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Chapter 36: Problem 38

Perhaps to confuse a predator, some tropical gyrinid beetles (whirligig beetles) are colored by optical interference that is due to scales whose alignment forms a diffraction grating (which scatters light instead of transmitting it). When the incident light rays are perpendicular to the grating, the angle between the first-order maxima (on opposite sides of the zeroth-order maximum) is about \(24^{\circ}\) in light with a wavelength of \(550 \mathrm{~nm}\). What is the grating spacing of the beetle?

Short Answer

Expert verified
The grating spacing is approximately \(2.645 \times 10^{-6}\) meters.

Step by step solution

01

Understanding the Problem

We need to find the grating spacing, which is the distance between lines or grooves on the diffraction grating. Given data includes the angle for the first-order maxima and the wavelength of the light.
02

Using the Diffraction Grating Formula

The formula for diffraction grating is given by \( d \sin \theta = m \lambda \), where \( d \) is the grating spacing, \( \theta \) is the angle of the maxima, \( m \) is the order of the maximum, and \( \lambda \) is the wavelength of light.
03

Identifying the Zero and First Order Maxima

The problem states the angle between the first-order maxima on opposite sides of the zeroth-order maximum is \(24^{\circ}\). Thus, the angle \( \theta \) for one side is \(12^{\circ}\), since 24 divided by 2 equals 12.
04

Substituting Values into the Formula

Set \( m = 1 \) for the first-order maximum and \( \lambda = 550 \) nm (or \(550 \times 10^{-9}\) meters). The angle \( \theta \) is \(12^{\circ}\). The equation becomes:\[ d \sin 12^{\circ} = 1 \times 550 \times 10^{-9} \]
05

Solving for Grating Spacing (d)

Rearrange the equation to solve for \( d \):\[ d = \frac{550 \times 10^{-9}}{\sin 12^{\circ}} \].Calculate \( \sin 12^{\circ} \), substitute in the value, and compute \(d\).
06

Final Calculation

Calculating \( \sin 12^{\circ} \approx 0.2079 \), substitute this back into the equation:\[ d = \frac{550 \times 10^{-9}}{0.2079} \approx 2.645 \times 10^{-6} \text{ meters} \].

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Key Concepts

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

Optical Interference
Optical interference is a fascinating phenomenon that occurs when waves, such as light, overlap and combine. The interference can be constructive or destructive, depending on the phase and amplitude of the interacting waves. Constructive interference occurs when the crest of one wave overlaps with the crest of another, resulting in a brighter intensity. On the contrary, destructive interference happens when the crest of one wave meets the trough of another, leading to cancellation and producing darker areas.
This concept explains why certain materials or objects might display iridescent colors, such as the scales on a whirligig beetle. By forming a diffraction grating, these scales generate optical interference patterns visible to our eyes. When light hits the surface perpendicular to the grating, specific colors intensify due to constructive interference, creating the beetle's unique coloration.
Wavelength of Light
The wavelength of light is a crucial factor in understanding optical phenomena, such as diffraction and interference. It defines the distance between consecutive peaks of a wave and determines the light's color within the visible spectrum. For example, light with a wavelength of 550 nm appears green-yellow to the human eye.
In the scenario of a diffraction grating like that found on the scales of a beetle, the wavelength plays a key role. As light waves interact with the grating, different wavelengths will diffract at different angles, creating a spectrum of colors like a rainbow. Calculating phenomena, such as the grating spacing, involves considering the wavelength, angle of diffraction, and orders of maxima, as seen in the beetle example.
Light Scattering
Light scattering is the process by which small particles or surfaces cause electromagnetic waves to change direction. This can happen through reflection, refraction, or diffraction. Scattering is responsible for various natural phenomena, such as the blue color of the sky, which is due to the scattering of shorter (blue) wavelengths of sunlight.
In the case of whirligig beetles, their scales act as a diffraction grating, causing light to scatter. When the light does not pass directly through but instead is diffracted at specific angles, it creates visible interference patterns. These patterns, due to light scattering, are why the beetles can confuse predators with their shimmering appearance.
Physics Problem Solving
Physics problem solving is essential in identifying the underlying principles of complex phenomena, such as diffraction grating. When approaching such a problem, follow these steps:
  • First, fully understand the problem by identifying what is given and what needs to be found.
  • Second, apply the relevant equations, like the diffraction grating formula: \[ d \sin \theta = m \lambda \]
  • Next, identify all variables, such as angles and wavelengths, and transform them into useful units*.
  • Finally, solve the equation step-by-step, substituting known values and solving for unknowns, like grating spacing \(d\).
Breaking the problem into comprehensible steps ensures an analytical approach and leads to accurate solutions.

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Most popular questions from this chapter

Light of wavelength \(420 \mathrm{~nm}\) is incident normally on a diffraction grating. Two adjacent maxima occur at angles given by \(\sin \theta=0.2\) and \(\sin \theta=0.3\). The fourth-order maxima are missing. (a) What is the separation between adjacent slits? (b) What is the smallest slit width this grating can have? For that slit width, what are the (c) largest, (d) second largest, and (e) third largest values of the order number \(m\) of the maxima produced by the grating?

An \(x\)-ray beam of a certain wavelength is incident on an \(\mathrm{NaCl}\) crystal, at \(30.0^{\circ}\) to a certain family of reflecting planes of spacing \(37.6 \mathrm{pm}\). If the reflection from those planes is of the first order, what is the wavelength of the \(x\) rays?

Suppose that the central diffraction envelope of a double-slit diffraction pattern contains 13 bright fringes and the first diffraction minima eliminate (are coincident with) bright fringes. How many bright fringes lie between the second and third minima of the diffraction envelope?

Find the separation of two points on the Moon's surface that can just be resolved by the \(200 \mathrm{in} .(=5.1 \mathrm{~m})\) telescope at Mount Palomar, assuming that this separation is determined by diffraction effects. The distance from Earth to the Moon is \(3.8 \times 10^{5} \mathrm{~km}\). Assume a wavelength of \(550 \mathrm{~nm}\) for the light.

A single slit is illuminated by light of wavelengths \(\lambda_{a}\) and \(\lambda_{\text {on }}\) chosen so that the first diffraction minimum of the \(\lambda_{2}\) component coincides with the second minimum of the \(\lambda_{b}\) component. (a) If \(\lambda_{b}=420 \mathrm{~nm}\), what is \(\lambda_{\alpha}\) ? For what order number \(m_{b}\) (if any) does a minimum of the \(\lambda_{b}\) component coincide with the minimum of the \(\lambda_{a}\) component in the order number (b) \(m_{a}=2\) and (c) \(m_{a}=3\) ?
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