Question E3C06:
How does the VHF/UHF radio horizon compare to the geographic horizon?  

Choices:  
A: It is approximately 15 percent farther  
B: It is approximately 20 percent nearer  
C: It is approximately 50 percent farther  
D: They are approximately the same    

Correct Answer is  A: It is approximately 15 percent farther  

Correct answer: A — It is approximately 15 percent farther.

Why this is correct:
- VHF/UHF signals in the lower atmosphere bend slightly downward due to normal tropospheric refraction (the refractive index decreases with height). This makes the effective Earth radius larger (the “4/3 Earth” model), so the radio horizon extends beyond the geometric horizon by about 15%.
- Mathematically:
  $$d_{\text{radio}} \approx \sqrt{\tfrac{4}{3}} \, d_{\text{geometric}} \approx 1.15 \, d_{\text{geometric}}$$

Why the other choices are incorrect:
- B (20 percent nearer): Refraction extends, not shortens, the horizon under standard conditions.
- C (50 percent farther): Standard refraction is not that strong; typical extension is about 15%, not 50%.
- D (They are approximately the same): They are close, but radio horizon is measurably farther (~15%).

Useful formulas / rules of thumb:
- “4/3 Earth” rule: Use an effective Earth radius of $kR$ with $k \approx \tfrac{4}{3}$ for VHF/UHF.
- Single-end horizon from height h:
  - Geometric (no refraction), h in meters, distance in km:
    $$d_{\text{geo}} \approx 3.57\,\sqrt{h}$$
  - Radio horizon (standard refraction), h in meters, distance in km:
    $$d_{\text{radio}} \approx 4.12\,\sqrt{h}$$
- For two stations (heights $h_1$, $h_2$ in meters), total path:
  - Geometric:
    $$d_{\text{geo}} \approx 3.57\,(\sqrt{h_1} + \sqrt{h_2})$$
  - Radio:
    $$d_{\text{radio}} \approx 4.12\,(\sqrt{h_1} + \sqrt{h_2})$$
- In miles/feet (useful alternative):
  - Geometric:
    $$d_{\text{geo}}(\text{mi}) \approx 1.23\,(\sqrt{h_1} + \sqrt{h_2})$$
  - Radio:
    $$d_{\text{radio}}(\text{mi}) \approx 1.415\,(\sqrt{h_1} + \sqrt{h_2})$$

Key takeaway: Standard atmospheric refraction makes the VHF/UHF radio horizon about 15% farther than the geographic horizon.