Problem 64 - Physics

Question

Concept Questions (a) Two diffraction gratings are located at the same distance from observation screens. Light with the same wavelength \(\lambda\) is used for each. The principal maxima of grating A are observed to be closer together on the screen than the principal maxima of grating B. Which grating diffracts the light to a greater extent? (b) Which grating has the smaller slit separation \(d ?\) (c) Which grating has the greater number of lines permeter? Justify each of your answers. Problem The separation between adjacent principal maxima for grating \(\mathrm{A}\) is \(2.7 \mathrm{~cm},\) and for grating \(\mathrm{B}\) it is \(3.2 \mathrm{~cm}\). Grating A has 2000 lines per meter. How many lines per meter does grating B have? The diffraction angles are small enough that \(\sin \theta \approx \tan \theta .\) Be sure that your answer is consistent with your answers to the Concept Questions.

Step-by-Step Solution

Answer

Grating B diffracts light more, has smaller slit separation and more lines per meter (approximately 2300 lines/m).

Step 1: Analyzing Diffraction Extent

The principal maxima are closer together for grating A, which indicates it diffracts less. Therefore, grating B diffracts to a greater extent as its maxima are further apart, suggesting larger angles of diffraction.

Step 2: Relating Slit Separation to Diffraction

According to the grating equation \(d \sin(\theta) = m \lambda\), a smaller slit separation \(d\) results in larger angles \(\theta\) for principal maxima, and thus further spaced maxima. Since B's maxima are further apart, grating B must have a smaller slit separation.

Step 3: Determining Lines per Meter

The number of lines per meter \(N\) and slit separation \(d\) are related by \(d = \frac{1}{N}\). Therefore, if B has a smaller slit separation, it has more lines per meter compared to grating A.

Step 4: Using Experimental Data for Grating B

Given grating A has 2000 lines/m, and the difference in separation of maxima is given, use the formula \(\Delta x = \frac{\lambda L}{d}\). Rearrange to find \(d_B\):\[ d_B = \frac{d_A \cdot \Delta x_B}{\Delta x_A} \] where \(d_A = \frac{1}{2000}\). Substitute \(\Delta x_A = 2.7\, \text{cm}\) and \(\Delta x_B = 3.2\, \text{cm}\).

Step 5: Calculate Number of Lines for Grating B

Finding \(d_B\), use it to calculate \(N_B = \frac{1}{d_B}\) for the number of lines per meter for grating B.

Key Concepts

Principal Maxima

In the context of diffraction gratings, **principal maxima** refer to the bright spots where constructive interference of light waves occurs. These maxima appear on a screen after light passes through a diffraction grating. The location of these maxima can be determined by using the equation: \[ d \sin(\theta) = m \lambda \] where:

• \(d\) is the slit separation,
• \(\theta\) is the angle of diffraction,
• \(m\) is the order of the maxima (an integer),
• \(\lambda\) is the wavelength of the light.

In a practical scenario, the position of principal maxima is observed as light and dark fringes on a screen.
The key point is that closer principal maxima indicate smaller angles of diffraction. This means the grating does not diffract light as much. Conversely, when maxima are further apart, it indicates greater diffraction.
By observing the distance between these bright spots, we can infer the diffraction behavior of the grating being examined.

Slit Separation

Slit separation, denoted as \(d\), is the distance between adjacent slits in a diffraction grating. It plays a crucial role in determining the diffraction pattern observed. In a diffraction experiment, the smaller the slit separation, the larger the angle \(\theta\) at which lines appear. This is due to the grating condition equation: \[ d \sin(\theta) = m \lambda \]
This equation implies that for a fixed wavelength \(\lambda\), a smaller \(d\) results in larger angles, causing the maxima (bright spots) to spread further apart on the observing screen. Hence, if a grating causes principal maxima to appear further apart, it likely has a smaller slit separation. Grating B, with its more widely spaced maxima than Grating A, would thus have a smaller \(d\). Understanding this concept helps in analyzing and comparing diffraction gratings based on their physical spacings.

Lines Per Meter

The concept of **lines per meter** is a way to describe the density or compactness of lines in a diffraction grating. It is the inverse of the slit separation \(d\), expressed as: \[ N = \frac{1}{d} \] where \(N\) represents lines per meter. A grating with a higher number of lines per meter has more closely spaced slits.
This increased density results in greater diffraction, as more lines create a larger cumulative interference effect. Therefore, a greater number of lines per meter will align with grating producing wider spaced maxima.
To find the number of lines per meter for a grating, if you know the slit separation from other physical parameters, just invert \(d\). For instance, if Grating B requires calculation, first determine \(d_B\) based on known parameters of Grating A and their separation differences, then use the above relationship to calculate \(N_B\).
Given the relationships covered, Grating B having maxima further apart reflects its higher lines per meter compared to Grating A, which has known 2000 lines/m.