Communication Systems: Application Problems (2)

Question

A TV transmitting antenna is $h_T = 100$ m tall. Estimate the maximum line-of-sight range over which a TV signal can be received by an antenna at ground level. Take Earth’s radius $R = 6.4 \times 10^6$ m.

Solution — Step by Step

Why This Works

A TV signal travels in a straight line (it’s in the VHF/UHF range, which doesn’t reflect off the ionosphere). The horizon distance from a tower of height $h$ is set by the tangent line touching Earth. Using the right triangle relation $(R + h)^2 = R^2 + d^2$ and approximating $h \ll R$ gives $d^2 \approx 2 R h$.

Alternative Method

If both transmitting and receiving antennas have heights $h_T$ and $h_R$, the range is $d = \sqrt{2 R h_T} + \sqrt{2 R h_R}$. Useful in NEET problems that give both heights.

Common Mistake

Students sometimes write $d = \sqrt{R h}$ (forgetting the factor of 2). The factor comes from the geometry — both the tangent on Earth and the height contribute. Always start from $(R+h)^2 = R^2 + d^2$ if you forget the formula.