In a Young's double slit experiment the intensity at a point where tha path difference is `(lamda)/(6)` (`lamda` being the wavelength of light used) is I. If `I_0` denotes the maximum intensity, `(I)/(I_0)` is equal to

Options

A. `(3)/(4)`
B. `(1)/(sqrt2)`
C. `(sqrt3)/(2)`
D. `(1)/(2)`

Explanation

To solve the problem, we need to find the ratio \( \frac{I}{I_0} \) where \( I \) is the intensity at a point with a path difference of \( \frac{\lambda}{6} \) and \( I_0 \) is the maximum intensity in a Young's double slit experiment.

### Step-by-Step Solution:

1. **Understanding the Intensity Formula**:
In a Young's double slit experiment, the intensity \( I \) at any point on the screen can be expressed as:
\[ I = I_0 \cos^2\left(\frac{\delta \phi}{2}\right) \]
where \( \delta \phi \) is the phase difference between the two waves arriving at that point.

2. **Calculating the Phase Difference**:
The phase difference \( \delta \phi \) is related to the path difference \( \Delta x \) by the formula:
\[ \delta \phi = \frac{2\pi}{\lambda} \Delta x \]
Given that the path difference \( \Delta x = \frac{\lambda}{6} \), we can substitute this into the equation:
\[ \delta \phi = \frac{2\pi}{\lambda} \cdot \frac{\lambda}{6} = \frac{2\pi}{6} = \frac{\pi}{3} \]

3. **Substituting Phase Difference into the Intensity Formula**:
Now, we substitute \( \delta \phi \) back into the intensity formula:
\[ I = I_0 \cos^2\left(\frac{\delta \phi}{2}\right) = I_0 \cos^2\left(\frac{\pi}{3 \cdot 2}\right) = I_0 \cos^2\left(\frac{\pi}{6}\right) \]

4. **Calculating \( \cos^2\left(\frac{\pi}{6}\right) \)**:
We know that:
\[ \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \]
Therefore:
\[ \cos^2\left(\frac{\pi}{6}\right) = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4} \]

5. **Final Expression for Intensity**:
Now we can express \( I \) in terms of \( I_0 \):
\[ I = I_0 \cdot \frac{3}{4} \]

6. **Finding the Ratio \( \frac{I}{I_0} \)**:
To find the ratio \( \frac{I}{I_0} \):
\[ \frac{I}{I_0} = \frac{3}{4} \]

### Conclusion:
Thus, the ratio \( \frac{I}{I_0} \) is:
\[ \frac{I}{I_0} = \frac{3}{4} \]