Phase difference at the central point changes by `pi//3` when as thick film having a refractive index 1.5 and thickness `0.4mum` is placed in front of upper slit of a YDSE set up. If the wavelength (in nm) of the light used is 600 k, find k.

Answer

To solve the problem step by step, we will follow the logical flow of the information provided in the question.

### Step 1: Understand the phase difference formula
The phase difference (ΔΦ) is given by the formula:
\[ \Delta \Phi = \frac{2\pi}{\lambda} \cdot \Delta x \]
where Δx is the path difference and λ is the wavelength of the light used.

### Step 2: Substitute the given phase difference
We know from the problem that the phase difference at the central point changes by \( \frac{\pi}{3} \). Therefore, we can set up the equation:
\[ \frac{\pi}{3} = \frac{2\pi}{\lambda} \cdot \Delta x \]

### Step 3: Simplify the equation
We can simplify the equation by canceling out \( \pi \) from both sides:
\[ \frac{1}{3} = \frac{2}{\lambda} \cdot \Delta x \]
This can be rearranged to find Δx:
\[ \Delta x = \frac{\lambda}{6} \]

### Step 4: Relate Δx to the thickness and refractive index
The path difference Δx can also be expressed in terms of the thickness (T) of the film and its refractive index (μ):
\[ \Delta x = (μ - 1) \cdot T \]
Given that \( μ = 1.5 \) and \( T = 0.4 \, \mu m = 0.4 \times 10^{-6} \, m \), we can substitute these values:
\[ \Delta x = (1.5 - 1) \cdot (0.4 \times 10^{-6}) = 0.5 \cdot (0.4 \times 10^{-6}) = 0.2 \times 10^{-6} \, m \]

### Step 5: Set the two expressions for Δx equal
Now we equate the two expressions for Δx:
\[ \frac{\lambda}{6} = 0.2 \times 10^{-6} \]

### Step 6: Solve for λ
To find λ, we multiply both sides by 6:
\[ \lambda = 6 \cdot (0.2 \times 10^{-6}) = 1.2 \times 10^{-6} \, m \]
Converting this to nanometers (1 m = \( 10^9 \) nm):
\[ \lambda = 1.2 \times 10^{3} \, nm = 1200 \, nm \]

### Step 7: Relate λ to k
We are given that \( \lambda = 600k \). Therefore, we can set up the equation:
\[ 600k = 1200 \]
Dividing both sides by 600 gives:
\[ k = \frac{1200}{600} = 2 \]

### Final Answer
Thus, the value of \( k \) is:
\[ \boxed{2} \]