In the Young's double slit experiment, the intensity of light at a point on the screen where the path difference is `lambda` is K, (`lambda` being the wave length of light used). The intensity at a point where the path difference is `lambda"/"4`, will be

Options

A. `K`
B. `K"/"4`
C. `K"/"2`
D. zero

Answer

To solve the problem, we need to find the intensity of light at a point on the screen in the Young's double slit experiment where the path difference is \( \frac{\lambda}{4} \), given that the intensity at a path difference of \( \lambda \) is \( K \).

### Step-by-Step Solution:

1. **Understanding Path Difference and Phase Difference:**
The phase difference \( \phi \) can be calculated using the formula:
\[ \phi = \frac{2\pi}{\lambda} \times \text{(path difference)} \]
For a path difference of \( \lambda \):
\[ \phi = \frac{2\pi}{\lambda} \times \lambda = 2\pi \]

2. **Intensity at Path Difference \( \lambda \):**
The intensity at this point is given as \( K \). The relationship between intensity and phase difference is given by:
\[ I = 4I_0 \cos^2\left(\frac{\phi}{2}\right) \]
For \( \phi = 2\pi \):
\[ I = 4I_0 \cos^2\left(\frac{2\pi}{2}\right) = 4I_0 \cos^2(\pi) = 4I_0 \times 1 = 4I_0 \]
Hence, we can equate:
\[ K = 4I_0 \implies I_0 = \frac{K}{4} \]

3. **Calculating Phase Difference for Path Difference \( \frac{\lambda}{4} \):**
Now, for a path difference of \( \frac{\lambda}{4} \):
\[ \phi = \frac{2\pi}{\lambda} \times \frac{\lambda}{4} = \frac{2\pi}{4} = \frac{\pi}{2} \]

4. **Intensity at Path Difference \( \frac{\lambda}{4} \):**
Using the intensity formula again:
\[ I = 4I_0 \cos^2\left(\frac{\phi}{2}\right) \]
Substituting \( \phi = \frac{\pi}{2} \):
\[ I = 4I_0 \cos^2\left(\frac{\pi}{4}\right) = 4I_0 \left(\frac{1}{\sqrt{2}}\right)^2 = 4I_0 \times \frac{1}{2} = 2I_0 \]

5. **Substituting \( I_0 \) Back:**
Now substituting \( I_0 = \frac{K}{4} \):
\[ I = 2 \times \frac{K}{4} = \frac{K}{2} \]

### Final Answer:
The intensity at a point where the path difference is \( \frac{\lambda}{4} \) is:
\[ \frac{K}{2} \]