Question

Your local AM radio station broadcasts at a frequency of f = 1100 kHz. The electric-field component of the signal you receive at your home has the time dependence E(t) = E0 sin(2πft), where the amplitude is E0 = 0.62 N/C. Radio waves travel through air at approximately the speed of light.

a) At what wavelength, in meters, docs this station broadcast?
b) What is the value of the radio wave's electric field, in newtons per coulomb, at your home at a time of t = 3.1 μs?

Answer 1

a) The wavelength of the radio station's broadcast is approximately 272.73 meters. b) At a time of 3.1 μs, the value of the radio wave's electric field is approximately 0.619 N/C.

Step-by-step explanation:

a) To calculate the wavelength of the radio station's broadcast, we can use the formula λ = c/f, where λ is the wavelength, c is the speed of light, and f is the frequency. Plugging in the given frequency of 1100 kHz (or 1100 x 10^3 Hz), we get: λ = (3 x 10^8 m/s) / (1100 x 10^3 Hz) = 272.73 m

b) To find the value of the radio wave's electric field at a specific time, we can use the given time dependence equation E(t) = E0 sin(2πft), where E0 is the amplitude, f is the frequency, and t is the time. Plugging in the given amplitude of E0 = 0.62 N/C, frequency of 1100 kHz (or 1100 x 10^3 Hz), and time of 3.1 μs (or 3.1 x 10^-6 s), we get: E(t) = 0.62 sin(2π x 1100 x 10^3 x 3.1 x 10^-6) ≈ 0.619 N/C

Answer 2

(a) 272.73 m

(b) 0.338 N/C

Step-by-step explanation:

frequency, f = 1100 kHz = 1100 x 1000 Hz

E(t) = Eo Sin(2πft)

Eo = 0.62 N/C

(a) Velocity of light, c = 3 x 10^8 m/s

wavelength, λ = c / f = (3 x 10^8) / (1100000) = 272.73 m

Thus, the wavelength is 272.73 m.

(b) at t = 3.1 microsecond = 3.1 x 10^-6 s

E = Eo Sin (2 π ft)

E = 0.62 Sin (2 x 3.14 x 1100 x 10^3 x 3.1 x 10^-6)

E = 0.62 Sin (21.4148)

E = 0.62 x 0.5449 = 0.338 N/C

Thus, the electric field at t = 3.1 microsecond s 0.338 N/C.