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. mV How is the emf related to the electric field produced at the 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. mV How is the emf related to the electric field produced at the antenna?
Solution
To solve this problem, we need to use the formula for the power radiated by an isotropic antenna, which is given by:
P = (1/2) * ε0 * c * E^2 * A
where: P is the power, ε0 is the permittivity of free space (8.85 x 10^-12 C^2/N*m^2), c is the speed of light (3 x 10^8 m/s), E is the electric field strength, and A is the area over which the power is spread.
First, we need to convert the power from kW to W and the distance from miles to meters:
P = 3.60 kW = 3600 W d = 4.00 mi = 6437 m (approximately)
The power is spread over the surface area of a sphere with radius equal to the distance from the transmitter to the receiver. So, we can calculate A as:
A = 4πd^2
Substituting the values into the formula and solving for E, we get:
E = sqrt((2P) / (ε0 * c * A))
Once we have the electric field strength, we can calculate the amplitude of the emf induced in the antenna using the formula:
emf = E * l
where l is the length of the antenna. Substituting the values and solving, we get the amplitude of the emf.
The emf is directly proportional to the electric field produced at the antenna. This means that the stronger the electric field, the larger the emf that is induced in the antenna.
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