If a diffraction grating produces its third-order bright...

If a diffraction grating produces its third-order bright band at an angle of 78.4$^\circ$ for light of wavelength 681 nm, find (a) the number of slits per centimeter for the grating and (b) the angular location of the first-order and second-order bright bands. (c) Will there be a fourth-order bright band? Explain.

Key Concepts

Diffraction Grating Equation

This key concept describes the fundamental equation d sin(?) = m? that governs the behavior of light when it passes through a diffraction grating. In this equation, d is the spacing between adjacent slits in the grating, ? is the angle at which a bright fringe occurs, m is the integer order of the fringe, and ? is the wavelength of the incident light. This equation underpins the analysis of the angular positions of bright bands and is crucial for understanding the interference patterns produced by the grating.

Slit Density and Grating Spacing

The slit density, typically expressed as the number of slits per unit length (for example, slits per centimeter), is directly related to the grating spacing d. Specifically, if d is determined from the diffraction equation, the number of slits per centimeter is the reciprocal of the inter-slit distance expressed in centimeters. This relationship enables the determination of the mechanical properties of the grating, such as its resolution and dispersive power.

Order of Diffraction

The order in diffraction refers to the integer multiple (m) in the grating equation that signifies the sequence of the bright bands that are observed. Higher orders correspond to larger angles of diffraction but are only possible if the condition sin(?) ? 1 is satisfied. Understanding the concept of diffraction order is essential for predicting which orders will produce observable bright bands and for explaining the limits on the maximum order that can be formed with a given grating.

Angular Limitations in Diffraction

This concept addresses the physical constraint on the sine function, specifically that sin(?) cannot exceed one. This limitation means that for a given wavelength and grating spacing, there will be a highest possible order beyond which no further bright bands can appear. This principle is used to explain why, under certain conditions, some orders (such as a potential fourth order in some cases) might not be observable if they require sin(?) to be greater than one.

Transcript

00:01 We are given for a diffraction grading.

00:04 The third order maxima occurs at an angle of 78 .4 degrees.

00:11 And the wavelength of the light used is 681 nanometers.

00:16 Now we know that the formula that governs diffraction gratings is d sine theta is equal to m lambda, where d is the distance between two slits.

00:29 But we can recast this formula in terms of the number of slits per cent, uh, per unit length using n equal to 1 by d.

00:38 If n is the number of slits per unit length, and then d is the distance between two slits.

00:44 And this is how they're related.

00:46 By using this, we can recast this formula as sine theta is equal n m lambda.

00:55 Now that we have this, for part a, we simply find out the value of n.

01:04 We know that n is equal to sine theta.

01:07 By m lambda.

01:09 So using the data we are given, sine 78 .4 degrees divided by m is 3 times 681 nanometers is 681 times 10 bar minus 9.

01:25 This comes out to be 4 .9795 into 10 bar 5.

01:32 But notice that this is an si units, and hence the units will be per meter.

01:37 To make it per centimeter, we divide this by 100.

01:42 Which gives us 4 ,800 slits per centimeter.

01:51 Notice that i have rounded this to three significant digits and you get this...

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