A GPS satellite develops a fault whereby it moves into an orbit 19 500 km above the Earth’s surface and also emits a weak signal whose power is only 24 W. What will be the power per square metre of the signal received on the Earth’s surface? Choose the nearest value from the list below. (W/m2 represents ‘watts per square metre’.)
To calculate the power per square meter (W/m2) of the signal received on the Earth's surface, we need to consider the inverse square law for signal power.
The inverse square law states that the power of a signal decreases proportionally to the square of the distance from the source. In this case, the GPS satellite has moved to an orbit 19,500 km above the Earth's surface.
To calculate the power per square meter, we need to compare the power at the satellite with the power at the Earth's surface. Let's denote the power at the satellite as Psatellite and the power at the Earth's surface as Psurface.
The relationship between these two powers is given by:
Psurface = Psatellite * (r_s / r_e)^2
where r_s is the distance from the satellite to the Earth's surface and r_e is the radius of the Earth.
Given that the satellite emits a weak signal with a power of 24 W and is located 19,500 km above the Earth's surface, we need to calculate the power per square meter at the Earth's surface.
First, we need to convert the distance from kilometers (km) to meters (m).
r_s = 19,500 km = 19,500,000 m
r_e is the radius of the Earth, which is approximately 6,371 km or 6,371,000 m.
Now, we can calculate the power per square meter:
Psurface = 24 W * (r_s / r_e)^2
= 24 W * (19,500,000 m / 6,371,000 m)^2
= 24 W * 9.618
= 230.832 W
Therefore, the power per square meter of the signal received on the Earth's surface is approximately 231 W/m2.