Ch 33: Wave Optics

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 33: Wave Optics

Problem 20

Chapter 33, Problem 20

In a single-slit experiment, the slit width is 200 times the wavelength of the light. What is the width (in mm) of the central maximum on a screen 2.0 m behind the slit?

Verified step by step guidance

Determine the angular position of the first minimum in the single-slit diffraction pattern using the formula for minima:

a sin(θ) = mλ

, where

a

is the slit width,

λ

is the wavelength, and

m

is the order of the minimum (for the first minimum,

m = 1

Express the slit width

a

in terms of the wavelength:

a = 200λ

. Substitute this into the formula for the first minimum:

200λ sin(θ) = λ

. Simplify to find

sin(θ) = 1/200

Calculate the angular position of the first minimum,

θ

, by taking the inverse sine:

θ = sin

^-1(1/200). This gives the angle to the first minimum on one side of the central maximum.

The total angular width of the central maximum is twice the angle to the first minimum:

2θ

. Use this to calculate the linear width of the central maximum on the screen using the small-angle approximation:

w = 2L tan(θ)

, where

L

is the distance to the screen (2.0 m). For small angles,

tan(θ) ≈ sin(θ)

, so

w ≈ 2L sin(θ)

Substitute the known values into the formula:

w ≈ 2(2.0 \, m)(1/200)

. Simplify to find the width of the central maximum in meters, then convert to millimeters by multiplying by 1000.

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Key Concepts

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

Single-Slit Diffraction

Single-slit diffraction occurs when light passes through a narrow slit and spreads out, creating a pattern of light and dark bands on a screen. The width of the central maximum is influenced by the slit width and the wavelength of the light, leading to a characteristic diffraction pattern that can be analyzed mathematically.

Wavelength and Slit Width Relationship

In diffraction, the relationship between the wavelength of light and the slit width is crucial. When the slit width is significantly larger than the wavelength, the diffraction pattern becomes less pronounced. In this case, with the slit width being 200 times the wavelength, the central maximum will be relatively narrow, affecting the overall spread of the light on the screen.

Angular Width of the Central Maximum

The angular width of the central maximum in a single-slit diffraction pattern can be calculated using the formula θ = λ/a, where λ is the wavelength and a is the slit width. This angle helps determine the physical width of the central maximum on a screen placed at a distance, allowing for practical calculations in experimental setups.

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