Non-Ionizing EM Field of a 50 kW Radio Transmitter: 17-Phys-B1 Radiation Physics, December 2016, Q3

Question 3: Non-Ionizing EM Field of a 50 kW Radio Transmitter (8 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Transmitter power $P=50$ kW, distance $r=50$ km; reference station: $P_{\text{ref}}=1$ kW gives reliable reception range $r_{\text{ref}}=50$ km.

Given data
Quantity	Symbol	Value
Transmitter power	$P$	50 kW
Distance	$r$	50 km
Free-space impedance	$Z_0$	$377\ \Omega$
Reference power / range	$P_{\text{ref}}/r_{\text{ref}}$	1 kW / 50 km

Find. (a) mean power density $S$ at 50 km; (b) maximum $E$-field; (c) maximum $H$-field; (d) listening (reliable-reception) range at 50 kW.

Fig. Q3 — non-directional antenna radiating isotropically; power density falls off as $1/(4\pi r^2)$.

Approach. A commercial broadcast antenna with no stated directionality is reasonably modelled as radiating isotropically (uniformly over a sphere) — consistent with the hint's own "non-directional antenna" reference station. Apply $S=P/(4\pi r^2)$, relate $S$ to the peak fields via the free-space impedance, then scale the reference reliable-reception range by $\sqrt{P/P_{\text{ref}}}$ since reception range is set by a fixed threshold power density, $S\propto P/r^2=\text{const}\Rightarrow r\propto\sqrt P$.

Part (a) — mean power density at 50 km. $$S = \frac{P}{4\pi r^2} = \frac{50{,}000\ \text{W}}{4\pi(50{,}000\ \text{m})^2}$$ $$\boxed{S \approx 1.59\times10^{-6}\ \text{W/m}^2 \ \ (1.59\ \mu\text{W/m}^2)}$$
Part (b) — maximum electric field strength. For a plane wave the time-averaged power density relates to the peak field by $S=E_{\max}^2/(2Z_0)$, so $E_{\max}=\sqrt{2 Z_0 S}$: $$E_{\max} = \sqrt{2(377\ \Omega)(1.59\times10^{-6}\ \text{W/m}^2)}$$ $$\boxed{E_{\max} \approx 0.0346\ \text{V/m} = 34.6\ \text{mV/m}}$$
Part (c) — maximum magnetic field strength. The peak $E$ and $H$ fields of a plane wave are linked by the free-space impedance, $H_{\max}=E_{\max}/Z_0$: $$H_{\max} = \frac{0.0346\ \text{V/m}}{377\ \Omega}$$ $$\boxed{H_{\max} \approx 9.19\times10^{-5}\ \text{A/m} = 91.9\ \mu\text{A/m}}$$
Part (d) — listening range at 50 kW. Reliable reception occurs down to a fixed minimum power density at the receiver, $S_{\min}=P/(4\pi r^2)=\text{const}$, so the reliable range scales as $r\propto\sqrt P$. Scaling the given 1 kW $\to$ 50 km reference: $$r_{50\text{kW}} = r_{\text{ref}}\sqrt{\frac{P}{P_{\text{ref}}}} = 50\ \text{km}\times\sqrt{\frac{50\ \text{kW}}{1\ \text{kW}}}$$ $$\boxed{r_{50\text{kW}} \approx 353.6\ \text{km}}$$

Question 3 — results
Part	Result
(a) $S$	$1.59\times10^{-6}$ W/m$^2$
(b) $E_{\max}$	34.6 mV/m
(c) $H_{\max}$	91.9 $\mu$A/m
(d) Listening range	$\approx354$ km

Topic. Isotropic radiation, plane-wave power density, and the free-space impedance relation between $E$ and $H$.

Key relations. $S=P/4\pi r^2$; $S=E_{\max}^2/2Z_0$; $H_{\max}=E_{\max}/Z_0$, $Z_0\approx377\ \Omega$; threshold-reception scaling $r\propto\sqrt P$.

Why this works. A non-directional antenna spreads its power uniformly over an expanding sphere, so power density obeys the same inverse-square law as any isotropic point source; the plane-wave impedance then fixes a unique ratio between the peak $E$ and $H$ fields at any given power density, and a fixed reception-threshold power density is what pins the reliable range to $\sqrt P$ scaling.

Common pitfall. Using the RMS power-density relation $S=E_{\text{rms}}^2/Z_0$ when the question asks for the peak field, or vice versa — the factor of 2 between $E_{\max}^2$ and $2E_{\text{rms}}^2$ is easy to drop.

Source. F. H. Attix, Introduction to Radiological Physics and Radiation Dosimetry, Ch. 2 (electromagnetic wave quantities); J. E. Turner, Atoms, Radiation, and Radiation Protection, 3rd ed., Ch. 15 (non-ionizing radiation).

Question 3 of 7: Non-Ionizing EM Field of a 50 kW Radio Transmitter

Question 3: Non-Ionizing EM Field of a 50 kW Radio Transmitter (8 marks)