In YDSE pattern with light of wavelength `lambda_1 = 500nm`, 15 fringes are obtained on a certain segment of screen. If number of fringes for light of wavelength `lambda_2` on same segment of screen is 10, then the value of `lambda_2` (in nm) is-

Answer

To solve the problem, we need to find the wavelength \( \lambda_2 \) given that 15 fringes are observed for a wavelength \( \lambda_1 = 500 \, \text{nm} \) and 10 fringes are observed for \( \lambda_2 \) on the same segment of the screen.

### Step-by-step Solution:

1. **Understand the relationship between fringe width and wavelength**:
The fringe width \( \beta \) in Young's Double Slit Experiment (YDSE) is given by the formula:
\[ \beta = \frac{\lambda d}{D} \]
where \( \lambda \) is the wavelength of light, \( d \) is the distance between the slits, and \( D \) is the distance from the slits to the screen.

2. **Calculate the length of the segment of the screen for the first wavelength**:
The length of the segment of the screen \( L \) where 15 fringes are observed can be expressed as:
\[ L = n_1 \cdot \beta_1 \]
where \( n_1 = 15 \) and \( \beta_1 = \frac{\lambda_1 d}{D} \). Thus,
\[ L = 15 \cdot \frac{\lambda_1 d}{D} \]

3. **Calculate the length of the segment of the screen for the second wavelength**:
Similarly, for the second wavelength \( \lambda_2 \) where 10 fringes are observed, the length of the segment of the screen can be expressed as:
\[ L = n_2 \cdot \beta_2 \]
where \( n_2 = 10 \) and \( \beta_2 = \frac{\lambda_2 d}{D} \). Thus,
\[ L = 10 \cdot \frac{\lambda_2 d}{D} \]

4. **Set the two expressions for \( L \) equal to each other**:
Since the length of the segment of the screen is the same for both wavelengths, we can equate the two expressions:
\[ 15 \cdot \frac{\lambda_1 d}{D} = 10 \cdot \frac{\lambda_2 d}{D} \]

5. **Cancel out common terms**:
Since \( d \) and \( D \) are the same for both cases, we can cancel them out:
\[ 15 \lambda_1 = 10 \lambda_2 \]

6. **Solve for \( \lambda_2 \)**:
Rearranging the equation gives:
\[ \lambda_2 = \frac{15}{10} \lambda_1 = 1.5 \lambda_1 \]
Substituting \( \lambda_1 = 500 \, \text{nm} \):
\[ \lambda_2 = 1.5 \times 500 \, \text{nm} = 750 \, \text{nm} \]

### Final Answer:
The value of \( \lambda_2 \) is \( 750 \, \text{nm} \).

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