Ch. 34 - The Wave Nature of Light: Interference and Polarization

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 34 - The Wave Nature of Light: Interference and Polarization

Problem 12

Chapter 33, Problem 12

(II) Light of wavelength λ passes through a pair of slits separated by 0.17 mm, forming a double-slit interference pattern on a screen located a distance 35 cm away. Suppose that the image in Fig. 34–9a is an actual-size reproduction of this interference pattern. Use a ruler to measure a pertinent distance on this image; then utilize this measured value to determine λ (nm) .

Verified step by step guidance

Step 1: Understand the double-slit interference pattern. The formula for the position of bright fringes (constructive interference) is given by:

y = \frac{m λ L}{d}

, where

y

is the distance between the central maximum and the m-th bright fringe,

m

is the fringe order,

λ

is the wavelength of light,

L

is the distance to the screen, and

d

is the slit separation.

Step 2: Measure the distance between two adjacent bright fringes (Δy) on the image using a ruler. This distance corresponds to the fringe spacing, which is related to the wavelength of light. Ensure the measurement is accurate and note the value.

Step 3: Use the relationship between fringe spacing and wavelength. The fringe spacing is given by:

Δy = \frac{λ L}{d}

. Rearrange this formula to solve for the wavelength:

λ = \frac{Δy d}{L}

Step 4: Substitute the measured value of Δy, the given slit separation

d

= 0.17 mm (convert to meters), and the screen distance

L

= 35 cm (convert to meters) into the formula. Ensure all units are consistent (meters).

Step 5: Perform the calculation to determine the wavelength

λ

in meters, then convert the result to nanometers (1 nm = 10^-9 m). This will give the wavelength of the light in nanometers.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Double-Slit Experiment

The double-slit experiment demonstrates the wave nature of light through interference patterns created when light passes through two closely spaced slits. When coherent light waves overlap, they can constructively or destructively interfere, resulting in bright and dark fringes on a screen. This phenomenon is fundamental in understanding wave behavior and is crucial for analyzing the interference pattern described in the question.

Wavelength (λ)

Wavelength (λ) is the distance between successive peaks of a wave, typically measured in nanometers (nm) for light. It is a key parameter in wave phenomena, influencing the color of light and the spacing of interference patterns. In the context of the double-slit experiment, the wavelength can be determined by analyzing the distance between interference fringes and the geometry of the setup.

Interference Pattern Measurement

The interference pattern produced in a double-slit experiment can be quantitatively analyzed by measuring the distance between bright or dark fringes on the screen. This measurement, combined with the known slit separation and distance to the screen, allows for the calculation of the wavelength of the light used. Understanding how to accurately measure and apply these distances is essential for solving the problem presented.

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Related Practice

Textbook Question

Monochromatic light falling on two slits 0.018 mm apart produces the fifth-order bright fringe at a 12° angle. What is the wavelength of the light used?

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Textbook Question

(II) Suppose a thin piece of glass is placed in front of the lower slit in Fig. 34–7 so that the two waves enter the slits 180° out of phase (Fig. 34–44). Describe in detail the interference pattern on the screen.

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Textbook Question

(II) Assume that light of a single color, rather than white light, passes through the two-slit setup described in Example 34–4. If the distance from the central fringe to a first-order fringe is measured to be 2.9 mm on the screen, determine the light’s wavelength (in nm) and color (see Fig. 34–11).

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Textbook Question

In a two-slit interference experiment, the path length to a certain point P on the screen differs for one slit in comparison with the other by 1.25λ.

(a) What is the phase difference between the two waves arriving at point P?

(b) Determine the intensity at P, expressed as a fraction of the maximum intensity Iₒ on the screen.

Textbook Question

Suppose that one slit of a double-slit apparatus is wider than the other so that the intensity of light passing through it is twice as great. Determine the intensity I as a function of position (θ) on the screen for coherent light.

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