Problem 35 - University Physics with Modern Physics

Question

A satellite 575 km above the earth's surface transmits sinusoidal electromagnetic waves of frequency 92.4 MHz uniformly in all directions, with a power of 25.0 kW. (a) What is the intensity of these waves as they reach a receiver at the surface of the earth directly below the satellite? (b) What are the amplitudes of the electric and magnetic fields at the receiver? (c) If the receiver has a totally absorbing panel measuring 15.0 cm by 40.0 cm oriented with its plane perpendicular to the direction the waves travel, what average force do these waves exert on the panel? Is this force large enough to cause significant effects?

Step-by-Step Solution

Answer

The wave intensity is \(4.12 \times 10^{-11} \text{ W/m}^2\), field amplitudes are \(E_0 = 0.176 \text{ V/m}\), \(B_0 = 5.87 \times 10^{-10} \text{ T}\), and the force on the panel is \(8.22 \times 10^{-21} \text{ N}\), too small for significant effects.

Step 1: Calculate the Distance from the Satellite to Earth

The satellite is 575 km above the Earth's surface. The Earth's radius is approximately 6371 km. Thus, the total distance \( R \) from the satellite to a point on Earth's surface directly below it is the sum of the Earth's radius and the satellite's height:\[R = 575 + 6371 = 6946 \text{ km} = 6.946 \times 10^6 \text{ m}\]

Step 2: Calculate the Intensity of the Waves

The intensity \( I \) of the electromagnetic waves spread uniformly in all directions is given by:\[ I = \frac{P}{A} \]where \( P = 25.0 \text{ kW} = 25000 \text{ W} \) and \( A \) is the surface area of a sphere centered on the satellite:\[ A = 4\pi R^2 \]Substituting for \( R \) gives:\[A = 4 \pi (6.946 \times 10^6)^2 \approx 6.07 \times 10^{14} \text{ m}^2\]Thus:\[I = \frac{25000}{6.07 \times 10^{14}} \approx 4.12 \times 10^{-11} \text{ W/m}^2\]

Step 3: Calculate the Amplitudes of the Electric and Magnetic Fields

The intensity \( I \) is related to the amplitude of the electric field \( E_0 \) by:\[ I = \frac{1}{2}\epsilon_0 c E_0^2 \]where \( \epsilon_0 = 8.85 \times 10^{-12} \text{ F/m} \) is the permittivity of free space, and \( c = 3 \times 10^8 \text{ m/s} \) is the speed of light. Solving for \( E_0 \):\[E_0 = \sqrt{\frac{2I}{\epsilon_0 c}} = \sqrt{\frac{2 \times 4.12 \times 10^{-11}}{8.85 \times 10^{-12} \times 3 \times 10^8}} \approx 0.176 \text{ V/m}\]The magnetic field \( B_0 \) is related to the electric field by:\[ B_0 = \frac{E_0}{c} \approx \frac{0.176}{3 \times 10^8} \approx 5.87 \times 10^{-10} \text{ T}\]

Step 4: Calculate the Average Force on the Panel

The pressure on the panel from the waves is related to the intensity by:\[ P = \frac{I}{c} \]The force \( F \) is then given by:\[ F = P \times A_{ ext{panel}} \]where the panel area \( A_{ ext{panel}} = 15.0 \times 10^{-2} \times 40.0 \times 10^{-2} = 0.060 \text{ m}^2 \). Using previous results:\[ P = \frac{4.12 \times 10^{-11}}{3 \times 10^8} \approx 1.37 \times 10^{-19} \text{ N/m}^2 \]Thus, the force is:\[ F = 1.37 \times 10^{-19} \times 0.060 \approx 8.22 \times 10^{-21} \text{ N} \]

Step 5: Conclusion and Significance

The calculated force on the panel is extremely small (\( 8.22 \times 10^{-21} \text{ N} \)). This is not large enough to cause any significant effects on the panel or the receiving equipment.

Key Concepts

Electromagnetic Wave Intensity

In the study of satellite communication, understanding wave intensity is crucial. Intensity tells us how much power flows through a unit area and is generally measured in watts per square meter (\( \text{W/m}^2 \)).
The power of the satellite's electromagnetic waves is dispersed uniformly in all directions, forming a spherical pattern around the satellite. The intensity of the wave at any point on this sphere is given by the formula:

• \[ I = \frac{P}{A} \]

where \( P \) is the power of the source (in this case, the satellite, which transmits 25.0 kW), and \( A \) is the surface area of the sphere through which the waves are spreading.
For our problem, the distance from the satellite to the Earth's surface is 6946 km, converting to meters for consistency in SI units. Using the formula for the surface area of a sphere \( A = 4\pi R^2 \), we can find \( A \) and thus the intensity \( I \). After calculations, the intensity of the waves reaching Earth is approximately \( 4.12 \times 10^{-11} \text{ W/m}^2 \), a very small number that reflects the vast distance and energy spread.

Electric and Magnetic Fields

Electromagnetic waves consist of oscillating electric and magnetic fields. The amplitude of these fields is related to the intensity of the wave. The electric field amplitude \( E_0 \) can be found using the relation between intensity and electric field:

• \[ I = \frac{1}{2}\epsilon_0 c E_0^2 \]

where \( \epsilon_0 \) is the permittivity of free space and \( c \) is the speed of light. By rearranging this formula and plugging in the known intensity, we find \( E_0 \) to be approximately 0.176 V/m.
Once \( E_0 \) is known, we can easily calculate the magnetic field amplitude \( B_0 \), because the electric and magnetic fields' amplitudes are related through the speed of light:

• \[ B_0 = \frac{E_0}{c} \]

This gives us a magnetic field amplitude of about \( 5.87 \times 10^{-10} \) Tesla. These amplitudes indicate the strengths of the fields arriving at Earth, showing the extremely dispersed nature of the signals by the time they reach their destination.

Average Force on a Panel

When electromagnetic waves hit a surface, they exert a pressure. This leads to a force which can be calculated if the area of the surface and the intensity of the wave are known. The pressure \( P \) exerted by electromagnetic waves is related to the intensity by:

• \[ P = \frac{I}{c} \]

Here, \( c \) is again the speed of light. To find the force \( F \), multiply the pressure \( P \) by the area \( A_{\text{panel}} \) of the receiver panel:

• \[ F = P \times A_{\text{panel}} \]

The panel's area is 0.060 m². Calculations show that the force exerted by the satellite's waves on the panel is approximately \( 8.22 \times 10^{-21} \text{ N} \).
Such a small force indicates that electromagnetic waves transmitted over vast distances result in minimal mechanical impact on receiving equipment. This is key to understanding satellite communications, showcasing how signals can be transmitted over great distances without exerting destructive forces on devices.