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Chapter 36: Q101P (page 1115)

Show that the dispersion of a grating is D=tanθλ

Short Answer

Expert verified

It is proved that the dispersion of a grating istanθλ

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 θof the mthorder diffraction pattern produced from a grating having line separation is

dsinθ=mλ …(i)

Here, λ is the wavelength of the incident light.

02

Showing that dispersion of a grating  D=tanθλ

The dispersion is given by

D=dθdλ

Differentiate equation (i) with respect to λ to get

dcosθdθdλ=mdθdλ=mdcosθ

Substitute form of m from equation (i) to get

dθdλ=dsinθλdcosθdθdλ=tanθλ

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Most popular questions from this chapter

In three arrangements, you view two closely spaced small objects that are the same large distance from you. The angles that the objects occupy in your field of view and their distances from you are the following: (1) 2ϕand ; (2) 2ϕand 2R; (3) ϕ/2and R/2. (a) Rank the arrangements according to the separation between the objects, with the greatest separation first. If you can just barely resolve the two objects in arrangement 2, can you resolve them in (b) arrangement 1 and (c) arrangement 3?

For three experiments, Fig.36-31 gives the parameter of Eq. 36-20 versus angle in two-slit interference using light of wavelength 500 nm. The slit separations in the three experiments differ. Rank the experiments according to (a) the slit separations and (b) the total number of two slit interference maxima in the pattern, greatest first.

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’s view, what is the (linear) diameter of the structure producing the diffraction?

Estimate the linear separation of two objects on Mars that can just be resolved under ideal conditions by an observer on Earth (a) using the naked eye and (b) using the 200in.(=5.1m) Mount Palomar telescope. Use the following data: distance to Mars = 8.0×107km, diameter of pupil = 5.0mm, wavelength of light 550nm.

If Superman really had x-ray vision at 0.10nm wavelength and a 4.0mm pupil diameter, at what maximum altitude could he distinguish villains from heroes, assuming that he needs to resolve points separated by 5.0cm to do this?
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