In Young's double slit experiment the 7th maximum with wavelength `lambda_(1)`is at a distance `d_(1)`, and that with wavelength `lambda_(2)` is at distance `d_(2)`. Then `d_(1)//d_(2)` is

Options

A. `lambda_(1)/lambda_(2)`
B. `lambda_(2)/sqrtlambda_(1)`
C. `lambda_(1)^(2)//lambda_(2)^(2)`
D. `lambda_(2)^(2)//lambda_(2)^(1)`

Answer

To solve the problem, we need to analyze the relationship between the distances of the maxima in Young's double slit experiment for two different wavelengths. Here's the step-by-step solution:

### Step 1: Understand the conditions for maxima in Young's double slit experiment
In Young's double slit experiment, the condition for maxima is given by the formula:
\[ \Delta x = n \lambda \]
where \( \Delta x \) is the path difference, \( n \) is the order of the maximum (an integer), and \( \lambda \) is the wavelength of the light used.

### Step 2: Relate the path difference to the distance from the central maximum
The path difference can also be expressed in terms of the angle \( \theta \) and the distance \( d \) between the slits:
\[ d \sin \theta = n \lambda \]
For small angles, we can use the approximation \( \sin \theta \approx \tan \theta \approx \frac{y}{D} \), where \( y \) is the distance from the central maximum to the \( n \)-th maximum and \( D \) is the distance from the slits to the screen.

### Step 3: Express the distance \( y \) for the \( n \)-th maximum
From the above relationships, we can express \( y \) as:
\[ y = \frac{n \lambda D}{d} \]
This means that the distance to the \( n \)-th maximum is directly proportional to the wavelength.

### Step 4: Apply the formula for both wavelengths
For the 7th maximum with wavelength \( \lambda_1 \):
\[ d_1 = \frac{7 \lambda_1 D}{d} \]
For the 7th maximum with wavelength \( \lambda_2 \):
\[ d_2 = \frac{7 \lambda_2 D}{d} \]

### Step 5: Find the ratio \( \frac{d_1}{d_2} \)
Now, we can find the ratio of the distances:
\[ \frac{d_1}{d_2} = \frac{\frac{7 \lambda_1 D}{d}}{\frac{7 \lambda_2 D}{d}} = \frac{\lambda_1}{\lambda_2} \]

### Conclusion
Thus, the ratio of the distances \( \frac{d_1}{d_2} \) is given by:
\[ \frac{d_1}{d_2} = \frac{\lambda_1}{\lambda_2} \]