Light from two sources, `lambda_(1)= 623 nm` and `lambda_(2) = 488 nm`, is incident on a diffraction grating that has `5550 "lines" // cm`. What is the angular separation, `lambda_(1)-lambda_(2)` of the second order maxima of the two waves?

Options

A. `11.0^(@)`
B. `25.0^(@)`
C. `15.0^(@)`
D. `32.8^(@)`

Answer

To find the angular separation of the second order maxima of two waves with wavelengths \( \lambda_1 = 623 \, \text{nm} \) and \( \lambda_2 = 488 \, \text{nm} \) incident on a diffraction grating with \( 5550 \, \text{lines/cm} \), we can follow these steps:

### Step 1: Calculate the grating spacing \( d \)
The number of lines per cm is given as \( 5550 \, \text{lines/cm} \). To find the grating spacing \( d \) (the distance between adjacent slits), we can use the formula:

\[ d = \frac{1}{\text{number of lines per cm}} = \frac{1}{5550 \, \text{lines/cm}} = \frac{1 \, \text{cm}}{5550} = \frac{10^{-2} \, \text{m}}{5550} \approx 1.80 \times 10^{-6} \, \text{m} \]

### Step 2: Use the diffraction grating formula
The diffraction grating formula is given by:

\[ d \sin \theta_n = n \lambda \]

For the second order maxima (\( n = 2 \)), we can write this for both wavelengths:

1. For \( \lambda_1 = 623 \, \text{nm} \):
\[ d \sin \theta_1 = 2 \lambda_1 \]

2. For \( \lambda_2 = 488 \, \text{nm} \):
\[ d \sin \theta_2 = 2 \lambda_2 \]

### Step 3: Calculate \( \theta_1 \)
Substituting \( \lambda_1 \) into the equation:

\[ \sin \theta_1 = \frac{2 \lambda_1}{d} = \frac{2 \times 623 \times 10^{-9} \, \text{m}}{1.80 \times 10^{-6} \, \text{m}} \approx 0.692 \]

Now, calculate \( \theta_1 \):

\[ \theta_1 = \sin^{-1}(0.692) \approx 43.7^\circ \]

### Step 4: Calculate \( \theta_2 \)
Now substituting \( \lambda_2 \):

\[ \sin \theta_2 = \frac{2 \lambda_2}{d} = \frac{2 \times 488 \times 10^{-9} \, \text{m}}{1.80 \times 10^{-6} \, \text{m}} \approx 0.541 \]

Now, calculate \( \theta_2 \):

\[ \theta_2 = \sin^{-1}(0.541) \approx 32.7^\circ \]

### Step 5: Calculate the angular separation
The angular separation \( \Delta \theta \) between the second order maxima of the two wavelengths is given by:

\[ \Delta \theta = \theta_1 - \theta_2 = 43.7^\circ - 32.7^\circ = 11.0^\circ \]

### Final Answer
The angular separation of the second order maxima of the two waves is approximately \( 11.0^\circ \).

---