This problem involves calculating the maximum detection range of a radar system considering Earth's curvature. The key concept is line-of-sight distance limited by the Earth's spherical shape.

Step 1: Understanding the Geometry
The radar is at height $h$ above Earth's surface. Due to Earth's curvature, the maximum distance $d$ to the horizon (where the radar can "see") is given by the formula:

$d=\sqrt{2Rh}$

where:
$R$ = Radius of Earth = $6.4\times {10}^{6}m$
$h$ = Height of radar = $500m$

Step 2: Why This Formula?
This formula comes from applying the Pythagorean theorem to the right triangle formed by:
- Earth's center
- Radar location
- Horizon point (tangent point)

The derivation shows that for small $h\ll R$, the distance simplifies to $\sqrt{2Rh}$.

Step 3: Important Note
The radar's power (1 kW) and frequency (10 GHz) are irrelevant for this particular calculation. These parameters would matter for detecting the actual signal strength, but the question specifically asks for the maximum distance limited by Earth's curvature, which is purely a geometric constraint.

Step 4: Calculation
Substitute the values into the formula:

$d=\sqrt{2\times 6.4\times {10}^6\times 500}$

$d=\sqrt{2\times 6.4\times 5\times {10}^8}=\sqrt{64\times {10}^8}$

$d=8\times {10}^{4}m=80\mathrm{km}$

Final Answer: 80 km

Related Topics

Horizon Distance: The maximum line-of-sight distance between two points at different heights above a spherical surface. For two points at heights $h_1$ and $h_2$, the maximum distance is $\sqrt{2R{h}_1}+\sqrt{2R{h}_2}$.

Radar Principles: While not needed here, radar systems use electromagnetic waves to detect objects. The power and frequency affect the system's ability to detect and resolve objects within the line-of-sight range.

Key Formula

Horizon distance: $d=\sqrt{2Rh}$