/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 32 A laser beam of wavelength \(\la... [FREE SOLUTION] | 91Ó°ÊÓ

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A laser beam of wavelength \(\lambda=632.8 \mathrm{nm}\) shines at normal incidence on the reflective side of a compact disc. (a) The tracks of tiny pits in which information is coded onto the CD are \(1.60 \mu \mathrm{m}\) apart. For what angles of reflection (measured from the normal) will the intensity of light be maximum? (b) On a DVD, the tracks are only \(0.740 \mu \mathrm{m}\) apart. Repeat the calculation of part (a) for the DVD.

Short Answer

Expert verified
The angle of maximum intensity for the CD is approximately 24 degrees and for the DVD is approximately 49 degrees.

Step by step solution

01

Calculate the angle of reflection for the CD.

To find the angles of maximum intensity for a CD, we will use the formula for constructive interference: \(m\lambda = d\sin(\Theta)\). The order \(m=1\) corresponds to first-order maxima. Therefore the angle of maximum intensity \(\Theta\) can be calculated as: \(\Theta = \sin^{-1}(\frac{m\lambda}{d}) = \sin^{-1}(\frac{1*632.8*10^{-9}}{1.60*10^{-6}})\)
02

Calculate the angle of reflection for the DVD.

For a DVD, again we will use the formula for constructive interference: \(m\lambda = d\sin(\Theta)\) with the order \(m=1\) for first maximum. Consequently, the angle of maximum intensity \(\Theta\) can be calculated as: \(\Theta = \sin^{-1}(\frac{m\lambda}{d}) = \sin^{-1}(\frac{1*632.8*10^{-9}}{0.740*10^{-6}})\)
03

Calculate both angles using a scientific calculator

Now we calculate the inverse sine of both expressions obtained in Step 1 and Step 2. This will give us angles in degrees.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Laser Beam
A laser beam is a highly concentrated beam of light consisting of photons that travel in the same direction. What makes laser light unique is its coherence and monochromatic nature. "Monochromatic" means that the light is of a single color or wavelength, and "coherent" implies that all the light waves are in phase with each other.

This uniformity allows lasers to create precise beams that can travel over long distances without spreading out significantly. Lasers are used in a variety of applications including communication, medicine, and, as in this exercise, reading data from optical discs like compact discs (CDs) and digital versatile discs (DVDs).
  • Highly directional – Most laser light travels in a direct line.
  • Minimal spread – Laser beams stay focused over a long distance.
  • Used in technology for reading data.
A laser's ability to precisely target tiny tracks on a CD or DVD is what makes it invaluable for reading and writing data.
Constructive Interference
Constructive interference is a principle that occurs when two or more light waves meet and coincide in such a way that they reinforce each other, resulting in increased intensity.

In the context of optical discs, it refers to laser light reflecting off the disc's surface and interacting in a manner that leads to bright spots or enhanced intensity. This happens when the conditions are just right for the waves to combine their crests and troughs in a way that amplifies their effect.
  • Occurs when the path difference between light waves is a multiple of the wavelength, specifically for the equation: \(m\lambda = d\sin(\Theta)\).
  • The "m" in the equation represents the order of the maximum; for the first order, \(m = 1\).
  • This principle is vital for determining the angles of maximum light intensity.
Reflection Angle
The reflection angle is an angle at which light reflects off a surface relative to the normal, which is an imaginary line perpendicular to the surface at the point of incidence.

In this exercise, finding the angle of reflection where constructive interference occurs is key to seeing maximum light intensity.

This is achieved using the formula \(\Theta = \sin^{-1}(\frac{m\lambda}{d})\). Here, \(\Theta\) is the angle we're trying to find.
  • Normal incidence means the laser hits at a 90-degree angle to the surface.
  • The reflection angle ensures that light behaves predictably, making it easier to design systems like optical drives.
  • Different materials may slightly alter the reflection angle due to their refractive indices, but here we assume ideal conditions for simplicity.
Wavelength
The wavelength of light is the distance between successive crests of a wave. For visible light such as a laser beam, this can be measured in nanometers (nm).

In this problem, the laser used has a wavelength of 632.8 nm. This is particularly relevant as the wavelength must match the conditions for constructive interference.
  • A shorter wavelength means higher energy and sometimes more precision.
  • The specified wavelength directly influences the interference and diffraction patterns on a CD or DVD.
  • Knowing the wavelength is crucial for calculating reflective properties and thus reading optical discs.
Compact Disc
A compact disc (CD) is an optical disc used to store data. It makes use of tiny pits and lands, which are ridges and flat spots, to encode digital information.

The tracks of pits are separated by a distance known as the track pitch. In the exercise, this distance is pivotal for determining the angles of maximum intensity for a laser beam reflecting off the CD's surface.
  • Information is read by laser light that reflects off the CD's mirrored surface.
  • The CD's standard track separation is around 1.60 μm, specifically relevant to our calculations.
  • Compact discs were revolutionary for data storage and are still used, albeit less frequently, today.

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Most popular questions from this chapter

On December 26, 2004, a violent earthquake of magnitude 9.1 occurred off the coast of Sumatra. This quake triggered a huge tsunami (similar to a tidal wave) that killed more than 150,000 people. Scientists observing the wave on the open ocean measured the time between crests to be \(1.0 \mathrm{~h}\) and the speed of the wave to be \(800 \mathrm{~km} / \mathrm{h}\). Computer models of the evolution of this enormous wave showed that it bent around the continents and spread to all the oceans of the earth. When the wave reached the gaps between continents, it diffracted between them as through a slit. (a) What was the wavelength of this tsunami? (b) The distance between the southern tip of Africa and northern Antarctica is about \(4500 \mathrm{~km},\) while the distance between the southern end of Australia and Antarctica is about \(3700 \mathrm{~km}\). As an approximation, we can model this wave's behavior by using Fraunhofer diffraction. Find the smallest angle away from the central maximum for which the waves would cancel after going through each of these continental gaps.

Light of wavelength \(633 \mathrm{nm}\) from a distant source is incident on a slit \(0.750 \mathrm{~mm}\) wide, and the resulting diffraction pattern is observed on a screen \(3.50 \mathrm{~m}\) away. What is the distance between the two dark fringes on either side of the central bright fringe?

When laser light of wavelength \(632.8 \mathrm{nm}\) passes through a diffraction grating, the first bright spots occur at \(\pm 17.8^{\circ}\) from the central maximum. (a) What is the line density (in lines/cm) of this grating? (b) How many additional bright spots are there beyond the first bright spots, and at what angles do they occur?

The VLBA (Very Long Baseline Array) uses a number of individual radio telescopes to make one unit having an equivalent diameter of about \(8000 \mathrm{~km}\). When this radio telescope is focusing radio waves of wavelength \(2.0 \mathrm{~cm}\), what would have to be the diameter of the mirror of a visible-light telescope focusing light of wavelength \(550 \mathrm{nm}\) so that the visible-light telescope has the same resolution as the radio telescope?

Although we have discussed single-slit diffraction only for a slit, a similar result holds when light bends around a straight, thin object, such as a strand of hair. In that case, \(a\) is the width of the strand. From actual laboratory measurements on a human hair, it was found that when a beam of light of wavelength \(632.8 \mathrm{nm}\) was shone on a single strand of hair, and the diffracted light was viewed on a screen \(1.25 \mathrm{~m}\) away, the first dark fringes on either side of the central bright spot were \(5.22 \mathrm{~cm}\) apart. How thick was this strand of hair?

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