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Chapter 25: Problem 49

Light from a red laser passes through a single slit to form a diffraction pattern on a distant screen. If the width of the slit is increased by a factor of two, what happens to the width of the central maximum on the screen?

Short Answer

Expert verified
Answer: The width of the central maximum becomes half of its original value when the width of the slit is increased by a factor of two.

Step by step solution

01

Understand the basics of single-slit diffraction patterns

Single-slit diffraction is an interference pattern formed when light passes through a single, narrow slit and spreads out in a series of bright and dark fringes on a distant screen. The central maximum is the brightest part of the pattern, and its width depends on the width of the slit and the wavelength of the light.
02

Use the formula for the angular width of the central maximum

The angular width (θ) of the central maximum in a single-slit diffraction pattern is given by the formula: θ ≈ \(\frac{2λ}{a}\) where λ is the wavelength of the light and a is the width of the slit.
03

Calculate the new angular width of the central maximum with the increased slit width

Let's denote the original slit width as a and the new slit width(after increasing by a factor of two) as a'. Then, a' = 2a Now, we will plug this new slit width into the formula for the angular width of the central maximum: θ' ≈ \(\frac{2λ}{a'}\) θ' ≈ \(\frac{2λ}{2a}\) θ' ≈ \(\frac{λ}{a}\)
04

Compare the new angular width with the original angular width

Now, let's compare the new angular width (θ') with the original angular width (θ). We have, θ ≈ \(\frac{2λ}{a}\) and θ' ≈ \(\frac{λ}{a}\) Dividing θ' by θ, we get: \(\frac{θ'}{θ}\) = \(\frac{\frac{λ}{a}}{\frac{2λ}{a}}\) \(\frac{θ'}{θ}\) = \(\frac{1}{2}\) This means that the width of the central maximum on the screen will be half of what it originally was when the width of the slit is increased by a factor of two.

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Most popular questions from this chapter

A thin film of oil \((n=1.50)\) is spread over a puddle of water \((n=1.33) .\) In a region where the film looks red from directly above $(\lambda=630 \mathrm{nm}),$ what is the minimum possible thickness of the film? (tutorial: thin film).
In a double-slit experiment, what is the linear distance on the screen between adjacent maxima if the wavelength is \(546 \mathrm{nm}\), the slit separation is \(0.100 \mathrm{mm},\) and the slit-screen separation is \(20.0 \mathrm{cm} ?\)
When a double slit is illuminated with light of wavelength \(510 \mathrm{nm},\) the interference maxima on a screen \(2.4 \mathrm{m}\) away gradually decrease in intensity on either side of the 2.40 -cm-wide central maximum and reach a minimum in a spot where the fifth-order maximum is expected. (a) What is the width of the slits? (b) How far apart are the slits?
A grating has 5000.0 slits/cm. How many orders of violet light of wavelength \(412 \mathrm{nm}\) can be observed with this grating? (tutorial: grating).
A spectrometer is used to analyze a light source. The screen-to-grating distance is \(50.0 \mathrm{cm}\) and the grating has 5000.0 slits/cm. Spectral lines are observed at the following angles: $12.98^{\circ}, 19.0^{\circ}, 26.7^{\circ}, 40.6^{\circ}, 42.4^{\circ}\( \)63.9^{\circ},\( and \)77.6^{\circ} .$ (a) How many different wavelengths are present in the spectrum of this light source? Find each of the wavelengths. (b) If a different grating with 2000.0 slits/cm were used, how many spectral lines would be seen on the screen on one side of the central maximum? Explain.
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