A 2.10 m diameter university communications satellite dish receives TV signals that have a maximum electric field strength (for one channel) of 7.90 µV/m.
(a) What is the intensity of this wave?
W/m2?
(b) What is the power received by the antenna?
W?
(c) If the orbiting satellite broadcasts uniformly over an area of 1.50 1013 m2 (a large fraction of North America), how much power does it radiate?
W?
(a) I =cε₀E²/2 =3•10⁸•8.85•10⁻¹²•(7.9•10⁻⁶)²/2=8.2•10⁻¹⁴W/m²
(b) P=I•A=I•πD²/4=8.2•10⁻¹⁴•π•2.1²/4=2.84•10⁻¹³ W
(c) P(total)=P•A(total) =2.84•10⁻¹³•1.5•10¹³=4.26 W
thats 10^13 m2
To calculate the intensity of the wave, we will use the formula:
Intensity = (Electric Field Strength)^2 / (2 * Permittivity of Free Space)
Where:
Electric Field Strength = 7.90 µV/m (convert to volts)
Permittivity of Free Space = 8.85 x 10^-12 F/m (Farads per meter)
(a) Calculating the Intensity:
Electric Field Strength = 7.90 µV/m = 7.90 x 10^-6 V/m
Permittivity of Free Space = 8.85 x 10^-12 F/m
Intensity = (7.90 x 10^-6 V/m)^2 / (2 * 8.85 x 10^-12 F/m)
Intensity = 1.755 x 10^-10 W/m^2
The intensity of the wave is approximately 1.755 x 10^-10 W/m^2.
(b) To calculate the power received by the antenna, we will use the formula:
Power = Intensity * Area
Where:
Intensity = 1.755 x 10^-10 W/m^2
Area = π * (Radius)^2
Radius = Diameter / 2 = 2.10 m / 2 = 1.05 m
Area = π * (1.05 m)^2 = 3.459 m^2
Power = 1.755 x 10^-10 W/m^2 * 3.459 m^2
Power = 6.067 x 10^-10 W
The power received by the antenna is approximately 6.067 x 10^-10 W.
(c) To calculate the power radiated by the satellite, we will multiply the power received by the antenna by the area over which the satellite broadcasts:
Power radiated = Power received * Broadcast Area
Broadcast Area = 1.50 x 10^13 m^2
Power radiated = 6.067 x 10^-10 W * 1.50 x 10^13 m^2
Power radiated = 9.10 x 10^3 W
The satellite radiates approximately 9.10 x 10^3 W of power.
To find the answers to these questions, we will use the formulas related to the intensity and power of electromagnetic waves.
(a) The intensity of an electromagnetic wave is given by the formula:
Intensity = (Electric Field Strength)^2 / (2 * Impedance of Free Space)
The Impedance of Free Space, denoted by Z0, has a value of approximately 377 ohms.
In this case, the electric field strength is given as 7.90 µV/m. To calculate the intensity, we need to convert this value to volts per meter (V/m).
7.90 µV/m = 7.90 × 10^(-6) V/m
Substituting these values into the formula, we get:
Intensity = (7.90 × 10^(-6) V/m)^2 / (2 * 377 ohms)
Calculating this expression will give us the intensity in watts per square meter (W/m^2).
(b) Power is the rate at which energy is transferred by the wave. The formula for power is:
Power = Intensity * Surface Area
In this case, we need to determine the power received by the antenna, so the surface area will be the cross-sectional area of the satellite dish. The cross-sectional area of a circular dish is given by:
Surface Area = π * (radius)^2
Since the diameter of the dish is given (2.10 m), we can calculate the radius (r) by dividing the diameter by 2:
r = 2.10 m / 2 = 1.05 m
Now, we can substitute the values of intensity and surface area into the power formula to find the power received by the antenna.
(c) To calculate the power radiated by the satellite over the given area, we can use a similar formula. The power radiated is equal to the intensity multiplied by the total surface area of the broadcast region.
Substituting the values of intensity and surface area into this formula will give us the required power.
By following these steps and applying the relevant formulas, you should be able to find the answers to all three questions.