In Young's double slit experiment with light of wavelength `lambda=600 nm`, intensity of central fringe is `I_(0)`. Now one of the slit is covered by glass plate of refractive index 1.4 and thickness t=`5mum`. The new intensity at the same point on screen will be :

Options

A. `(I_(0))/(4)`
B. `(3I_(0))/(4)`
C. `I_(0)`
D. `(I_(0))/(2)`

Answer

To solve the problem, we will follow these steps:

### Step 1: Identify the given values
- Wavelength of light, \( \lambda = 600 \, \text{nm} = 600 \times 10^{-9} \, \text{m} \)
- Refractive index of the glass plate, \( \mu = 1.4 \)
- Thickness of the glass plate, \( t = 5 \, \mu m = 5 \times 10^{-6} \, \text{m} \)

### Step 2: Calculate the path difference introduced by the glass plate
The path difference \( \Delta x \) caused by the glass plate can be calculated using the formula:
\[ \Delta x = (\mu - 1) \cdot t \]
Substituting the values:
\[ \Delta x = (1.4 - 1) \cdot (5 \times 10^{-6}) = 0.4 \cdot 5 \times 10^{-6} = 2 \times 10^{-6} \, \text{m} \]

### Step 3: Calculate the phase difference
The phase difference \( \phi \) corresponding to the path difference can be calculated using the formula:
\[ \phi = \frac{2\pi}{\lambda} \cdot \Delta x \]
Substituting the values:
\[ \phi = \frac{2\pi}{600 \times 10^{-9}} \cdot (2 \times 10^{-6}) \]
Calculating this:
\[ \phi = \frac{4\pi \times 10^{-6}}{600 \times 10^{-9}} = \frac{4\pi}{0.6} = \frac{20\pi}{3} \]

### Step 4: Calculate the new intensity
The intensity \( I \) at the central fringe is given by:
\[ I = 4I_0 \cos^2\left(\frac{\phi}{2}\right) \]
Substituting the value of \( \phi \):
\[ I = 4I_0 \cos^2\left(\frac{20\pi}{6}\right) = 4I_0 \cos^2\left(\frac{10\pi}{3}\right) \]
Since \( \frac{10\pi}{3} \) is equivalent to \( \frac{10\pi}{3} - 3\pi = \frac{10\pi - 9\pi}{3} = \frac{\pi}{3} \):
\[ \cos\left(\frac{10\pi}{3}\right) = \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \]
Thus:
\[ I = 4I_0 \left(\frac{1}{2}\right)^2 = 4I_0 \cdot \frac{1}{4} = I_0 \]

### Final Answer
The new intensity at the same point on the screen will be:
\[ I = I_0 \]