Red light of wavelength 625 nm is incident normally on a optical diffraction grating with `2xx10^(5)` lines/m. Including central principal maxima, how many maxima may be observed on a screen which is far from the grating?

Options

A. 15
B. 17
C. 8
D. 16

Answer

To solve the problem of how many maxima can be observed on a screen when red light of wavelength 625 nm is incident normally on a diffraction grating with 2 x 10^5 lines/m, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Values:**
- Wavelength of red light, \( \lambda = 625 \, \text{nm} = 625 \times 10^{-9} \, \text{m} \)
- Number of lines per meter in the grating, \( n = 2 \times 10^5 \, \text{lines/m} \)

2. **Calculate the Grating Spacing (d):**
The grating spacing \( d \) is the distance between adjacent lines and can be calculated as:
\[ d = \frac{1}{n} = \frac{1}{2 \times 10^5} = 5 \times 10^{-6} \, \text{m} \]

3. **Use the Diffraction Grating Equation:**
The condition for maxima in a diffraction grating is given by:
\[ d \sin \theta = n \lambda \]
where \( n \) is the order of the maxima.

4. **Determine the Maximum Order of Maxima (n):**
Since \( \sin \theta \) cannot exceed 1, we set:
\[ d \sin \theta \leq d \]
This leads to:
\[ n \lambda \leq d \]
Rearranging gives:
\[ n \leq \frac{d}{\lambda} \]
Substituting the values:
\[ n \leq \frac{5 \times 10^{-6}}{625 \times 10^{-9}} = \frac{5}{625} \times 10^{3} = 8 \]
Thus, the maximum order \( n \) is 8.

5. **Calculate Total Number of Maxima:**
The total number of observable maxima includes the central maximum and the maxima on both sides. Therefore:
\[ \text{Total Maxima} = 2n + 1 \]
Substituting \( n = 8 \):
\[ \text{Total Maxima} = 2 \times 8 + 1 = 16 + 1 = 17 \]

### Final Answer:
The total number of maxima that may be observed on the screen is **17**.

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