Ch 36: Diffraction

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610당신이 사용하는 게 아니라요?교과서 변경

Young & Freedman Calc 14th Edition

Ch 36: Diffraction

문제 3

36장, 문제 3

Light of wavelength 585 nm falls on a slit 0.0666 mm wide. (a) On a very large and distant screen, how many totally dark fringes (indicating complete cancellation) will there be, including both sides of the central bright spot? Solve this problem without calculating all the angles! (Hint: What is the largest that sin u can be? What does this tell you is the largest that m can be?) (b) At what angle will the dark fringe that is most distant from the central bright fringe occur?

검증된 단계별 안내

Step 1: Begin by understanding the diffraction condition for dark fringes in a single-slit experiment. The condition for destructive interference is given by the equation:

a \sin u = m λ

, where

a

is the slit width,

u

is the angle of the dark fringe,

m

is the order of the fringe (an integer), and

λ

is the wavelength of the light.

Step 2: To determine the maximum number of dark fringes, consider the physical constraint that

\sin u

cannot exceed 1. Rearrange the equation to find the maximum value of

m

m = \frac{a}{λ}

. Substitute the given values for

a

(0.0666 mm = 6.66 × 10⁻⁵ m) and

λ

(585 nm = 5.85 × 10⁻⁷ m) to calculate the maximum integer value of

m

. This will give the total number of dark fringes on one side of the central bright spot.

Step 3: Multiply the maximum value of

m

by 2 to account for both sides of the central bright spot. Add 1 to include the central bright spot itself in the count. This will give the total number of dark fringes.

Step 4: For part (b), use the diffraction condition

a \sin u = m λ

to find the angle

u

for the most distant dark fringe. Substitute the maximum value of

m

into the equation and solve for

\sin u

. Then use the inverse sine function to calculate the angle

u

Step 5: Summarize the results conceptually: The total number of dark fringes is determined by the maximum value of

m

, and the angle of the most distant dark fringe is calculated using the diffraction condition. Ensure that the physical constraints (e.g.,

\sin u

≤ 1) are satisfied.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Wave Interference

Wave interference occurs when two or more waves overlap, resulting in a new wave pattern. In the context of light, this can lead to constructive interference (bright fringes) or destructive interference (dark fringes). The condition for destructive interference in a single slit is given by the equation a sin(θ) = mλ, where 'a' is the slit width, 'm' is the order of the dark fringe, and 'λ' is the wavelength of light.

Sinusoidal Function Limits

The sine function, sin(θ), has a maximum value of 1, which occurs at θ = 90 degrees. This limit is crucial when determining the maximum order 'm' of dark fringes in a diffraction pattern. If sin(θ) exceeds 1, it indicates that no further dark fringes can be formed, thus providing a boundary for the possible values of 'm' in the interference equation.

Dark Fringe Position

The position of dark fringes in a diffraction pattern can be calculated using the formula a sin(θ) = mλ. The angle θ for the most distant dark fringe corresponds to the maximum value of 'm' that satisfies the sine function's limit. This angle indicates where the first minimum occurs on either side of the central bright fringe, providing insight into the distribution of light and dark regions on the observation screen.

관련 실천

교과서 질문

Diffraction occurs for all types of waves, including sound waves. High-frequency sound from a distant source with wavelength 9.00 cm passes through a slit 12.0 cm wide. A microphone is placed 8.00 m directly in front of the center of the slit, corresponding to point O in Fig. 36.5a . The microphone is then moved in a direction perpendicular to the line from the center of the slit to point O. At what distances from O will the intensity detected by the microphone be zero?

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교과서 질문

A series of parallel linear water wave fronts are traveling directly toward the shore at 15.0 cm/s on an otherwise placid lake. A long concrete barrier that runs parallel to the shore at a distance of 3.20 m away has a hole in it. You count the wave crests and observe that 75.0 of them pass by each minute, and you also observe that no waves reach the shore at ±61.3 cm from the point directly opposite the hole, but waves do reach the shore everywhere within this distance. At what other angles do you find no waves hitting the shore?

Parallel rays of green mercury light with a wavelength of 546 nm pass through a slit covering a lens with a focal length of 60.0 cm. In the focal plane of the lens, the distance from the central maximum to the first minimum is 8.65 mm. What is the width of the slit?

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