The electric field of an electromagnetic wave traveling in...

The electric field of an electromagnetic wave traveling in the positive $x$ -direction is given by $$\overrightarrow{\mathbf{E}}=E_{0} \hat{\mathbf{j}} \sin \frac{\pi z}{z_{0}} \cos (k x-\omega t)$$ (a) Describe the field verbally. (b) Determine an expression for $k$ (c) Find the phase speed of the wave.

The electric field of an electromagnetic wave traveling in the positive $x$ -direction is given by
$$\overrightarrow{\mathbf{E}}=E_{0} \hat{\mathbf{j}} \sin \frac{\pi z}{z_{0}} \cos (k x-\omega t)$$ (a) Describe the field verbally. (b) Determine an expression for $k$
(c) Find the phase speed of the wave.

Key Concepts

Wave Number (k)

The wave number is a fundamental parameter that quantifies the spatial frequency of the wave, defined as the number of wave cycles per unit distance. It is typically related to the wavelength by the relation k = 2?/? and is a critical component in the wave’s phase expression, connecting spatial variation with the temporal evolution.

Electromagnetic Wave Propagation

This concept involves the transmission of energy through space in the form of electromagnetic radiation, where the electric and magnetic fields oscillate perpendicular to each other and to the direction of propagation. In these waves, the spatial and temporal variations are combined in a way that describes how the wave moves, here specifically traveling in the positive x-direction.

Electric Field Polarization

Electric field polarization describes the orientation of the electric field vector in an electromagnetic wave. In the given context, the electric field oscillates along a fixed axis, indicating a specific polarization state; this feature is crucial in determining how the wave interacts with matter and boundary conditions.

Spatial and Temporal Variation in Waves

This concept refers to how wave functions are structured as products of functions that depend separately on space and on time. In the presented case, one factor determines the amplitude variation in a direction perpendicular to propagation (illustrating a standing-wave pattern), while another factor describes the oscillatory behavior along the direction of propagation, indicative of traveling wave characteristics.

Phase Speed of the Wave

Phase speed is the rate at which a given phase of the wave propagates through space. It is defined as the ratio of the angular frequency to the wave number (v_phase = ?/k), providing insight into how quickly the oscillatory pattern—the crests and troughs of the wave—moves along the propagation direction.

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