Numerical: cordless phone free space path loss

Free space path loss (with unity gain antennas) is given by the Friis equation:

PL(dB) = 10 log(Pt/Pr) = 20 log(4πd/λ)

Given: f = 52 000 kHz = 52 MHz, d = 50 m (range of the phone).

Step 1: Wavelength

λ = c/f = 3×10⁸ / 52×10⁶ = 5.769 m

Step 2: Check far field

The phone's antenna is small (much less than λ), so the far field begins within a few wavelengths; 50 m is about 8.7λ, so the Friis equation can be used.

Step 3: Path loss

4πd/λ = 4π × 50 / 5.769 = 108.91

PL = 20 log(108.91) = 40.74 dB

Check using the practical form:

PL = 32.44 + 20 log f(MHz) + 20 log d(km)
   = 32.44 + 20 log 52 + 20 log 0.05
   = 32.44 + 34.32 − 26.02
   = 40.74 dB

Answer: Free space path loss at the 50 m range ≈ 40.7 dB (the received power is about 1/11 860 of the transmitted power for unity-gain antennas).