1 \begin{problem*}{24.22}
2 An AM radio station broadcasts isotropically (equally in all
3 directions) with an average power of $4.00\U{kW}$. A dipole recieving
4 antenna $65.0\U{cm}$ long is at a location $4.00\U{miles}$ from the
5 transmitter. Compute the amplitude of the emf that is induced by this
6 signal between the ends of the recieving antenna.
7 \end{problem*} % problem 24.22
10 To find the signal intensity at our antenna, we note that the power
11 broadcast from the station is spread out over a sphere of radius
12 $R=4.00\U{miles}$. The average intensity is then
14 I = S_\text{avg} = \frac{P}{A} = \frac{P}{4\pi R^2}
15 = \frac{4.00\E{3}\U{W}}{4\pi(4.00\U{miles}\cdot 1.609\E{3}\U{m/mile})^2}
16 = 7.68\U{$\mu$W/m$^2$} \;.
18 From Equation 24.27, we see
20 S_\text{avg} &= \frac{E_\text{max}^2}{2\mu_0 c} \tag{24.27} \\
21 E_\text{max} &= \sqrt{2\mu_0 c S_\text{avg}} = 76.1\U{mV/m}
23 The total voltage difference produced across our length $L=65.0\U{cm}$
26 \Delta V = LE_\text{max} = \ans{49.4\U{mV}}