Question

You are making a diffraction grating that is required to separate the two spectral lines in the sodium \(D\) doublet, at wavelengths 588.9950 and \(589.5924 \mathrm{nm}\), by at least \(2.00 \mathrm{~mm}\) on a screen that is \(80.0 \mathrm{~cm}\) from the grating. The lines are to be ruled over a distance of \(1.50 \mathrm{~cm}\) on the grating. What is the minimum number of lines you should have on the grating?

Short answer

Based on the given information about the sodium doublet and the distances involved, we have calculated the minimum number of lines on a diffraction grating required to separate the two spectral lines by at least 2.00 mm on an 80.0 cm distant screen. The minimum number of lines is approximately 6275.

Step-by-step solution

List the given information

We have the following data: 1. \(\lambda_1 = 588.9950 \mathrm{nm}\) 2. \(\lambda_2 = 589.5924 \mathrm{nm}\) 3. Separation on the screen, \(X = 2.00 \mathrm{mm}\) 4. Distance between screen and grating, \(D = 80.0 \mathrm{~cm}\) 5. Distance ruled on the grating, \(L = 1.50 \mathrm{~cm}\)

Calculate the angular separation

The angular separation (\(\Delta \theta\)) between the two wavelengths can be found using the small angle approximation since \(\tan(\Delta \theta) \approx \sin(\Delta \theta) \approx \Delta \theta\) for small angles. So, \(\Delta \theta = \frac{X}{D} = \frac{2.00 \mathrm{mm}}{80.0 \mathrm{~cm}} = 0.0025 \mathrm{rad}\)

Utilize the diffraction grating equation

The diffraction grating equation tells us the relationship between the wavelength (\(\lambda\)), ruling distance (\(d\)), and angle (\(\theta\)) for the maximums in the interference pattern: \(m\lambda = d\sin(\theta)\) where \(m\) is the order number.

Calculate the change in the wavelength

Given that the angular separation is small, we can assume that both wavelengths will have the same order (\(m\)) and the diffraction grating will have the same ruling distance for both wavelengths (\(d\)). We can then express the change in angle (\(\Delta \theta\)) in terms of the change in their wavelengths (\(\Delta \lambda\)). \(m(\lambda_1 + \Delta \lambda) = d(\sin(\theta + \Delta \theta))\) Considering the small angle approximation, we can write the equation as: \(m(\lambda_1 + \Delta \lambda) \approx d(\theta + \Delta \theta)\) Now subtract the grating equation for \(\lambda_1\): \(m\lambda_1 = d\theta\) This gives us: \(m\Delta \lambda \approx d\Delta \theta\) Solving for \(m\), we get: \(m = \frac{d\Delta \theta}{\Delta \lambda}\)

Calculate the number of lines required

The number of lines (\(N\)) on the grating is related to the ruling distance (\(d\)) as: \(d = \frac{L}{N}\), where \(L = 1.50 \mathrm{~cm}\). Substituting this into the equation for \(m\), we have: \(m = \frac{\frac{L}{N} \Delta \theta}{\Delta \lambda}\) Now, we need to solve for \(N\): \(N = \frac{L \Delta \theta}{m\Delta \lambda}\)

Find the minimum number of lines on the grating

We can now find the minimum number of lines by substituting the given values into the above equation: \(N = \frac{(1.50 \mathrm{~cm}) (0.0025 \mathrm{rad})}{(1)(589.5924 \mathrm{nm} - 588.9950 \mathrm{nm})} = \frac{0.00375}{0.0000005974}\) \(N \approx 6275\) To ensure that the spectral lines are separated by at least 2.00 mm on the screen, we should have a minimum of 6275 lines on the diffraction grating.

Key concepts

Spectral Lines Separation

The process of distinguishing two closely spaced spectral lines, such as those seen in the sodium D doublet, is known as spectral lines separation. The objective is to achieve a clear resolution of these lines on a screen using a diffraction grating.

The separation depends on the diffraction grating's ability to disperse light into its component wavelengths, producing distinct patterns for each line. The more lines per unit distance (lines/mm) on the diffraction grating, the greater the angular separation between the two lines. However, increasing the number of lines on the grating also requires more precise manufacturing to maintain uniform spacing.

Using the provided exercise, a significant concept is the angular separation, denoted as \(\text{\Delta}\theta\), which can be calculated given the distance from the grating to the screen, and the desired separation on the screen. This calculated angular distance is critical in determining the resolution of the spectral lines and is inherently connected to the number of lines on the grating—the more lines, the greater the separation and consequently, the resolution.

Wavelength Calculation

Understanding how to calculate the wavelength of light using a diffraction grating is central to interpreting diffraction patterns. The core equation that governs the relationship between the wavelength of light and the diffraction pattern is the grating equation:

\(m\lambda = d\sin(\theta)\)

This equation links the observed wavelength \(\lambda\) to the distance between adjacent grating lines, \(d\), and the angle \(\theta\) at which the diffracted light is observed.

Determining the Finer Details

Even small differences in wavelengths can result in different angles of diffraction, emphasizing the grating’s ability to separate them into distinct lines. The exercise shows that to resolve the close wavelengths in the sodium D doublet, the precise calculation of \(m\), or the order number, and \(d\), the grating spacing, is crucial. Solving such problems often involves applying approximations for small angles, making the math more manageable and focusing on the relationship between the physical properties and the observed spectral pattern.

Interference Pattern

An interference pattern arises when waves overlap and either reinforce or cancel each other out. In the context of diffraction gratings, the overlapping waves are those of light, and the resulting pattern is a spectrum of bright and dark areas — these bright areas are the maxima, and they are where the light waves have constructively interfered.

The condition for constructive interference is that the path difference between the light from adjacent slits equals an integer multiple of the wavelength. As a result, for each wavelength of light, the interference pattern will be at a slightly different angle, leading to the separation of different colors or spectral lines on a screen.

Implications on Observations

In the exercise, the goal was to calculate the grating necessary to achieve a specific separation of spectral lines, which ultimately boils down to manipulating and measuring the interference pattern. By understanding the principles of wave interference, one can better comprehend how diffraction patterns are formed and predict the behavior of light as it interacts with various gratings, allowing for accurate measurements and the effective separation of wavelengths.