Suppose that the electric field part of an electromagnetic...

Suppose that the electric field part of an electromagnetic wave in vacuum is $\mathbf{E}=\left\{(3.1 \mathrm{~N} / \mathrm{C}) \cos \left[(1.8 \mathrm{rad} / \mathrm{m}) y+\left(5.4 \times 10^{6} \mathrm{rad} / \mathrm{s}\right) t\right]\right\} \hat{\mathbf{i}}$. (a) What is the direction of propagation? (b) What is the wavelength $\lambda$ ? (c) What is the frequency $v ?$ (d) What is the amplitude of the magnetic field part of the wave? (e) Write an expression for the magnetic field part of the wave.

Suppose that the electric field part of an electromagnetic wave in vacuum is $\mathbf{E}=\left\{(3.1 \mathrm{~N} / \mathrm{C}) \cos \left[(1.8 \mathrm{rad} / \mathrm{m}) y+\left(5.4 \times 10^{6} \mathrm{rad} / \mathrm{s}\right) t\right]\right\} \hat{\mathbf{i}}$.
(a) What is the direction of propagation?
(b) What is the wavelength $\lambda$ ?
(c) What is the frequency $v ?$
(d) What is the amplitude of the magnetic field part of the wave?
(e) Write an expression for the magnetic field part of the wave.

Key Concepts

Electromagnetic Waves in Vacuum

Electromagnetic waves in vacuum consist of interdependent electric and magnetic fields oscillating perpendicular to one another and to the direction of propagation. They move at the speed of light and are described by sinusoidal functions that express both spatial and temporal variation, reflecting the wave’s inherent periodicity and energy transport characteristics.

Phase Factor and Direction of Propagation

The phase factor in the wave function, typically written as (k·r ± ?t), combines the spatial and temporal components of the wave. The sign between the spatial (involving the wave number) and temporal (involving the angular frequency) parts indicates the direction in which the wave propagates, making it a crucial concept for understanding wave travel.

Wave Number and Wavelength

The wave number (k) quantifies the number of radians per unit distance and is directly related to the wavelength of the wave by the formula ? = 2?/k. This relationship is key in linking the spatial characteristics of the wave to its sinusoidal mathematical representation.

Angular Frequency and Frequency

Angular frequency (?) indicates how fast the wave oscillates in time in terms of radians per second, while the regular frequency (v) measures the number of oscillations per second (in Hertz). These are connected by the relationship v = ?/(2?), which is fundamental in relating time-domain behavior with the oscillatory properties of the wave.

Relationship Between Electric and Magnetic Fields

In electromagnetic waves, the electric and magnetic fields are interrelated through Maxwell's equations, leading to the relation E = cB in free space, where c is the speed of light. This relationship not only shows how the magnitude of one field determines the magnitude of the other but also plays a critical role in understanding energy propagation and field dynamics.

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