Problem 14 - University Physics with Modern Physics

Question

Coherent light with wavelength 400 nm passes through two very narrow slits that are separated by \(0.200 \mathrm{mm},\) and the interference pattern is observed on a screen 4.00 \(\mathrm{m}\) from the slits. (a) What is the width (in mm) of the central interference maximum? (b) What is the width of the first-order bright fringe?

Step-by-Step Solution

Answer

(a) 8 mm; (b) 4 mm

Step 1: Understand the Young's Double Slit Experiment

The problem involves the interference pattern created by coherent light passing through two slits. The interference pattern consists of bright and dark fringes on a screen. We need to find the width of the central maximum and the first-order bright fringe.

Step 2: Determine the Central Maximum Width

The central maximum is formed between the two first-order dark fringes. Each dark fringe is a path difference of half a wavelength off the path difference for a bright fringe. For small angles, the position of these fringes is given by: \( y_n = \frac{n \lambda L}{d} \).Here, for the first dark fringe, \( n=0.5 \) and \( n=-0.5 \), wavelength \( \lambda = 400 \, \mathrm{nm} = 400 \times 10^{-9} \, \mathrm{m} \), distance \( L = 4.00 \, \mathrm{m} \), and slit separation \( d = 0.200 \, \mathrm{mm} = 0.200 \times 10^{-3} \, \mathrm{m} \).

Step 3: Calculate the Position for First-order Dark Fringes

Substituting in the formula, find the positions for \( n=0.5 \) and \( n=-0.5 \):\( y_{0.5} = \frac{0.5 \times 400 \times 10^{-9} \times 4 }{0.200 \times 10^{-3}} = 4 \times 10^{-3} \, \mathrm{m} \) and \( y_{-0.5} = -4 \times 10^{-3} \, \mathrm{m} \).

Step 4: Find the Width of the Central Maximum

The width of the central maximum is the distance between these two positions, \( \text{Width} = y_{0.5} - y_{-0.5} = 8 \times 10^{-3} \, \mathrm{m} \) or \( 8 \, \mathrm{mm} \).

Step 5: Determine the Width of the First-order Bright Fringe

The width of the first-order bright fringe is the distance between the first and second \(n=1\) and \(n=-1\) order dark fringes on one side:\( y_1 = \frac{1 \times 400 \times 10^{-9} \times 4 }{0.200 \times 10^{-3}} = 8 \times 10^{-3} \, \mathrm{m} \).Therefore, width of the first-order bright fringe on one side is \( y_1 - y_{0.5} = 8 \times 10^{-3} - 4 \times 10^{-3} = 4 \times 10^{-3} \, \mathrm{m} \) or \( 4 \, \mathrm{mm} \).

Step 6: Verify and Summarize the Solution

We computed the width of the central maximum to be 8 mm, and for a first-order bright fringe, as measured from the middle of one fringe to the same position on the next, it can be understood by the symmetry as half of the central maximum width; hence, 4 mm. This contains the deviation between consecutive bright or dark peaks and midpoints.

Key Concepts

Interference Pattern

The interference pattern is a mesmerizing phenomenon seen in Young's Double Slit Experiment. When coherent light passes through two closely spaced slits, it diffracts and overlaps on a screen, creating regions of alternating light and dark bands.

This pattern results from the principle of superposition, where the crests and troughs of light waves add together. When the crests line up with crests, we see bright bands called fringes. Conversely, when crests meet troughs, the waves cancel out, creating dark bands.

• Bright Fringes: Constructive interference occurs where the paths of light differ by an integer multiple of wavelengths.
• Dark Fringes: Destructive interference happens when the path difference is an odd half-wavelength multiple.

Understanding this pattern helps in distinguishing the different regions on the interference screen.

Coherent Light

In Young's Double Slit Experiment, the concept of coherent light is crucial to understanding how an interference pattern is formed. Coherent light refers to light waves that maintain a constant phase difference. This consistency is key to producing clear and stable interference patterns.

Coherent light is usually produced using lasers, as they provide light of a single wavelength and in a single phase across space and time. This uniformity ensures that when the light passes through the slits, it maintains its orderly wave pattern, necessary for consistent constructive and destructive interference.

Central Maximum

The central maximum is the brightest and widest part of the interference pattern in Young's Double Slit Experiment. This region appears directly opposite the light source, between the two slits.

To understand its size, consider how the light waves from the slits travel the same path length to this point on the screen, resulting in maximum constructive interference. This creates the broadest bright band compared to other fringes.

• It spans from the first-order dark fringe on one side to the same on the opposite side.
• In this exercise, its width is calculated as 8 mm using the formula: \[ y_n = \frac{n \lambda L}{d} \]at the specific conditions given.

Recognizing the central maximum helps in defining reference points for measuring other parts of the pattern.

First-Order Bright Fringe

First-order bright fringes are the next set of bright fringes found alongside the central maximum. They are the brightest fringes immediately adjacent to it and help define the structure of the interference pattern.

In terms of measurement, each first-order bright fringe occurs at a path difference of one full wavelength from the reference central bright fringe position.

• To calculate their position, use the same interference formula as for the central maximum but adjust for the first order (n=1): \[ y_1 = \frac{1 \times 400 \times 10^{-9} \times 4 }{0.200 \times 10^{-3}} \]
• The width of this fringe (about 4 mm in this exercise) is determined by finding the difference in position between its adjacent dark fringes.

This visual pattern distinguishes itself by showing how light behaves under diffraction and interference principles.