Problem 38 - College Physics

Question

(a) Only a limited number of maxima can be observed with a diffraction grating. The factor(s) that limit(s) the number of maxima seen is (are) (a) (1) the wavelength, (2) the grating spacing, (3) both. Explain. (b) How many maxima appear when monochromatic light of wavelength 560 nm illuminates a diffraction grating that has 10000 lines \(/ \mathrm{cm},\) and what are their order numbers?

Step-by-Step Solution

Answer

Both wavelength and grating spacing limit the maxima; orders 0, 1, -1 appear.

Step 1: Understand the Limiting Factors for Maxima

In diffraction grating, the number of observable maxima is affected by the geometry of the grating and the properties of light. The main factors that limit this number are: (1) the wavelength of the light used and (2) the spacing of the grating. These parameters are crucial as they determine the angles at which constructive interference occurs, thereby forming maxima. Both wavelength and grating spacing are thus important.

Step 2: Consider the Diffraction Equation

To find the number of observable maxima, use the diffraction grating formula:\[ d \sin \theta = m \lambda \]where \(d\) is the grating spacing, \(\theta\) is the diffraction angle, \(m\) is the order of maximum, and \(\lambda\) is the wavelength of the light. Each order \(m\) represents a possible maximum.

Step 3: Calculate the Grating Spacing

Given that the grating has 10000 lines per cm, the spacing \(d\) is calculated by inverting the line density:\[ d = \frac{1}{10000}\, \text{cm} = \frac{1 \times 10^{-2}}{10000} \, \text{m} = 1 \times 10^{-6} \text{ m}\].

Step 4: Solve for Possible Maxima

Substitute \(\lambda = 560 \text{ nm} = 560 \times 10^{-9} \text{ m}\) into the equation. Assuming \(\sin \theta \leq 1\) for realistic angles:\[ m \leq \frac{d}{\lambda} = \frac{1 \times 10^{-6}}{560 \times 10^{-9}} \approx 1.79\]The maximum integer value for \(m\) is 1, indicating possible maximas from \(m = 0, \pm 1\).

Step 5: Conclusion

Only two maxima (\(+1\) and \(-1\)) can be observed apart from the zeroth order. Order numbers of observable maxima are 0, 1, and -1.

Key Concepts

Maxima Limitations

In a diffraction grating, the observable maxima depend on factors such as the wavelength of light and the spacing between the grating lines. These elements determine the angles where constructive interference creates bright spots. Let's break it down further:

• Wavelength: Different wavelengths of light spread differently. A longer wavelength may lead to fewer maxima, as each maximum location corresponds to a specific wavelength.


• Grating Spacing: The distance between each line in a grating affects how light diffracts. A closer spacing will mean shorter paths for light to travel, impacting the number of maxima observed.

Both factors, in unison, restrict the number of maxima that can be viewed. Thus, it's both the wavelength and grating spacing that play key roles in this feature of diffraction gratings.

Diffraction Equation

The diffraction of light through a grating can be described using the diffraction equation:\[ d \sin \theta = m \lambda \]where:

• \( d \) represents the spacing between the lines on the grating.


• \( \theta \) is the angle at which the maxima occur.


• \( m \) signifies the order of the maximum.


• \( \lambda \) stands for the wavelength of the incident light.

This formula is instrumental in predicting where maxima appear on a screen. Each value of \( m \) gives a unique angle \( \theta \), leading to distinct positions for the bright spots. The equation showcases the influence of both the grating's physical characteristics and the light's properties on the resulting diffraction pattern.

Wavelength Effect

Wavelength decides how the light behaves as it passes through the diffraction grating. Longer wavelengths result in broadly spaced interference patterns, while shorter wavelengths produce more narrowly packed maxima.

• A longer wavelength creates a wider spacing between maxima, potentially reducing the number of observed bright spots.


• Conversely, a shorter wavelength leads to narrower spacing, thus allowing more maxima to be visible within the same angular range.

Thus, the choice of light wavelength can greatly affect the number and clarity of maxima that are visible when observing diffraction patterns.

Grating Spacing

Grating spacing refers to the distance between the lines of a diffraction grating. This spacing, usually represented as \( d \), is fundamental in determining diffraction patterns:

• Smaller spacing causes the light to spread more, potentially increasing the number of maxima.


• Larger spacing results in a tighter diffraction pattern, which may limit maxima to just a few visible orders.

In practice, the grating spacing is calculated by taking the inverse of the line density. For example, if a grating has 10,000 lines per cm, the spacing is \( 1 \times 10^{-6} \text{ m} \). This directly ties into the diffraction equation and impacts where maxima will appear on the detection screen.

Interference Orders

Interference orders denote the different maxima observed when light diffracts through a grating. Each integer value of \( m \) indicates a distinct order. The central maximum is always the zeroth order (\( m = 0 \)), and each subsequent integer—positive or negative—represents a new maximum order on either side.

• First-order maxima: Occur when \( m = +1 \) and \( m = -1 \), appearing symmetrically about the central maximum.


• Higher orders: These are less intense and occur at larger angles, such as \( m = +2, -2, \) and soon thereafter. Often, the number of observable orders is limited by both the wavelength and grating spacing.

Understanding these orders helps in analyzing diffraction patterns and determining the optimal conditions under which maximum diffraction clarity is obtained.