Interference fringes are obtained due to interference of waves from two coherent sources of light having amplitude `a_1` and `2a_1`. The ratio of maximum and minimum intensities in the interference pattern will be

Options

A. 2
B. 3
C. 4
D. 9

Answer

To solve the problem of finding the ratio of maximum and minimum intensities in the interference pattern created by two coherent sources of light with amplitudes \( a_1 \) and \( 2a_1 \), we can follow these steps:

### Step 1: Understand the relationship between amplitude and intensity
The intensity \( I \) of a wave is proportional to the square of its amplitude \( A \):
\[ I \propto A^2 \]

### Step 2: Define the amplitudes
Let the amplitudes of the two coherent sources be:
- \( A_1 = a_1 \)
- \( A_2 = 2a_1 \)

### Step 3: Calculate maximum intensity
The maximum intensity \( I_{\text{max}} \) occurs when the two waves are in phase. The resultant amplitude \( A_{\text{max}} \) can be calculated as:
\[ A_{\text{max}} = A_1 + A_2 = a_1 + 2a_1 = 3a_1 \]
Now, using the relationship between intensity and amplitude:
\[ I_{\text{max}} \propto (A_{\text{max}})^2 = (3a_1)^2 = 9a_1^2 \]

### Step 4: Calculate minimum intensity
The minimum intensity \( I_{\text{min}} \) occurs when the two waves are out of phase. The resultant amplitude \( A_{\text{min}} \) can be calculated as:
\[ A_{\text{min}} = |A_1 - A_2| = |a_1 - 2a_1| = | - a_1 | = a_1 \]
Now, using the relationship between intensity and amplitude:
\[ I_{\text{min}} \propto (A_{\text{min}})^2 = (a_1)^2 = a_1^2 \]

### Step 5: Calculate the ratio of maximum and minimum intensities
Now, we can find the ratio of maximum intensity to minimum intensity:
\[ \frac{I_{\text{max}}}{I_{\text{min}}} = \frac{9a_1^2}{a_1^2} = 9 \]

### Final Answer
The ratio of maximum and minimum intensities in the interference pattern is:
\[ \boxed{9} \]