers

The resultant slit separation \( d \) is in meters. To make it more comprehensible, convert it to micrometers (1 m = 1,000,000 µm):\[ d = 1.93 \times 10^{-4} \times 1,000,000 = 193 \text{ µm} \]

Key Concepts

Coherent Light

Coherent light is crucial for the double slit experiment. Coherence refers to the waves being in phase with each other. This means they have a constant phase difference. To visualize this, imagine waves of light traveling side by side, never differing in their peaks and troughs as they move.

For a double slit experiment to work effectively, coherence ensures the interference of the light waves can occur predictably. This creates a clear and steady interference pattern of alternating bright and dark fringes on the screen.

• Coherent waves originated from a single source.
• Important for producing discernible patterns in experiments.

Understanding coherence helps grasp why only certain patterns occur during the experiment.

Wavelength

Wavelength is a measure of the distance between consecutive peaks of a wave. In this context, it is represented by the Greek letter \( \lambda \). For the double slit experiment, we used a wavelength of 450 nm, which is in the visible light spectrum, giving it a bluish color.
The wavelength determines the spacing and nature of the fringes seen in the interference pattern:

• Shorter wavelengths lead to closer fringe spacing.
• Longer wavelengths spread the fringes further apart.

In our problem, the wavelength was key to calculating how the light interacted with the slits, ultimately determining the fringe pattern on the screen.

Interference Pattern

An interference pattern arises when waves overlap and combine. In the double slit experiment, this combination is due to the coherent light passing through two slits. The light waves coming from each slit interfere with each other, creating regions of constructive and destructive interference.

Constructive interference leads to bright fringes, where waves peak together, amplifying their brightness. Conversely, destructive interference results in dark fringes, where the peak of one wave meets the trough of another, canceling each other out. This alternating sequence of bright and dark bands forms the characteristic interference pattern.

• Bright fringes = Constructive interference.
• Dark fringes = Destructive interference.

By studying the position of these fringes, as seen in experiments, we can learn about the properties of waves and how different factors influence wave behavior.

Fringe Spacing

Fringe spacing is the distance between two consecutive similar fringes, such as bright to bright or dark to dark. This is crucial for calculating any unknowns in the double slit experiment, like the separation of the slits.
In the exercise, the formula \( \Delta y = \frac{\lambda L}{d} \) was used, where:

• \( \Delta y \) is fringe spacing, 4.20 mm in this case.
• \( \lambda \) is the light's wavelength, 450 nm.
• \( L \) is the distance from slits to the screen, 1.80 m.
• \( d \) is the slit separation.

Substituting these values helps pinpoint the space between the slits, calculated here as 193 µm. Understanding fringe spacing helps interpret the results and make calculations in wave-based experiments.