At 1.5 km from the transmitter, the peak electric field of a radio wave is 350 mV/m. (a) What is the transmitter’s power output, assuming it broadcasts uniformly in all directions ? (b) What is the peak electric field 10 km from the transmitter ?
To answer these questions, we need to use the concept of the inverse square law for electromagnetic radiation. According to this law, the intensity or amplitude of a wave decreases as the square of the distance from the source increases.
Let's break down each question step by step:
(a) What is the transmitter’s power output, assuming it broadcasts uniformly in all directions?
To calculate the power output of the transmitter, we need to use the formula:
Power = (Intensity/Area) * Area
In our case, the intensity is the square of the peak electric field strength, and the area can be calculated as the surface area of a sphere with a radius of 1.5 km.
Step 1: Calculate the area:
Area = 4 * pi * radius^2
= 4 * pi * (1.5 km)^2
Step 2: Calculate the power output:
Power = (Intensity/Area) * Area
= (350 mV/m)^2 / [4 * pi * (1.5 km)^2]
Now, convert the units to make them consistent:
1 km = 1000 m
1 mV = 10^-3 V
Power = (350 * 10^-3 V/m)^2 / [4 * pi * (1.5 * 10^3 m)^2]
You can calculate this expression using a calculator or a mathematical software to find the power output in watts (W).
(b) What is the peak electric field 10 km from the transmitter?
Using the inverse square law, we can find the peak electric field 10 km from the transmitter by comparing it to the initial peak electric field, which is given as 350 mV/m at 1.5 km.
Step 1: Calculate the decrease in intensity due to the increased distance:
(Initial distance / New distance)^2 = (1.5 km / 10 km)^2
Step 2: Calculate the new peak electric field:
New peak electric field = Initial peak electric field * (Initial distance / New distance)
Plug in the values:
New peak electric field = 350 mV/m * (1.5 km / 10 km)^2
Again, you can calculate this expression to find the new peak electric field in mV/m.
To calculate the transmitter's power output, we can use the formula:
Power = (Electric Field)^2 / (2 * Impedance)
Given:
- Electric Field (E) = 350 mV/m = 350 * 10^-3 V/m
- Distance (r) = 1.5 km = 1.5 * 10^3 m
- Impedance (Z) = 377 Ω (characteristic impedance of free space)
Now, let's calculate the power output of the transmitter:
(a) Power = (350 * 10^-3 V/m)^2 / (2 * 377 Ω)
Power = 0.309 W
So, the transmitter's power output is approximately 0.309 Watts.
To find the peak electric field 10 km from the transmitter, we can use the inverse square law, which states that the intensity of the wave decreases with the square of the distance.
Given:
- Distance (r) = 10 km = 10 * 10^3 m
- Electric Field at 1.5 km (E_1.5) = 350 mV/m
Let's calculate the peak electric field at 10 km:
(b) Electric Field at 10 km (E_10) = E_1.5 * (r_1.5 / r_10)^2
Using the inverse square law, E_10 = 350 * (1.5 / 10)^2 mV/m
E_10 = 350 * (1.5 / 10)^2 mV/m
E_10 ≈ 7 mV/m
So, the peak electric field 10 km from the transmitter is approximately 7 mV/m.