Problem 29 - University Physics with Modern Physics

Question

(a) What is the wavelength of light that is deviated in the first order through an angle of 13.5\(^\circ\) by a transmission grating having 5000 slits/cm? (b) What is the second-order deviation of this wavelength? Assume normal incidence.

Step-by-Step Solution

Answer

(a) 468 nm, (b) 27.9°

Step 1: Understand the Problem

We need to find the wavelength of light that is deviated in the first order and the second-order deviation angle given a transmission grating with 5000 slits/cm. The angle for the first order is 13.5°.

Step 2: Determine the Grating Equation

The grating equation is given by \( d\sin(\theta) = m\lambda \), where \( d \) is the distance between slits (grating spacing), \( \theta \) is the angle of deviation, \( m \) is the order of diffraction, and \( \lambda \) is the wavelength. For this problem, \( m = 1 \) for the first-order deviation.

Step 3: Calculate Grating Spacing

The number of slits per cm is 5000. Hence, the grating spacing \( d \) is the inverse of this number. \( d = \frac{1}{5000 \text{ cm}^{-1}} = 2 \times 10^{-4} \text{ cm} = 2 \times 10^{-6} \text{ m}.\)

Step 4: Solve for First Order Wavelength

Given \( \theta = 13.5^\circ \), convert it to radians for calculation: \( \theta = 13.5 \times \frac{\pi}{180} \). Use the equation \( d\sin(\theta) = m\lambda \). For \( m = 1 \), \( \lambda = d\sin(\theta) \). Computing gives \( \lambda = 2 \times 10^{-6} \sin(13.5^\circ) \).

Step 5: Calculate for First Order Wavelength

Compute \( \lambda = 2 \times 10^{-6} \times \sin(13.5^\circ) \approx 4.68 \times 10^{-7} \text{ m} \) or 468 nm.

Step 6: Calculate Second Order Deviation

Using the same formula, for \( m = 2 \), we rearrange to find \( \sin(\theta_2) = \frac{2 \lambda}{d} \). Substitute \( \lambda = 468\text{ nm} = 4.68 \times 10^{-7} \text{ m} \) to get \( \sin(\theta_2) = \frac{2 \times 4.68 \times 10^{-7}}{2 \times 10^{-6}} \approx 0.468 \).

Step 7: Solve for Second Order Angle

Calculate \( \theta_2 \) by taking the inverse sine. \( \theta_2 = \arcsin(0.468) \approx 27.9^\circ \).

Key Concepts

Wavelength Calculation

Wavelength calculation is a necessary step in various optical phenomena, including diffraction. In this exercise, we calculated the wavelength of light that undergoes diffraction through a grating. To find the wavelength (\(\lambda\)), we use the grating equation, but first, we need the distance between the slits, known as grating spacing (\(d\)). The spacing is the inverse of the number of slits per unit length. In this case, with 5000 slits per cm, \(d = \frac{1}{5000 \text{ cm}^{-1}} = 2 \times 10^{-6} \text{ m}\). The angle of deviation is provided in degrees, but it's useful for calculations to convert it to radians: \(\theta = 13.5 \times \frac{\pi}{180}\).Once you have these, the wavelength is calculated using \(\lambda = d\sin(\theta)\) for the first order (\(m = 1\)). Plug in the values:

• \(d = 2 \times 10^{-6} \text{ m}\)
• The sine of 13.5 degrees is used for \(\theta\)

This results in a wavelength of approximately 468 nm, which is a typical wavelength in the visible range.

Order of Diffraction

The order of diffraction is a significant concept in wave interference phenomena. It refers to the integer \(m\) in the grating equation that represents the sequence or hierarchy of the maxima.In this exercise, the "first order" implies that \(m = 1\), meaning the first level of constructive interference. For the given angle of 13.5 degrees, the calculation determines the wavelength at this initial order.The order of diffraction affects both the deviation angle and brightness of the diffracted waves. As \(m\) increases, the maxima become less intense, and diffraction patterns spread out. The second-order diffraction (\(m = 2\)) shows where this same wavelength of light deviates next. Calculations in this task were extended up to the second order to find an additional angle for this wavelength.

Grating Equation

The grating equation is the fundamental formula used to determine different properties of light as it passes through a diffraction grating. It is expressed as:\[ d\sin(\theta) = m\lambda \]where \(d\) is the grating spacing, \(\theta\) is the diffraction angle, \(m\) is the order of diffraction, and \(\lambda\) is the wavelength of light. This equation helps understand how different wavelengths are separated by a grating.For example, substituting into this equation helps calculate either the wavelength, the angle, or the order, depending on which values are known. The grating equation balances the optical characteristics and the physical arrangement of the grating, allowing for precise measurements. This equation shows why gratings are critical in applications like spectroscopy, where separating light into its component wavelengths is necessary.

Angle of Deviation

The angle of deviation is crucial in describing the behavior of light as it diffracts through a grating. It denotes the angle at which light emerges after passing through the grating slits, compared to the incident direction. In our exercise, the first order deviation occurred at an angle of 13.5 degrees. Calculating further, the second order angle was found to be approximately 27.9 degrees, showing a straightforward link between the order of diffraction and deviation angles. These angles are essential when designing optical instruments, as they dictate how separated the wavelengths become and how to align components to capture them. Notably, the angle of deviation depends not only on the order but also on the wavelength and grating spacing, as represented in the grating equation. This angle can further be fine-tuned by adjusting these parameters, enabling multiple experimental setups and analyses in optics.