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Chapter 36: Problem 30

If a diffraction grating produces a third-order bright spot for red light (of wavelength 700 \(\mathrm{nm}\) ) at \(65.0^{\circ}\) from the central maximum, at what angle will the second-order bright spot be for violet light (of wavelength 400 \(\mathrm{nm} ) ?\)

Short Answer

Expert verified
The second-order angle for violet light is approximately \(20.2^{\circ}\).

Step by step solution

01

Identify the formula to use

In a diffraction grating, the position of the bright spots (maxima) is determined by the grating equation: \[ d \sin(\theta) = m\lambda \]where \(d\) is the distance between the slits (grating spacing), \(\theta\) is the angle of the maxima, \(m\) is the order of the maxima, and \(\lambda\) is the wavelength of the light.
02

Solve for grating spacing using red light

We know that for the third-order bright spot of red light (\(\lambda = 700\,\text{nm}\)), the angle \(\theta\) is \(65.0^{\circ}\). Thus, the formula becomes:\[ d \sin(65.0^{\circ}) = 3 \times 700 \times 10^{-9}\,\text{m} \]Solve for \(d\):\[ d = \frac{3 \times 700 \times 10^{-9}}{\sin(65.0^{\circ})}\,\text{m} \]
03

Calculate the second-order angle for violet light

Using the grating spacing \(d\) found in Step 2, replace the wavelength with violet light (\(\lambda = 400\,\text{nm}\)) and order with 2:\[ d \sin(\theta) = 2 \times 400 \times 10^{-9}\,\text{m} \]Using the same \(d\) value, solve for \(\theta\) using:\[ \sin(\theta) = \frac{2 \times 400 \times 10^{-9}}{d} \]Find \(\theta\) by calculating the inverse sine.
04

Compute final angle

Plug the value of \(d\) from Step 2 into the equation from Step 3. Let's calculate it step-by-step for better understanding. Using the calculator we find:- Compute \(\sin(65.0^{\circ}) \approx 0.9063\)- Then, \(d = \frac{3 \times 700 \times 10^{-9}}{0.9063} \approx 2.317 \times 10^{-6}\,\text{m}\)- Next, \( \sin(\theta) = \frac{2 \times 400 \times 10^{-9}}{2.317 \times 10^{-6}} \approx 0.3446 \)- Finally, \(\theta \approx \sin^{-1}(0.3446) = 20.2^{\circ}\)

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Key Concepts

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

Grating Equation
A diffraction grating works by having numerous slits that separate light into different components through interference. The pattern of light spread, or maxima, is determined by the grating equation: \[ d \sin(\theta) = m\lambda \]- **\(d\)** is the distance between consecutive slits or the grating spacing.- **\(\theta\)** is the angle at which you observe a particular maxima.- **\(m\)** is the order number of the maxima, indicating how many wavelengths fit into the path difference.- **\(\lambda\)** is the wavelength of the light.The grating equation is essential for predicting where different colors of light will appear when encountering a grating. Expanding our understanding, this equation helps in practical experiments where precise measurements of light dispersion are needed. Diving into our exercise, the third-order maxima for red light appears at an angle, using this equation we solve for the grating spacing \(d\) which is crucial for predicting the behavior of other wavelengths.
Order of Maxima
When light shines through a diffraction grating, it creates patterns of light and dark called maxima and minima. The order of maxima \(m\) shows how many full wavelengths separate each bright spot. These orders are labeled as first-order, second-order, and so on:- **First-order maxima (\(m=1\))** occurs where the path difference is equal to one full wavelength.- **Second-order maxima (\(m=2\))** where this difference becomes two full wavelengths.- Similarly, for third-order maxima (\(m=3\)), the difference is three wavelengths.Each of these orders results in light appearing at specific angles. Higher orders occur at greater angles and are typically fainter due to spreading over a larger area. In our problem, the shift in order from three to two signifies a tighter pattern for the same initial grating setup, highlighting how essential it is to understand this concept when analyzing diffraction outcomes.
Wavelength
Wavelength (\(\lambda\)) is a fundamental property of light, playing a crucial role in diffraction and interference. It is the distance over which the wave's shape repeats. In the context of diffraction gratings, different wavelengths separate at distinct angles due to their differing path lengths:- **Red light** has longer wavelengths, typical around 700 nm and above.- **Violet light** has shorter wavelengths, generally around 400 nm or lower.Because wavelength affects the angle (\(\theta\)) in diffraction patterns, shorter wavelengths like violet will produce maxima at smaller angles compared to longer wavelengths like red. In our exercise, understanding how red and violet light differ in behavior with the same grating helps explain the smaller angle for violet light in the second-order than the third-order of red light. This showcases how essential precise wavelength measurements are in various scientific and practical applications, including spectrometry and optical communications.

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Most popular questions from this chapter

An interference pattern is produced by light of wave- length 580 \(\mathrm{nm}\) from a distant source incident on two identical parallel slits separated by a distance (between centers) of 0.530 \(\mathrm{mm}\) . (a) If the slits are very narrow, what would be the angular positions of the first-order and second- order, two-slit, interference maxima? (b) Let the slits have width 0.320 \(\mathrm{mm}\) . In terms of the intensity \(I_{0}\) at the center of the central maximum, what is the intensity at each of the angular positions in part (a)?

In a large vacuum chamber, monochromatic laser light passes through a narrow slit in a thin aluminum plate and forms a diffraction pattern on a screen that is 0.620 \(\mathrm{m}\) from the slit. When the aluminum plate has a temperature of \(20.0^{\circ} \mathrm{C},\) the width of the central maximum in the diffraction pattern is 2.75 \(\mathrm{mm}\) . What is the change in the width of the central maximum when the temperature of the plate is raised to \(520.0^{\circ} \mathrm{C}\) ? Does the width of the central diffraction maximum increase or decrease when the temperature is increased?

Monochromatic light from a distant source is incident on a slit 0.750 \(\mathrm{mm}\) wide. On a screen 2.00 \(\mathrm{m}\) away, the distance from the central maximum of the diffraction pattern to the first minimum is measured to be 1.35 \(\mathrm{mm}\) . Calculate the wavelength of the light.

Monochromatic \(\mathrm{x}\) rays are incident on a crystal for which the spacing of the atomic planes is 0.440 \(\mathrm{nm} .\) The first-order maximum in the Bragg reflection occurs when the incident and reflected \(x\) rays make an angle of \(39.4^{\circ}\) with the crystal planes. What is the wavelength of the \(\mathrm{x}\) rays?

If you can read the bottom row of your doctor's eye chart, your eye has a resolving power of 1 arcminute, equal to \(\frac{1}{60}\) degree. If this resolving power is diffraction limited, to what effective diameter of your eye's optical system does this correspond? Use Rayleigh's criterion and assume \(\lambda=550 \mathrm{nm} .\)
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