Two towers on the top of two hills are 40 km apart. The line joining them presses 50 m above a hill half way between the towers. What is the longest wavelength of radiowaves which can be send between the towers without apprecialbe fiffraction effects?

Options

A. 1.25 m
B. 0.125 m
C. 2.50 m
D. 0.250 m

Explanation

To solve the problem, we need to determine the longest wavelength of radio waves that can be sent between two towers without experiencing significant diffraction effects. Here’s a step-by-step solution:

### Step 1: Understand the Problem
We have two towers that are 40 km apart, and the line joining the tops of these towers is 50 m above the hill located halfway between them. We need to find the longest wavelength of radio waves that can be sent between the towers without appreciable diffraction.

### Step 2: Identify the Key Parameters
- Distance between the towers (D) = 40 km = 40,000 m
- Height above the hill (h) = 50 m
- Halfway distance (Zp) = D/2 = 20 km = 20,000 m

### Step 3: Use the Fresnel's Distance Formula
The Fresnel distance (Zp) is given by the formula:
\[ Zp = \frac{a^2}{\lambda} \]
where:
- \( a \) is the maximum height (50 m in this case),
- \( \lambda \) is the wavelength.

### Step 4: Rearrange the Formula to Solve for Wavelength
We can rearrange the formula to find the wavelength \( \lambda \):
\[ \lambda = \frac{a^2}{Zp} \]

### Step 5: Substitute the Values
Now, substituting the values into the formula:
- \( a = 50 \) m
- \( Zp = 20,000 \) m

Thus,
\[ \lambda = \frac{(50)^2}{20,000} \]

### Step 6: Calculate the Wavelength
Calculating the above expression:
\[ \lambda = \frac{2500}{20,000} = 0.125 \, \text{m} \]

### Step 7: Convert to Centimeters
To express the wavelength in centimeters:
\[ \lambda = 0.125 \, \text{m} \times 100 = 12.5 \, \text{cm} \]

### Conclusion
The longest wavelength of radio waves that can be sent between the towers without appreciable diffraction effects is **12.5 cm**.

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