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 x 10^13 m2 (a large fraction of North America), how much power does it radiate?
W?
To find the answers to these questions, we'll need to use some formulas related to the intensity and power of electromagnetic waves.
(a) Intensity (I) is defined as the power (P) delivered per unit area (A). The formula for intensity is:
I = P / A
To find the intensity, we'll first need to calculate the area (A) of the satellite dish. The diameter (D) is given as 2.10 m, so we can use the formula for the area of a circle to calculate it:
A = πr^2
where r is the radius of the dish. Since the diameter is given, we can calculate the radius as D/2:
r = D / 2 = 2.10 m / 2 = 1.05 m
Now, we can substitute the values into the area formula:
A = π(1.05 m)^2 ≈ 3.459 m^2
Next, we can calculate the intensity using the given power per unit area:
I = (7.90 µV/m)^2 / 3.459 m^2
First, convert microvolts to volts: 1 µV = 1×10^-6 V
I ≈ (7.90×10^-6 V/m)^2 / 3.459 m^2
I ≈ 4.7619×10^-11 V^2/m^2
Therefore, the intensity of the wave is approximately 4.7619×10^-11 V^2/m^2 or 4.7619×10^-11 W/m^2.
(b) Power (P) is given by the formula:
P = IA
where I is the intensity and A is the area. We already calculated the area in part (a), so we can substitute the values in:
P = (4.7619×10^-11 W/m^2) × 3.459 m^2
P ≈ 1.6477×10^-10 W
Therefore, the power received by the antenna is approximately 1.6477×10^-10 W.
(c) To find the power radiated by the satellite, we need to calculate the total intensity (I_total) over the broadcasted area (A_total). Since the satellite broadcasts uniformly over the area, the total intensity is the same as the intensity from part (a).
I_total = 4.7619×10^-11 W/m^2
To find the power radiated, we can use the same formula as in part (b):
P_total = I_total × A_total
Given that the broadcasted area (A_total) is 1.50×10^13 m^2, we can substitute the values:
P_total = (4.7619×10^-11 W/m^2) × (1.50×10^13 m^2)
P_total ≈ 7.1428×10^2 W
Therefore, the satellite radiates approximately 7.1428×10^2 Watts of power.