In the Young's double slit experiment , a mica slip of thickness t and refractive index `mu` is introduced in the ray from first source `S_(1)` . By how much distance fringes pattern will be displaced ? (d = distance between the slits and D is the distance between slits and screen)

Options

A. `(d)/(D)(mu-1)t`
B. `(D)/(d)(mu-1)t`
C. `(d)/((mu-1)D)`
D. `(D)/(d)(mu-1)`

Explanation

To solve the problem of how much the fringe pattern will be displaced when a mica slip of thickness \( t \) and refractive index \( \mu \) is introduced in the ray from the first source \( S_1 \) in Young's double slit experiment, we can follow these steps:

### Step 1: Understand the Setup
In Young's double slit experiment, we have two slits \( S_1 \) and \( S_2 \) that emit coherent light. When a mica slip is introduced in the path of light from \( S_1 \), it alters the optical path length of the light coming from that slit.

### Step 2: Calculate the Optical Path Length
The optical path length for light passing through a medium is given by the product of the physical length and the refractive index. For the light passing through the mica slip, the optical path length is:
\[ \text{Optical Path Length from } S_1 = \mu t \]
For the light from slit \( S_2 \) (which is in air), the optical path length remains:
\[ \text{Optical Path Length from } S_2 = t \]

### Step 3: Determine the Path Difference
The path difference \( \Delta x \) between the two rays reaching a point \( P \) on the screen can be expressed as:
\[ \Delta x = (\text{Optical Path Length from } S_1) - (\text{Optical Path Length from } S_2) \]
Substituting the values we found:
\[ \Delta x = (\mu t) - t = t(\mu - 1) \]

### Step 4: Relate Path Difference to Fringe Displacement
In Young's double slit experiment, the fringe position on the screen is related to the path difference by the formula:
\[ y = \frac{D}{d} \Delta x \]
where \( D \) is the distance from the slits to the screen and \( d \) is the distance between the slits.

### Step 5: Substitute the Path Difference
Now, substituting the expression for \( \Delta x \) into the fringe displacement formula:
\[ y = \frac{D}{d} \cdot t(\mu - 1) \]

### Final Result
Thus, the distance by which the fringe pattern will be displaced is:
\[ y = \frac{D}{d} \cdot t(\mu - 1) \]