At a certain point on a screen the path difference for the two interfering rays is `(1//8)^(th)` of a wavelength. Find the ratio of the intensity at this point to that at the centre of a bright fringe

Answer

To solve the problem, we need to find the ratio of the intensity at a point on the screen where the path difference is \( \frac{1}{8} \) of a wavelength to the intensity at the center of a bright fringe.

### Step-by-Step Solution:

1. **Identify the Path Difference**:
The path difference given is \( \Delta x = \frac{\lambda}{8} \).

2. **Calculate the Phase Difference**:
The phase difference \( \Delta \phi \) can be calculated using the formula:
\[ \Delta \phi = \frac{2\pi}{\lambda} \Delta x \]
Substituting the path difference:
\[ \Delta \phi = \frac{2\pi}{\lambda} \cdot \frac{\lambda}{8} = \frac{2\pi}{8} = \frac{\pi}{4} \]

3. **Intensity Formula**:
The intensity \( I \) at a point in an interference pattern can be expressed in terms of the maximum intensity \( I_{\text{max}} \) and the phase difference:
\[ I = I_{\text{max}} \cos^2\left(\frac{\Delta \phi}{2}\right) \]

4. **Substituting the Phase Difference**:
We substitute \( \Delta \phi = \frac{\pi}{4} \) into the intensity formula:
\[ I = I_{\text{max}} \cos^2\left(\frac{\pi}{4} \cdot \frac{1}{2}\right) = I_{\text{max}} \cos^2\left(\frac{\pi}{8}\right) \]

5. **Finding the Ratio of Intensities**:
The ratio of the intensity at the point to the maximum intensity is:
\[ \frac{I}{I_{\text{max}}} = \cos^2\left(\frac{\pi}{8}\right) \]

6. **Calculating \( \cos^2\left(\frac{\pi}{8}\right) \)**:
We can use the cosine value:
\[ \cos\left(\frac{\pi}{8}\right) \approx 0.92388 \]
Therefore,
\[ \cos^2\left(\frac{\pi}{8}\right) \approx (0.92388)^2 \approx 0.85355 \]

7. **Final Result**:
Thus, the ratio of the intensity at the point to that at the center of a bright fringe is approximately:
\[ \frac{I}{I_{\text{max}}} \approx 0.85355 \]

### Summary:
The ratio of the intensity at the given point to that at the center of a bright fringe is approximately \( 0.85355 \).