Two coherent radio point sources that are separated by 2.0 m are radiating in phase with a wavelength of 0.25m. If a detector moves in a large circle around their mid-point. At how many points will the detector show a maximum signal?

Explanation

To solve the problem of how many points a detector will show a maximum signal when moving in a large circle around two coherent radio point sources, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Distance between the two sources (d) = 2.0 m
- Wavelength (λ) = 0.25 m

2. **Understanding the Condition for Maximum Signal:**
- For constructive interference (maximum signal), the path difference (Δx) between the waves from the two sources must be an integer multiple of the wavelength:
\[ \Delta x = n\lambda \quad (n = 0, 1, 2, 3, \ldots) \]

3. **Calculate the Maximum Path Difference:**
- The maximum path difference occurs when the detector is at the farthest points from the midpoint of the two sources. The maximum path difference can be calculated as:
\[ \Delta x_{max} = d = 2.0 \text{ m} \]

4. **Determine the Maximum Number of Wavelengths:**
- To find how many wavelengths fit into the maximum path difference:
\[ n_{max} = \frac{\Delta x_{max}}{\lambda} = \frac{2.0 \text{ m}}{0.25 \text{ m}} = 8 \]

5. **Count the Number of Maxima:**
- The maxima occur at integer values of n from 0 to n_max. Therefore, the number of maxima is:
\[ n_{max} + 1 = 8 + 1 = 9 \]
- However, since the detector moves in a circle, we need to consider the symmetry and the fact that maxima will occur on both sides of the midpoint. Thus, we will have:
\[ \text{Total Maxima} = 2 \times n_{max} + 1 = 2 \times 8 + 1 = 17 \]

6. **Final Count of Maxima:**
- The total number of points where the detector will show a maximum signal is:
\[ \text{Total Maxima} = 17 \]

### Conclusion:
The detector will show a maximum signal at **17 points** as it moves in a large circle around the midpoint of the two coherent radio point sources.

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