Ch. 35 - Diffraction

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 35 - Diffraction

Problem 69

Chapter 34, Problem 69

What is the highest spectral order that can be seen if a grating with 6800 slits per cm is illuminated with 633-nm laser light? Assume normal incidence.

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Calculate the slit spacing (d) by taking the reciprocal of the number of slits per unit length. Since there are 6800 slits per cm, convert this to meters for standard SI units.

Use the grating equation for normal incidence, which is given by \(m \lambda = d \sin \theta\). Here, \(m\) is the order of the spectrum, \(\lambda\) is the wavelength of the light, and \(\theta\) is the angle of diffraction. For normal incidence, \(\theta\) is the angle relative to the normal of the grating.

Since the maximum angle \(\theta\) can be is 90 degrees (beyond which no light is diffracted), set \(\sin \theta = 1\) in the grating equation to find the maximum possible order \(m\).

Substitute the values of \(\lambda\) (633 nm converted to meters) and \(d\) into the modified grating equation \(m \lambda = d\) to solve for \(m\).

The highest possible order, \(m\), is the largest integer value that satisfies the equation without exceeding the value calculated in the previous step.

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Key Concepts

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

Diffraction Grating

A diffraction grating is an optical component with a periodic structure that disperses light into its component wavelengths. The number of slits per unit length determines the grating's resolving power and the angles at which different wavelengths are diffracted. In this case, the grating has 6800 slits per cm, which affects the maximum order of diffraction that can be observed.

Order of Diffraction

The order of diffraction refers to the integer multiples of the wavelength that can be observed in the diffraction pattern produced by a grating. The first order (m=1) corresponds to the angle where the first maximum occurs, and higher orders (m=2, m=3, etc.) correspond to subsequent maxima. The maximum observable order is limited by the grating equation and the wavelength of the light used.

Grating Equation

The grating equation, given by d sin(θ) = mλ, relates the angle of diffraction (θ), the wavelength of light (λ), the grating spacing (d), and the order of diffraction (m). Here, d is the distance between adjacent slits, which can be calculated from the number of slits per cm. This equation is essential for determining the maximum order of diffraction that can be achieved with a given wavelength and grating.

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Textbook Question

(II) X-rays of wavelength 0.138 nm fall on a crystal whose atoms, lying in planes, are spaced 0.315 nm apart. At what angle Φ (relative to the surface, Fig. 35–28) must the X-rays be directed if the first diffraction maximum is to be observed?

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Textbook Question

A slit of width D = 22 μm is cut through a thin aluminum plate. Light with wavelength λ = 620nm passes through this slit and forms a single-slit diffraction pattern on a screen a distance ℓ = 2.0 m away. Defining x to be the distance between the two first minima on either side of the center in this diffraction pattern ( m = +1 and m = -1), find the change ∆x in this distance when the temperature T of the metal plate is changed by an amount ∆T = 55 C°. [Hint: Since λ ≪ D, the first minima occur at a small angle.]

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Textbook Question

You want to design a spy satellite to photograph license plate numbers. Assuming it is necessary to resolve points separated by 2 cm with 550-nm light, and that the satellite orbits at a height of 130 km, what minimum lens aperture (diameter) is required?

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Textbook Question

X-rays of wavelength 0.10 nm fall on a microcrystalline powder sample. The sample is located 15 cm from a photographic sensor. The crystal structure of the sample has an atomic spacing of 0.22 nm. Calculate the radii of the diffraction rings corresponding to first- and second-order scattering. Note in Fig. 35–28 that the X-ray beam is deflected through an angle 2Φ.

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Textbook Question

(II) (a) Suppose for a conventional X-ray image that the X-ray beam consists of parallel rays. What would be the magnification of the image? (b) Suppose, instead, that the X-rays come from a point source (as in Fig. 35–31) that is 15 cm in front of a human body which is 25 cm thick, and the film is pressed against the person’s back. Determine and discuss the range of magnifications that result.

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