Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 25: Problem 27

You are given a slide with two slits cut into it and asked how far apart the slits are. You shine white light on the slide and notice the first-order color spectrum that is created on a screen \(3.40 \mathrm{m}\) away. On the screen, the red light with a wavelength of 700 nm is separated from the violet light with a wavelength of 400 nm by 7.00 mm. What is the separation of the two slits?

Short Answer

Expert verified
Based on the given information that the distances between the primary red and violet colors on the screen is 7.00 mm, and the distance from the slide to the screen is 3.40 meters, we calculated the separation between the two slits using the double-slit interference formula. The separation between the two slits is approximately 146 micrometers.

Step by step solution

01

Understand the double-slit interference formula

The double-slit interference formula is given by: \(\sin \theta = \frac{m \lambda}{d}\) where \(\theta\) is the angle between the central maximum and the m-th order maximum on the screen, \(m\) is the order of the interference pattern, \(\lambda\) is the wavelength of light, and \(d\) is the separation between the slits.
02

Calculate the angles for red and violet light

In this problem, we are given the distances between the red and violet light on the screen and the distance from the slide to the screen. We can use the small angle approximation (\(\sin \theta \approx \tan \theta\)) to calculate the angles for red and violet light: \(\tan \theta_{red} = \frac{distance_{red}}{distance_{screen}}\) \(\tan \theta_{violet} = \frac{distance_{violet}}{distance_{screen}}\) \(\tan \theta_{red} - \tan \theta_{violet} = \frac{7.00\,\mathrm{mm}}{3.40\,\mathrm{m}}\)
03

Use the double-slit interference formula to find the separation between slits

Since we are dealing with the first-order maximum (\(m=1\)), we can substitute the wavelengths and the calculated angle difference into the double-slit interference formula: \(\frac{\sin \theta_{red} - \sin \theta_{violet}}{1} = \frac{\lambda_{red} - \lambda_{violet}}{d}\) Using the small angle approximation, we get: \(\frac{\tan \theta_{red} - \tan \theta_{violet}}{1} = \frac{\lambda_{red} - \lambda_{violet}}{d}\) Plugging in the known values, we have: \(\frac{7.00\,\mathrm{mm}}{3.40\,\mathrm{m}} = \frac{700\,\mathrm{nm} - 400\,\mathrm{nm}}{d}\) Now we can solve for the separation between the slits, \(d\).
04

Solve for the slit separation, \(d\)

Rearrange the equation to find \(d\): \(d = \frac{(700\,\mathrm{nm} - 400\,\mathrm{nm})(3.40\,\mathrm{m})}{7.00\,\mathrm{mm}}\) Perform the calculation: \(d = \frac{300\,\mathrm{nm}(3.40\,\mathrm{m})}{7.00\,\mathrm{mm}}\) \(d = 0.000146\,\mathrm{m} = 146\,\mathrm{\mu m}\)
05

Conclusion

The separation between the two slits is approximately \(146\,\mathrm{\mu m}\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Suppose a transparent vessel \(30.0 \mathrm{cm}\) long is placed in one arm of a Michelson interferometer, as in Example 25.2. The vessel initially contains air at \(0^{\circ} \mathrm{C}\) and 1.00 atm. With light of vacuum wavelength \(633 \mathrm{nm}\), the mirrors are arranged so that a bright spot appears at the center of the screen. As air is slowly pumped out of the vessel, one of the mirrors is gradually moved to keep the center region of the screen bright. The distance the mirror moves is measured to determine the value of the index of refraction of air, \(n .\) Assume that, outside of the vessel, the light travels through vacuum. Calculate the distance that the mirror would be moved as the container is emptied of air.
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.
About how close to each other are two objects on the Moon that can just barely be resolved by the \(5.08-\mathrm{m}-\) (200-in.)-diameter Mount Palomar reflecting telescope? (Use a wavelength of \(520 \mathrm{nm}\).)
The radio telescope at Arecibo, Puerto Rico, has a reflecting spherical bowl of \(305 \mathrm{m}(1000 \mathrm{ft})\) diameter. Radio signals can be received and emitted at various frequencies with appropriate antennae at the focal point of the reflecting bowl. At a frequency of \(300 \mathrm{MHz}\), what is the angle between two stars that can barely be resolved? (Tutorial:radio telescope).
In a double-slit interference experiment, the wavelength is \(475 \mathrm{nm}\), the slit separation is \(0.120 \mathrm{mm},\) and the screen is $36.8 \mathrm{cm}$ away from the slits. What is the linear distance between adjacent maxima on the screen? [Hint: Assume the small-angle approximation is justified and then check the validity of your assumption once you know the value of the separation between adjacent maxima.] (tutorial: double slit 1 ).
See all solutions

Recommended explanations on Physics Textbooks

Particle Model of Matter

Read Explanation

Fluids

Read Explanation

Geometrical and Physical Optics

Read Explanation

Turning Points in Physics

Read Explanation

Dynamics

Read Explanation

Fundamentals of Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free