Let `S_1 and S_2` be the two slits in Young's double slit experiment. If central maxima is observed at P and angle `S_1 P S_2 = theta`, (`theta` is small) find the y-coordinates of the 3rd minima assuming the origin at the central maxima. (`lambda` = wavelength of monochromatic light used).`

Options

A. `+-(2lambda)/theta`
B. `+-(5lambda)/(2 theta)`
C. `+- 5/2 lambda theta`
D. `+-2 lambda theta `

Explanation

To find the y-coordinate of the third minima in Young's double slit experiment, we can follow these steps:

### Step 1: Understand the condition for minima
In Young's double slit experiment, minima occur when the path difference between the light coming from the two slits is an odd multiple of half the wavelength, i.e.,
\[ \Delta d = \left(m + \frac{1}{2}\right) \lambda \]
where \( m \) is an integer (0, 1, 2, ...). For the third minima, \( m = 2 \):
\[ \Delta d = \left(2 + \frac{1}{2}\right) \lambda = \frac{5}{2} \lambda \]

### Step 2: Relate path difference to geometry
The path difference \( \Delta d \) can also be expressed in terms of the angle \( \theta \) and the distance between the slits \( d \):
\[ \Delta d = d \sin(\theta) \]
For small angles, we can use the approximation \( \sin(\theta) \approx \theta \) (in radians). Thus, we have:
\[ \Delta d \approx d \theta \]

### Step 3: Set up the equation for the third minima
From the previous steps, we can equate the two expressions for the path difference:
\[ d \theta = \frac{5}{2} \lambda \]

### Step 4: Solve for \( \theta \)
Rearranging the equation gives:
\[ \theta = \frac{5 \lambda}{2d} \]

### Step 5: Find the y-coordinate
In the geometry of the setup, the y-coordinate of the minima can be expressed as:
\[ y = D \tan(\theta) \]
For small angles, \( \tan(\theta) \approx \theta \). Thus,
\[ y \approx D \theta \]
Substituting the expression for \( \theta \):
\[ y \approx D \left(\frac{5 \lambda}{2d}\right) \]
This gives:
\[ y = \frac{5D \lambda}{2d} \]

### Step 6: Consider both directions for minima
Since minima can occur above and below the central maxima, the y-coordinates for the third minima are:
\[ y = \pm \frac{5D \lambda}{2d} \]

### Final Result
The y-coordinates of the third minima are:
\[ y = \frac{5D \lambda}{2d} \quad \text{and} \quad y = -\frac{5D \lambda}{2d} \]