Problem 16 - University Physics with Modern Physics

Question

Coherent light of frequency \(6.32 \times10^{14}\) Hz passes through two thin slits and falls on a screen 85.0 cm away. You observe that the third bright fringe occurs at \(\pm\)3.11 cm on either side of the central bright fringe. (a) How far apart are the two slits? (b) At what distance from the central bright fringe will the third dark fringe occur?

Step-by-Step Solution

Answer

(a) The slit separation is 3.89 mm. (b) The third dark fringe occurs at 3.63 cm.

Step 1: Understanding the Problem

In this problem, we have coherent light passing through two slits and creating an interference pattern on a screen. We need to find the slit separation (a) and the location of the third dark fringe (b). The third bright fringe is observed at 3.11 cm from the central bright fringe.

Step 2: Use the Formula for Bright Fringes

The formula for the position of the m-th bright fringe is given by \( y_m = \frac{m\lambda L}{d} \), where \( y_m \) is the position of the m-th bright fringe, \( m \) is the fringe number, \( \lambda \) is the wavelength of light, \( L \) is the distance from slits to screen, and \( d \) is the distance between the slits.

Step 3: Calculate Wavelength from Frequency

Use the formula \( \lambda = \frac{c}{f} \) where \( c = 3.00 \times 10^8 \) m/s is the speed of light and \( f = 6.32 \times 10^{14} \) Hz is the frequency of the light. This gives:\[\lambda = \frac{3.00 \times 10^8}{6.32 \times 10^{14}} \approx 4.75 \times 10^{-7} \text{ m}\]

Step 4: Calculate Slit Separation (d)

For the third bright fringe \( (m = 3) \), located at 3.11 cm:\[y_m = \frac{m\lambda L}{d} \quad \Rightarrow \quad 0.0311 = \frac{3 \times 4.75 \times 10^{-7} \times 0.85}{d}\]Rearranging for \( d \), we find:\[d \approx \frac{3 \times 4.75 \times 10^{-7} \times 0.85}{0.0311} \approx 3.89 \times 10^{-5} \text{ m}\]

Step 5: Use the Formula for Dark Fringes

The formula for the position of the m-th dark fringe is given by \( y_m = \frac{(m+0.5)\lambda L}{d} \). We use this for the third dark fringe \( (m = 3) \).

Step 6: Calculate Position of Third Dark Fringe

Substitute our values into the formula for the dark fringe:\[y_m = \frac{(3+0.5) \times 4.75 \times 10^{-7} \times 0.85}{3.89 \times 10^{-5}}\]Simplifying, we get:\[y_m \approx 3.63 \times 10^{-2} \text{ m} \approx 3.63 \text{ cm}\]

Key Concepts

Coherent Light

Coherent light is vital for creating interference patterns, like those observed in the two-slit experiment. It consists of waves that have the same frequency and a constant phase difference. This stability allows the light waves to superimpose constructively and destructively at predictable intervals.
In the context of the two-slit experiment, when coherent light passes through the slits, it's crucial for maintaining consistent fringes on the screen. Because of their synchronized nature, the light beams can either combine to amplify the intensity for bright fringes or cancel each other out for dark fringes.
Understanding coherent light helps us grasp how small differences and changes in phase and path length result in the varied interference patterns we observe.

Fringe Pattern

A fringe pattern is the series of alternating bright and dark bands that appear on a screen when coherent light passes through two slits. This phenomenon results from the interference of light waves. When the peaks of waves align, they create bright fringes due to constructive interference.
For dark fringes, destructive interference occurs, where a peak and a trough meet, canceling each other's effects and resulting in minimal to no light on the screen. The position and spacing of these fringes depend on several factors, such as the wavelength of the light, distance between the slits (d), and the distance from slits to the screen (L).
Bright fringes, also called maxima, typically denote points of maximum energy, whereas dark fringes, or minima, indicate points of minimal energy. Understanding the fringe pattern helps in finding specific measurements such as slit separation.

Wavelength Calculation

To determine the wavelength (\( \lambda \)) of the light used in interference experiments, knowledge of the light's frequency (\( f \)) is vital. For light, the speed (\( c \)) is nearly constant in a vacuum at approximately \( 3.00 \times 10^8 \, \text{m/s} \). The wavelength can be calculated using the formula:

• \( \lambda = \frac{c}{f} \)

For this exercise, given the frequency \( 6.32 \times 10^{14} \, \text{Hz} \), we find:

• \( \lambda \approx \frac{3.00 \times 10^8}{6.32 \times 10^{14}} \approx 4.75 \times 10^{-7} \, \text{m} \)

Calculating the wavelength helps in determining where the interference patterns align and is crucial for solving the other facets of the problem, such as fringe distance.

Interference Formulas

In experimental physics, interference formulas aid in predicting and understanding the positions of fringes on a screen. For the two-slit experiment, the essential formulas are

• Bright Fringes: \( y_m = \frac{m\lambda L}{d} \), where \( y_m \) is the position of the m-th bright fringe, \( m \) is the fringe number, \( \lambda \) is the wavelength, \( L \) is the screen distance, and \( d \) is slit separation.
• Dark Fringes: \( y_m = \frac{(m+0.5)\lambda L}{d} \), used to find the m-th dark fringe.

Applying these formulas, you can deduce where both bright and dark fringes will occur. For instance, when asked to find the position of the third bright or dark fringe, these formulas determine where they manifest along the screen via the relationships between these variables.