91Ó°ÊÓ

The electric field (\( \mathbf{E} \)) is a crucial part of understanding electromagnetic waves. It is a vector field that represents the force a positive charge would experience at any point in space. In the context of electromagnetic waves like light, the electric field is time-varying and spatially oscillating. This means it changes with time and position, moving through space in the form of waves.

In the given exercise, the electric field of the wave is specified to be perpendicular to the \( \hat{\mathbf{z}} \) direction. This implies that the field can only have components in the \( \mathbf{y} \) direction if we consider a standard Cartesian coordinate system. Therefore, it can be mathematically represented as:

• \( \mathbf{E} = E_0\cos(kx - \omega t + \phi)\hat{\mathbf{y}} \)

Where \( E_0 \) is the maximum amplitude of the electric field, \( k \) is the wave number, \( \omega \) is the angular frequency, and \( \phi \) is the phase constant.

The magnetic field (\( \mathbf{B} \)) is another essential component of electromagnetic waves. Just like the electric field, it oscillates in a wave form. However, it is always perpendicular to both the electric field and the direction of wave propagation. This configuration helps maintain the integrity of the wave's predefined energy patterns.

In our exercise, the magnetic field must also follow these perpendicularity rules. It is expressed in terms of the electric field over the speed of light because of the connection between their magnitudes in electromagnetic waves:

• \( \mathbf{B} = \frac{E_0}{c}\cos(kx - \omega t + \phi)\hat{\mathbf{z}} \)

Here, \( c \) is the speed of light in a vacuum, making this relation crucial for calculating \( \mathbf{B} \) given \( \mathbf{E} \). This formula shows that the peak amplitudes of \( \mathbf{B} \) and \( \mathbf{E} \) are proportionately connected.

Wave propagation is the phenomenon by which waves travel through different mediums. For electromagnetic waves, such propagation occurs even in a vacuum. These waves are characterized by their dynamic oscillations of electric and magnetic fields at right angles to one another and also orthogonal to their direction of travel.

In this exercise, the wave travels in the negative \( \hat{\mathbf{x}} \) direction. This directionality influences how the fields are oriented, as both \( \mathbf{E} \) and \( \mathbf{B} \) must sustain perpendicularity both mutually and in relation to the wave's travel path. Understanding this angular relation is essential for grasping how energy transmits across space.

The frequency of an electromagnetic wave is fundamentally how many oscillations it completes per second. It is usually measured in Hertz (Hz). Frequency is a pivotal property because it determines the wave's energy and the spectrum it belongs to, such as radio waves, microwaves, visible light, etc.

In our scenario, the wave's frequency is given as \( 100 \) megahertz (MHz), equivalent to \( 10^8 \) Hz. High-frequency waves like this possess higher energy and shorter wavelengths, which influence their behavior in various mediums. Understanding frequency is key because it's directly related to the wave's angular frequency \( \omega \), calculated as:

• \( \omega = 2 \pi \times \text{frequency} \)

This connection helps determine the oscillatory motions of the electric and magnetic fields.

The speed of light (\( c \)) is a fundamental constant of nature, representing how fast electromagnetic waves move through a vacuum. It’s an incredibly high speed, approximately \( 3 \times 10^8 \) meters per second. Recognizing this speed is crucial for determining the relationship between the electric and magnetic fields in electromagnetic waves.

In your exercise, the speed of light is instrumental when relating field magnitudes as noted in \( \mathbf{B} = \frac{\mathbf{E}}{c} \). This shows that a change in the electric field is reciprocated proportionately in the magnetic field due to their intrinsic linkage through \( c \). Understanding the speed of light allows you to make accurate predictions and computations about wave behavior and properties across different scenarios.