Radio station WCCO in Minneapolis broadcasts at a frequency of 830 kHz . At a point some distanc

step 1 Part A of this problem is asking for the wavelength of this wave. And so we can get that by doin

Radio station WCCO in Minneapolis broadcasts at a frequency...

Radio station WCCO in Minneapolis broadcasts at a frequency of 830 $\mathrm{kHz}$ . At a point some distance from the transmitter, the magnetic-field amplitude of the electromagnetic wave from $\mathrm{WCCO}$ is $4.82 \times 10^{-11}$ T. Calculate (a) the wavelength; (b) the wave number; (c) the angular frequency; (d) the electric-field amplitude.

Key Concepts

Electromagnetic Wave Propagation

Electromagnetic waves travel through space at the speed of light in vacuum, and their behavior is characterized by properties such as wavelength, frequency, and phase. The fundamental relation c = ?f provides the basis for understanding how these properties interrelate, linking the frequency of the wave (f) with its wavelength (?) through the constant speed of light (c).

Dispersion Relation

The dispersion relation in free space for electromagnetic waves asserts that the angular frequency (?) is related to the wave number (k) via ? = ck. This relation is essential for describing the wave’s phase evolution across space and time, and it underscores how frequency and spatial periodicity are interconnected.

Wave Number

The wave number (k) quantifies the number of wave cycles per unit distance and is mathematically defined as k = 2?/?. It is a key parameter in wave analysis, as it indicates how rapidly the phase of the wave changes in space, which is vital for understanding wave propagation and interference phenomena.

Angular Frequency

Angular frequency (?) measures how quickly the wave oscillates in time and is defined by ? = 2?f. This parameter is significant for describing the temporal behavior of the wave, including its energy and phase variations, and is used extensively in wave analysis and the study of oscillatory systems.

Electric and Magnetic Field Amplitude Relationship

In a plane electromagnetic wave propagating through free space, the amplitudes of the electric (E) and magnetic (B) fields are intrinsically linked by the speed of light, expressed as E = cB. This relationship reflects the proportionality between the field strengths and is fundamental to understanding electromagnetic energy transport and wave dynamics.

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