In a Young's double-slit experiment, if the incident light consists of two wavelengths `lambda_(1)` and `lambda_(2)`, the slit separation is d, and the distance between the slit and the screen is D, the maxima due to each wavelength will coincide at a distance from the central maxima, given by

Options

A. `(lambda_(1)+lambda_(2))/(2Dd)`
B. LCM of `(lambda_(1))/(d)` and `(lambda_(2)D)/(d)`
C. `(lambda_(1)-lambda_(2))(2D)/(d)`
D. HCF of `(lambda_(1)D_(2))/(d)` and `(lambda_(2)D)/(d)`

Answer

To solve the problem of finding the distance from the central maxima where the maxima of two wavelengths coincide in a Young's double-slit experiment, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Condition for Maxima**:
In a Young's double-slit experiment, the position of the maxima on the screen is given by the formula:
\[ y = \frac{n \lambda D}{d} \]
where:
- \( y \) is the distance from the central maxima,
- \( n \) is the order of the maxima (an integer),
- \( \lambda \) is the wavelength of the light,
- \( D \) is the distance from the slits to the screen,
- \( d \) is the distance between the slits.

2. **Setting Up the Equations for Two Wavelengths**:
For two wavelengths \( \lambda_1 \) and \( \lambda_2 \), the positions of the maxima can be expressed as:
- For \( \lambda_1 \):
\[ y_1 = \frac{n_1 \lambda_1 D}{d} \]
- For \( \lambda_2 \):
\[ y_2 = \frac{n_2 \lambda_2 D}{d} \]
where \( n_1 \) and \( n_2 \) are the respective orders of maxima for each wavelength.

3. **Finding the Condition for Coinciding Maxima**:
For the maxima to coincide, we need:
\[ y_1 = y_2 \]
This leads to:
\[ \frac{n_1 \lambda_1 D}{d} = \frac{n_2 \lambda_2 D}{d} \]
Simplifying this gives:
\[ n_1 \lambda_1 = n_2 \lambda_2 \]

4. **Finding the Relationship Between Orders**:
Rearranging gives us:
\[ \frac{n_1}{n_2} = \frac{\lambda_2}{\lambda_1} \]
This indicates that the ratio of the orders of maxima \( n_1 \) and \( n_2 \) is equal to the ratio of the wavelengths.

5. **Using the Least Common Multiple (LCM)**:
To find the distance where the maxima coincide, we can express the distance in terms of the least common multiple (LCM) of the two wavelengths:
\[ y = \text{LCM}\left(\frac{n_1 \lambda_1 D}{d}, \frac{n_2 \lambda_2 D}{d}\right) \]
This means that the distance \( y \) where the maxima coincide can be expressed as:
\[ y = \frac{\text{LCM}(n_1 \lambda_1, n_2 \lambda_2) D}{d} \]

### Final Answer:
Thus, the distance from the central maxima where the maxima due to each wavelength coincide is given by:
\[ y = \frac{\text{LCM}(n_1 \lambda_1, n_2 \lambda_2) D}{d} \]