Problem — The AM antenna at home

Problem

An AM radio receives a $1\ \text{MHz}$ signal from a station $30\ \text{km}$ away radiating $50\ \text{kW}$. Estimate the electric field at the receiver and the emf induced on a loop antenna of area $A\approx 100\ \text{cm}^2$.

Solution

Intensity at the receiver. The power is spread over a hemisphere (the ground reflects), of area $2\pi r^2$: $I=\frac{P}{2\pi r^2}=\frac{50\cdot 10^{3}}{2\pi\,(3\cdot 10^{4})^2}=\frac{50\cdot 10^{3}}{5{,}65\cdot 10^{9}}\approx 8{,}8\cdot 10^{-6}\ \text{W/m}^2$

Electric field. From $I=E_0^2/(2\mu_0 c)$, with the impedance of free space $Z_0=\mu_0 c\approx 377\ \Omega$, i.e. $2\mu_0 c\approx 754\ \Omega$: $E_0=\sqrt{2\mu_0 c\,I}=\sqrt{754\cdot 8{,}8\cdot 10^{-6}}=\sqrt{6{,}6\cdot 10^{-3}}\approx 0{,}08\ \text{V/m}$ $\ev{E_0\sim 0{,}06\ \text{V/m}\ \text{(order of magnitude)}}$

Emf on the loop. The magnetic field is $B=E_0/c=0{,}06/(3\cdot 10^8)\approx 2\cdot 10^{-10}\ \text{T}$; the flux oscillates at $\omega=2\pi f=6{,}28\cdot 10^{6}\ \text{rad/s}$ over area $A=100\ \text{cm}^2=0{,}01\ \text{m}^2$: $E_\text{emf}\sim B\,A\,\omega=2\cdot 10^{-10}\cdot 0{,}01\cdot 6{,}28\cdot 10^{6}\approx 1{,}3\cdot 10^{-5}\ \text{V}$ $\ev{E_\text{emf}\sim 12\ \mu\text{V}}$ A few tens of microvolts: it is the tiny signal that the radio’s amplifier stage must process.

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Topics: Induzione elettromagnetica Concepts: Legge di Faraday-Neumann-Lenz Skills: Stime di Fermi