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34. Wave Optics

Young's Double Slit Experiment

Physics

34. Wave Optics

Young's Double Slit Experiment

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5:49 minutes

Problem 20

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.

Verified step by step guidance

To determine the phase difference between the two waves arriving at point P, recall that the phase difference Δϕ is related to the path difference Δx by the formula:

Δ ϕ =2 π Δ x / λ

. Here, the path difference Δx is given as 1.25λ. Substitute this value into the formula to calculate Δϕ.

Simplify the expression for Δϕ:

Δ ϕ =2 π (1.25)

. This will give the phase difference in radians.

To determine the intensity at point P, use the formula for intensity in a two-slit interference pattern:

I = I

_o(cos²(Δϕ/2)), where

I

_o is the maximum intensity and

Δ ϕ

is the phase difference calculated earlier.

Substitute the value of

Δ ϕ

from step 2 into the intensity formula. Simplify the expression to find the intensity at point P as a fraction of the maximum intensity

I

_o.

The final result will express the intensity at point P as a fraction of the maximum intensity. Ensure that the cosine function is evaluated correctly, and the fraction is simplified if possible.

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Key Concepts

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

Interference of Waves

Interference occurs when two or more waves overlap and combine to form a new wave pattern. In the context of the two-slit experiment, constructive interference happens when waves from both slits arrive in phase, while destructive interference occurs when they arrive out of phase. The resulting intensity pattern on the screen is a direct consequence of these interference effects.

Phase Difference

Phase difference refers to the difference in the phase of two waves at a given point in time. It is typically measured in radians or wavelengths (λ). In the two-slit experiment, a path length difference of 1.25λ corresponds to a phase difference of 2.5π radians, which indicates how much one wave is ahead or behind the other when they reach point P.

Intensity of Light

The intensity of light at a point in an interference pattern is related to the amplitude of the resultant wave at that point. The maximum intensity (Iₒ) occurs at points of constructive interference, while the intensity at point P can be calculated using the formula I = Iₒ * (cos²(Δφ/2)), where Δφ is the phase difference. This relationship allows us to express the intensity at P as a fraction of the maximum intensity.

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Textbook Question

(II) If 720-nm and 660-nm light passes through two slits 0.61 mm apart, how far apart are the second-order fringes for these two wavelengths on a screen 1.0 m away?

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Textbook Question

A parallel beam of light from a He–Ne laser, with a wavelength 633 nm, falls on two very narrow slits 0.068 mm apart. How far apart are the fringes in the center of the pattern on a screen 4.2 m away?

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Textbook Question

Two narrow slits separated by 1.4 mm are illuminated by 544-nm light. Find the distance between adjacent bright fringes on a screen 5.0 m from the slits.

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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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Textbook Question

Design a double-slit apparatus so that the central diffraction peak contains precisely fifteen fringes. Assume the first diffraction minimum occurs at (a) an interference minimum, (b) an interference maximum.

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Textbook Question

In a double-slit experiment, let d = 5.00D = 40.0λ. Compare (as a ratio) the intensity of the third-order interference maximum with that of the zero-order maximum.

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Textbook Question

Imagine the following Young’s double-slit experiment using matter rather than light: electrons are accelerated through a potential difference of 18 V, pass through two closely spaced slits separated by a distance d, and create an interference pattern. (a) Using Example 37–11 and Section 34–3 as guides, find the required value for d if the first-order interference fringe is to be produced at an angle of 10°. (b) Given the approximate size of atoms, would it be possible to construct the required two-slit set-up for this experiment?

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