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Chapter 36: Q101P (page 1115)

Show that the dispersion of a grating is D=tan胃位

Short Answer

Expert verified

It is proved that the dispersion of a grating istan胃位

Step by step solution

01

Definition and concept of diffraction from a grating

An optical element with a periodic structure that divides light into a number of beams that move in diverse directions is known as a diffraction grating.

The angular distance 胃of the mthorder diffraction pattern produced from a grating having line separation is

dsin胃=m位 鈥�(颈)

Here, 位 is the wavelength of the incident light.

02

Showing that dispersion of a grating  D=tanθλ

The dispersion is given by

D=d胃d位

Differentiate equation (i) with respect to 位 to get

dcos胃d胃d位=md胃d位=mdcos胃

Substitute form of m from equation (i) to get

d胃d位=dsin胃位dcos胃d胃d位=tan胃位

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Most popular questions from this chapter

Light at wavelength 589 nm from a sodium lamp is incident perpendicularly on a grating with 40,000 rulings over width 76 mm. What are the first-order (a) dispersion Dand (b) resolving power R, the second-order (c) Dand (d) R,and the third-order (e) Dand (f) R?

Question:If someone looks at a bright outdoor lamp in otherwise dark surroundings, the lamp appears to be surrounded by bright and dark rings (hence halos) that are actually a circular diffraction pattern as in Fig. 36-10, with the central maximum overlapping the direct light from the lamp. The diffraction is produced by structures within the cornea or lens of the eye (hence entoptic). If the lamp is monochromatic at wavelength 550nm and the first dark ring subtends angular diameter 2.5o in the observer鈥檚 view, what is the (linear) diameter of the structure producing the diffraction?

X rays of wavelength0.12鈥�nm are found to undergo second order reflection at a Bragg angle of28掳from a lithium fluoride crystal. What is the interplanar spacing of the reflecting planes in the crystal?

In a single-slit diffraction experiment, the top and bottom rays through the slit arrive at a certain point on the viewing screen with a path length difference of 4.0 wavelengths. In a phasor representation like those in Fig 36-7, how many overlapping circles does the chain of phasors make?

Assume that the limits of the visible spectrum are arbitrarily chosen as 430and680nm. Calculate the number of rulings per millimeter of a grating that will spread the first-order spectrum through an angle of 20.0掳20.0掳
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