A Active Figure 34.4b shows a plane electromagnetic...

A Active Figure 34.4b shows a plane electromagnetic sinusoidal wave propagating in the $x$ direction. Suppose the wavelength is 50.0 m and the electric field vibrates in the xy plane with an amplitude of $22.0 \mathrm{~V} / \mathrm{m}$. Calculate (a) the frequency of the wave and (b) the magnitude and direction of $\overrightarrow{\mathbf{B}}$ when the electric field has its maximum value in the negative $y$ direction. (c) Write an expression for $\overrightarrow{\mathbf{B}}$ with the correct unit vector, with numerical values for $B_{\text {max }}, k$, and $\omega$, and with its magnitude in the form $$ B=B_{\max } \cos (k x-\omega t) $$

Key Concepts

Electromagnetic Waves

Electromagnetic waves consist of oscillating electric and magnetic fields that are perpendicular to each other and to the direction of propagation. These waves travel through a vacuum at the speed of light and are described by sinusoidal functions in both space and time.

Wave Properties

The relationship between frequency, wavelength, and speed (c = ?f) is fundamental in wave physics. Angular frequency is related to the frequency by ? = 2?f. These relationships are used to determine the frequency and phase characteristics of electromagnetic waves.

Field Relationships

In free space, the electric and magnetic fields in an electromagnetic wave are related by their magnitudes as E = cB, where c is the speed of light. This relationship is essential for establishing the amplitude of one field given the amplitude of the other.

Vector Directions and Right-Hand Rule

The right-hand rule dictates that the directions of the electric field, magnetic field, and the wave's propagation are mutually perpendicular. Specifically, if the electric field points in a particular direction at its maximum value, the magnetic field orientation can be determined by ensuring that the cross product of the electric field with the magnetic field gives the direction of wave propagation.

Wave Function Expression

Electromagnetic fields are often represented using cosine (or sine) functions that depend on both position and time, expressed in the form A cos(kx - ?t). Here, k is the wave number related to wavelength by k = 2?/?, and ? is the angular frequency, providing a mathematically convenient way to describe the oscillatory behavior of the fields.

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