The distance between the tew slits in a Young's double slit experiment is d and the distance of the screen from the plane of the slits is b, P is a point on the screen directly infront of one of the slits. The path difference between the waves arriving at P from the two slits is

Options

A. `(d^(2))/(b)`
B. `(d^(2))/(2b)`
C. `(2d^(2))/(b)`
D. `(d^(2))/(4b)`

Explanation

To find the path difference between the waves arriving at point P from the two slits in Young's double slit experiment, we can follow these steps:

### Step 1: Understand the Setup
In a Young's double slit experiment, we have two slits separated by a distance \(d\) and a screen located at a distance \(b\) from the plane of the slits. Point P is directly in front of one of the slits.

### Step 2: Identify the Geometry
We can visualize the situation. Let’s denote:
- The distance between the slits as \(d\).
- The distance from the slits to the screen as \(b\).
- The vertical distance from the central maximum (the midpoint between the two slits on the screen) to point P as \(y\).

### Step 3: Calculate the Path Difference
The path difference (\(\Delta x\)) between the two waves reaching point P can be derived from the geometry of the situation. The wave from the slit farther from point P will have to travel a longer distance compared to the wave from the slit directly in front of point P.

Using the small angle approximation (valid when \(b\) is much larger than \(d\)), the path difference can be expressed as:
\[ \Delta x = \frac{d \cdot y}{b} \]

### Step 4: Substitute for y
In the case where point P is directly in front of one of the slits, the distance \(y\) can be expressed in terms of the distance from the central maximum to point P. If we consider the position of point P directly in front of one of the slits, we can take \(y\) as the vertical distance from the central maximum to point P.

### Step 5: Final Expression
Thus, the path difference becomes:
\[ \Delta x = \frac{d \cdot y}{b} \]
If we consider the specific case where \(y\) is half the distance between the slits (i.e., \(y = \frac{d}{2}\)), we can substitute this into the equation:
\[ \Delta x = \frac{d \cdot \frac{d}{2}}{b} = \frac{d^2}{2b} \]

### Conclusion
The path difference between the waves arriving at point P from the two slits is:
\[ \Delta x = \frac{d^2}{2b} \]