An AM radio station broadcasts isotropically (equally in all directions) with an average power of 3.60 kW. A receiving antenna 75.0 cm long is at a location 4.00 mi from the transmitter. Compute the amplitude of the emf that is induced by this signal between the ends of the receiving antenna.
Question
An AM radio station broadcasts isotropically (equally in all directions) with an average power of 3.60 kW. A receiving antenna 75.0 cm long is at a location 4.00 mi from the transmitter. Compute the amplitude of the emf that is induced by this signal between the ends of the receiving antenna.
Solution
To solve this problem, we need to use the formula for the power density (S) of an isotropic radiator, which is given by:
S = P / (4πr²)
where P is the power of the source and r is the distance from the source.
Step 1: Convert the power from kilowatts to watts and the distance from miles to meters.
P = 3.60 kW = 3600 W r = 4.00 mi = 6437.38 m (1 mile is approximately 1609.34 meters)
Step 2: Substitute these values into the formula to find the power density.
S = 3600 W / (4π(6437.38 m)²) = 4.43 x 10^-12 W/m²
The power density gives us the power per unit area, but we want to find the electric field strength (E). The power density and electric field strength are related by the formula:
S = 1/2 ε₀cE²
where ε₀ is the permittivity of free space (8.85 x 10^-12 C²/N·m²) and c is the speed of light (3.00 x 10^8 m/s).
Step 3: Rearrange this formula to solve for E and substitute the known values.
E = sqrt(2S / ε₀c) = sqrt((2 * 4.43 x 10^-12 W/m²) / (8.85 x 10^-12 C²/N·m² * 3.00 x 10^8 m/s)) = 1.00 x 10^-6 N/C
The electric field strength gives us the force per unit charge, but we want to find the amplitude of the emf (V), which is the work done per unit charge. The emf and electric field strength are related by the formula:
V = El
where l is the length of the antenna.
Step 4: Substitute the known values into this formula to find the emf.
V = 1.00 x 10^-6 N/C * 0.75 m = 7.50 x 10^-7 V
So, the amplitude of the emf that is induced by this signal between the ends of the receiving antenna is 7.50 x 10^-7 V.
Similar Questions
An AM radio station broadcasts with a wavelength of 484 m. Its transmitting antenna is vertical, resulting in vertically polarized radio waves (i.e, the electric field of the waves is vertical). You observe the station's signal at a point where it's propagating due east, and where the amplitude of the wave's electric field is 347 mV/m. Find (a) the wave frequency, (b) the amplitude of the wave's magnetic field, and (c) the direction of the magnetic field.
You are given a VHF television antenna (the antenna type is a magnetic loop antenna) and asked to determine the signal strength going into the receiver from the antenna. The antenna is a circular 1-turn loop of diameter 49.9 cm. The magnetic field of the TV signal is perpendicular to the plane of the loop and is essentially uniform over the area of the loop at any given instant. If the magnetic field is given by 𝐵=𝐵0sin(2𝜋𝑓𝑡) where 𝐵0=5.1 nT and 𝑓=109.8 MHz (megacycles/sec), the maximum value of the emf induced in the loop, 𝐸, is the strength of the signal going into the receiver in mV. What is this signal strength 𝐸?
In a plane EM wave, the electric field oscillates sinusoidally at a frequency of 5×1010 Hz and an amplitude of 50Vm−1. The total average energy density of the electromagnetic field of the wave is : [Use ε0=8.85×10−12C2/Nm2 ]
A radiated power of 5 W is produced by a resonant antenna. Its directional radiation intensity represented by: U = B0cos3θ (W/unit solid angle) 0 ≤ θ ≤ π/2; 0 ≤ ϕ ≤ 2πFind the(a) maximum power density (in W/m2) at a distance of 1500m (assuming far field distance).(b) Directivity of the antenna (dimensionless and in dB)(c) Gain of the antenna with an efficiency of 85% (dimensionless and in dB)
electromagnetic transmission
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.