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Chapter 26: Problem 36

\(\bullet\) When laser light of wavelength 632.8 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?

Short Answer

Expert verified
Line density: 4831 lines/cm. Two additional bright spots at 37.7掳 and 66.4掳.

Step by step solution

01

Convert the Wavelength

Convert the wavelength of the laser light from nanometers to centimeters: 632.8 nm is equivalent to \( 632.8 \times 10^{-7} \) cm, which simplifies to \( 6.328 \times 10^{-5} \) cm.
02

Identify the Diffraction Grating Equation

Use the diffraction grating formula: \( d \sin\theta = m\lambda \), where \( d \) is the distance between slits, \( \theta \) is the angle of diffraction, \( m \) is the order of the maximum, and \( \lambda \) is the wavelength in cm.
03

Solve for the Line Density

For the first bright spot, \( m = 1 \) and \( \theta = 17.8^{\circ} \). Plug these into the equation \( d \sin 17.8^{\circ} = 1 \times 6.328 \times 10^{-5} \). Solve for \( d \): \( d = \frac{6.328 \times 10^{-5}}{\sin 17.8^{\circ}} \approx 2.07 \times 10^{-4} \) cm. The line density \( n = \frac{1}{d} \approx \frac{1}{2.07 \times 10^{-4}} \approx 4831 \) lines/cm.
04

Determine the Maximum Order

Calculate the highest order \( m \) by using \( m\lambda \leq d \). Since \( d \sin 90^{\circ} = d \), \( m_{max} = \frac{d}{\lambda} \approx \frac{2.07 \times 10^{-4}}{6.328 \times 10^{-5}} \approx 3.27 \). The maximum order is 3.
05

Calculate Additional Bright Spots and Their Angles

For orders \( m = 2 \) and \( m = 3 \), use \( d \sin\theta = m\lambda \) to find the angles: - For \( m = 2 \), \( \sin\theta_2 = \frac{2 \times 6.328 \times 10^{-5}}{2.07 \times 10^{-4}} \approx 0.611 \). Therefore, \( \theta_2 \approx \sin^{-1}(0.611) = 37.7^{\circ} \).- For \( m = 3 \), \( \sin\theta_3 = \frac{3 \times 6.328 \times 10^{-5}}{2.07 \times 10^{-4}} \approx 0.916 \). Thus, \( \theta_3 \approx \sin^{-1}(0.916) = 66.4^{\circ} \). Therefore, there are two additional bright spots on either side of the central maximum.

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Key Concepts

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

Laser Light
Laser light is a special kind of light that is very different from ordinary light sources, like a light bulb.
  • It is coherent, meaning the light waves are not only aligned with each other but also remain in phase over long distances.
  • The light is monochromatic, consisting of exactly one wavelength. In this exercise, the laser light used has a wavelength of 632.8 nanometers.
  • Lasers are highly collimated, which means their beams are very narrow and do not spread much as they travel.
This makes laser light ideal for experiments with precise optical instruments like diffraction gratings. When laser light passes through a diffraction grating, it creates a pattern of dark and bright spots due to the interference of its coherent light waves.
Wavelength Conversion
Wavelength conversion is a necessary step when working with different measurement systems. Many scientific equations require consistency in units, which is why the wavelength needs to be converted from nanometers to centimeters. - 1 nanometer (nm) is equal to 1 x 10鈦烩伖 meters, which is necessary to understand as students often encounter nanometer units in optical physics. - With the laser light's wavelength of 632.8 nm, converting to centimeters involves multiplying by 10鈦烩伔, resulting in 6.328 x 10鈦烩伒 cm. This conversion allows us to use the diffraction grating formula effectively when calculating the line density and angles in the exercise.
Angles of Diffraction
The angles of diffraction are vital in understanding the behavior of light as it passes through a diffraction grating.- A diffraction grating splits and diffracts light into several beams traveling in different directions.- These directions depend on the spacing between the lines on the grating and the light鈥檚 wavelength.For our exercise:
  • The angle of diffraction for the first bright spot is given as \(17.8^{\circ}\).
  • For further bright spots, you calculate using the grating equation: \(d \sin\theta = m\lambda\), where \(m\) is the order number (1, 2, 3, etc.).
Understanding these angles helps visualize the spectrum patterns created, which is critical for applications like spectroscopy, where different wavelengths are separated for analysis.
Line Density
The line density of a diffraction grating refers to the number of lines per unit length.It is a measure of how many lines there are in a centimeter on the grating's surface, commonly expressed in lines per centimeter (lines/cm).- In our example, using the diffraction grating formula and solving for \(d\), the line density \(n\) is calculated as \(\frac{1}{d}\).For a line density of 4831 lines/cm, it indicates very closely spaced lines, crucial for achieving higher resolution in diffraction patterns.High line densities mean the grating can resolve very close wavelengths, making them preferable in applications requiring detailed spectral analysis.

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Most popular questions from this chapter

\( (f鈥� # \)\bullet$$\bullet\( Two small loudspeakers that are 5.50 \)\mathrm{m}\( apart are emitting sound in phase. From both of them, you hear a singer singing \)\mathrm{C} \\#\( (frequency 277 \)\mathrm{Hz} )\( , while the speed of sound in the room is 340 \)\mathrm{m} / \mathrm{s}\( . Assuming that you are rather far from these speak- ers, if you start out at point \)P\( equidistant from both of them and walk around the room in front of them, at what angles (measured relative to the fine from \)P$ to the midpoint between the speakers) will you hear the sound (a) maximally enhanced, (b) cancelled? Neglect any reflections from the walls.

\(\bullet\) Electromagnetic waves of wavelength 0.173 nm fall on a crystal surface. As the angle from the plane is gradually increased, starting at \(0^{\circ},\) you find that the first strong interfer- ence maximum occurs when the beam makes an angle of \(22.4^{\circ}\) with the surface of the crystal planes in the Bragg reflection. (a) What is the distance between the crystal planes? (b) At what other angles will interference maxima occur?

\(\bullet\) A thin film of polystyrene of refractive index 1.49 is used as a nonreflecting coating for Fabulite (strontium titanate) of refractive index \(2.409 .\) What is the minimum thickness of the film required? Assume that the wavelength of the light in air is 480 \(\mathrm{nm}\) .

\(\bullet\) Young's experiment is performed with light of wavelength 502 nm from excited helium atoms. Fringes are measured care- fully on a screen 1.20 \(\mathrm{m}\) away from the double slit, and the cen- ter of the 20 th fringe (not counting the central bright fringe) is found to be 10.6 \(\mathrm{mm}\) from the center of the central bright fringe. What is the separation of the two slits?

\(\bullet$$\bullet\) An oil tanker spills a large amount of oil \((n=1.45)\) into the sea \((n=1.33)\) . (a) If you look down onto the oil spill from overhead, what predominant wavelength of light do you see at a point where the oil is 380 nm thick? (b) In the water under the slick, what visible wavelength (as measured in air) is predominant in the transmitted light at the same place in the slick as in part (a)?
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