Ch 33: Wave Optics

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796당신이 사용하는 게 아니라요?교과서 변경

Knight Calc 5th Edition

Ch 33: Wave Optics

문제 33

33장, 문제 33

Infrared light of wavelength 2.5 μm illuminates a 0.20-mm-diameter hole. What is the angle of the first dark fringe in radians? In degrees?

검증된 단계별 안내

Step 1: Recognize that this is a diffraction problem involving a circular aperture. The angle to the first dark fringe for a circular aperture is given by the formula:

θ = 1.22 \times \frac{λ}{D}

, where

λ

is the wavelength of the light and

D

is the diameter of the aperture.

Step 2: Convert the given values into consistent SI units. The wavelength

λ

is given as 2.5 μm, which is equivalent to

2.5 \times 10 ⁻ 6

meters. The diameter

D

is given as 0.20 mm, which is equivalent to

2.0 \times 10 ⁻ 4

meters.

Step 3: Substitute the values of

λ

and

D

into the formula for

θ

θ = 1.22 \times \frac{2.5 \times 10 ⁻ 6}{2.0 \times 10 ⁻ 4}

. Simplify the fraction to find

θ

in radians.

Step 4: To convert the angle from radians to degrees, use the conversion factor:

1 rad = \frac{180}{π} \deg

. Multiply the angle in radians by this factor to find the angle in degrees.

Step 5: Verify the units and ensure the calculations are consistent. The final answers will be the angle of the first dark fringe in both radians and degrees.

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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.

Diffraction

Diffraction is the bending of waves around obstacles and the spreading of waves when they pass through small openings. In the context of light, diffraction patterns are created when light waves encounter an aperture, leading to regions of constructive and destructive interference. This phenomenon is crucial for understanding how light behaves when it passes through the hole mentioned in the question.

Single-Slit Diffraction

Single-slit diffraction refers to the pattern of light and dark fringes produced when light passes through a narrow slit or hole. The angle of the dark fringes can be calculated using the formula sin(θ) = mλ/a, where m is the order of the dark fringe, λ is the wavelength of light, and a is the width of the slit. This concept is essential for determining the angle of the first dark fringe in the given problem.

Wavelength

Wavelength is the distance between successive peaks of a wave, typically measured in meters or micrometers for light. It is a fundamental property of waves that influences their behavior, including diffraction patterns. In this question, the wavelength of the infrared light (2.5 μm) is critical for calculating the angle of the dark fringe, as it directly affects the spacing of the diffraction pattern.