The average path difference between two waves coming from third and fifth fresnel zones of a wave front at the centre of the screen is

Options

A. `(lambda)/(2)`
B. `2lambda`
C. `lambda`
D. `4lambda`

Answer

To find the average path difference between two waves coming from the third and fifth Fresnel zones at the center of the screen, we can follow these steps:

### Step 1: Understand Fresnel Zones
Fresnel zones are regions in the wavefront that contribute to the interference pattern. Each zone is a three-dimensional elliptical region, and the nth Fresnel zone is defined by its radius and the distance from the source.

### Step 2: Determine the Path Difference
The path difference between two waves coming from different Fresnel zones can be calculated using the formula:
\[ \Delta x = \frac{(n_2^2 - n_1^2) \lambda}{2d} \]
where:
- \( n_1 \) and \( n_2 \) are the zone numbers (in this case, 3 and 5),
- \( \lambda \) is the wavelength of the waves,
- \( d \) is the distance from the source to the observation point.

### Step 3: Substitute the Values
For this problem:
- \( n_1 = 3 \)
- \( n_2 = 5 \)
- The average path difference becomes:
\[ \Delta x = \frac{(5^2 - 3^2) \lambda}{2d} \]
Calculating \( 5^2 - 3^2 \):
\[ 5^2 = 25 \quad \text{and} \quad 3^2 = 9 \quad \Rightarrow \quad 25 - 9 = 16 \]
Thus, the path difference simplifies to:
\[ \Delta x = \frac{16 \lambda}{2d} = \frac{8 \lambda}{d} \]

### Step 4: Conclusion
The average path difference between the waves coming from the third and fifth Fresnel zones at the center of the screen is given by:
\[ \Delta x = \frac{8 \lambda}{d} \]