`lambda_(1)` and `lambda_(2)` are used to illuminated the slits. `beta_(1)` and `beta_(2)` are the corresponding fringe widths. The wavelength `lambda_(1)` can produce photoelectric effect when incident on a metal But the wavelength `lambda_(2)` cannot produce photoelectric effect. The correct relation between `beta_(1)` and `beta_(2)` is

Options

A. `beta_(1) lt beta_(2)`
B. `beta_(1)=beta_(2)`
C. `beta_(1) gt beta_(2)`
D. `beta_(1) ne beta_(2)`

Explanation

To solve the problem, we need to analyze the relationship between the fringe widths (β₁ and β₂) and the wavelengths (λ₁ and λ₂) of the light used to illuminate the slits in a double-slit interference experiment.

### Step-by-Step Solution:

1. **Understanding Fringe Width**:
The fringe width (β) in a double-slit interference pattern is given by the formula:
\[ \beta = \frac{\lambda D}{d} \]
where:
- \( \lambda \) is the wavelength of the light,
- \( D \) is the distance from the slits to the screen,
- \( d \) is the distance between the slits.

2. **Relating Fringe Widths to Wavelengths**:
From the formula, we can see that the fringe width is directly proportional to the wavelength:
\[ \beta \propto \lambda \]
This means that if the wavelength increases, the fringe width also increases.

3. **Given Conditions**:
- It is given that \( \lambda_1 \) can produce the photoelectric effect, while \( \lambda_2 \) cannot.
- The photoelectric effect occurs when the energy of the incident photons is above a certain threshold, which is related to the wavelength. This implies that \( \lambda_1 \) must be shorter than \( \lambda_2 \) (since shorter wavelengths correspond to higher energy photons).

4. **Comparing Wavelengths**:
Since \( \lambda_1 \) can produce the photoelectric effect and \( \lambda_2 \) cannot, we conclude:
\[ \lambda_1 < \lambda_2 \]

5. **Comparing Fringe Widths**:
Since \( \beta \) is directly proportional to \( \lambda \), we can relate the fringe widths:
\[ \beta_1 \propto \lambda_1 \quad \text{and} \quad \beta_2 \propto \lambda_2 \]
Therefore, since \( \lambda_1 < \lambda_2 \):
\[ \beta_1 < \beta_2 \]

6. **Conclusion**:
The correct relation between the fringe widths is:
\[ \beta_1 < \beta_2 \]

### Final Answer:
The correct relation between \( \beta_1 \) and \( \beta_2 \) is:
\[ \beta_1 < \beta_2 \]