If the intensities of the two interfering beams in Young.s double -Slit experiment be `I_1" and "I_2` then the contrast between the maximum and minimum intensity is good when

Options

A. `I_1` is much greater than `I_2`
B. `I_1` is much smaller than `I_2`
C. `I_(1)= I_(2)`
D. either `I_(1)= 0" or "I_(2)=0`

Answer

To determine when the contrast between the maximum and minimum intensity in Young's double-slit experiment is good, we need to analyze the relationship between the intensities of the two interfering beams.

### Step-by-Step Solution:

1. **Understanding Intensity in Interference:**
In Young's double-slit experiment, two coherent light beams with intensities \( I_1 \) and \( I_2 \) interfere with each other. The resulting intensity at any point on the screen can be calculated using the formula:
\[ I = I_1 + I_2 + 2\sqrt{I_1 I_2} \cos(\phi) \]
where \( \phi \) is the phase difference between the two beams.

2. **Maximum and Minimum Intensities:**
- The **maximum intensity** \( I_{max} \) occurs when \( \cos(\phi) = 1 \):
\[ I_{max} = I_1 + I_2 + 2\sqrt{I_1 I_2} \]
- The **minimum intensity** \( I_{min} \) occurs when \( \cos(\phi) = -1 \):
\[ I_{min} = I_1 + I_2 - 2\sqrt{I_1 I_2} \]

3. **Calculating Contrast:**
The contrast \( C \) between the maximum and minimum intensity can be defined as:
\[ C = \frac{I_{max} - I_{min}}{I_{max} + I_{min}} \]

4. **Substituting Values:**
Substituting \( I_{max} \) and \( I_{min} \) into the contrast formula:
\[ C = \frac{(I_1 + I_2 + 2\sqrt{I_1 I_2}) - (I_1 + I_2 - 2\sqrt{I_1 I_2})}{(I_1 + I_2 + 2\sqrt{I_1 I_2}) + (I_1 + I_2 - 2\sqrt{I_1 I_2})} \]
Simplifying this gives:
\[ C = \frac{4\sqrt{I_1 I_2}}{2(I_1 + I_2)} = \frac{2\sqrt{I_1 I_2}}{I_1 + I_2} \]

5. **Condition for Good Contrast:**
For the contrast to be good (i.e., close to 1), the ratio \( \frac{I_1}{I_2} \) should be close to 1. This means that the intensities of the two beams should be nearly equal:
\[ I_1 \approx I_2 \]

### Conclusion:
The contrast between the maximum and minimum intensity is good when the intensities of the two interfering beams are approximately equal, i.e., \( I_1 \approx I_2 \).