Answer :
To solve this physics problem, we will address each part step by step.
(a) Wavelength of the Signal:
The wavelength [tex]\lambda[/tex] of a radio signal can be calculated using the speed of light [tex]c[/tex] and the frequency [tex]f[/tex]. The formula is:
[tex]\lambda = \frac{c}{f}[/tex]
Where:
- [tex]c = 3 \times 10^8 \text{ m/s}[/tex] (speed of light),
- [tex]f = 9.13 \times 10^7 \text{ Hz}[/tex] (frequency).
Substituting the given values:
[tex]\lambda = \frac{3 \times 10^8 \text{ m/s}}{9.13 \times 10^7 \text{ Hz}} \approx 3.29 \text{ meters}[/tex]
(b) Average Intensity at the Radio Receiver:
The intensity [tex]I[/tex] of the broadcast signal can be calculated using the power [tex]P[/tex] and the area [tex]A[/tex] over which it spreads. Since the signal spreads over a hemisphere:
[tex]I = \frac{P}{A} = \frac{65 \times 10^4 \text{ W}}{2 \pi r^2}[/tex]
Where [tex]r = 45 \times 10^3 \text{ m}[/tex]. Substituting these values:
[tex]I = \frac{65 \times 10^4}{2 \pi (45 \times 10^3)^2}[/tex]
Calculating the denominator:
[tex]2 \pi (45 \times 10^3)^2 \approx 1.27 \times 10^{10} \text{ m}^2[/tex]
Thus:
[tex]I \approx \frac{65 \times 10^4}{1.27 \times 10^{10}} \approx 0.051 \text{ W/m}^2[/tex]
(c) Maximum Amplitude of the Electric Field:
The intensity is related to the amplitude of the electric field [tex]E_0[/tex] by:
[tex]I = \frac{c \varepsilon_0 E_0^2}{2}[/tex]
Solving for [tex]E_0[/tex]:
[tex]E_0 = \sqrt{\frac{2I}{c \varepsilon_0}}[/tex]
Where [tex]\varepsilon_0 = 8.85 \times 10^{-12} \text{ F/m}[/tex] is the permittivity of free space. Substituting the values:
[tex]E_0 = \sqrt{\frac{2 \times 0.051}{3 \times 10^8 \times 8.85 \times 10^{-12}}}\approx 0.19 \text{ V/m}[/tex]
(d) Radiation Pressure on the Receiver:
Radiation pressure [tex]P_r[/tex] for totally absorbed radiation is given by:
[tex]P_r = \frac{I}{c}[/tex]
Substituting the intensity:
[tex]P_r = \frac{0.051}{3 \times 10^8} \approx 1.7 \times 10^{-10} \text{ N/m}^2[/tex]
(e) Magnitude of the Maximum Electric Force on an Electron:
The electric force [tex]F[/tex] on an electron is given by:
[tex]F = e \cdot E_0[/tex]
Where [tex]e \approx 1.6 \times 10^{-19} \text{ C}[/tex] is the elementary charge. Using the electric field amplitude:
[tex]F \approx 1.6 \times 10^{-19} \times 0.19 \approx 3.04 \times 10^{-20} \text{ N}[/tex]
These calculations provide insights into the nature and effects of the radio transmission on the receiver.
