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We often encounter light that is polarized, meaning that its electric field oscillates in specific directions. In the given problem, the magnetic field of the electromagnetic wave is provided along the x-axis, indicated by the term \( B_x \).
This suggests that this component of the wave oscillates in the x-direction.

For electromagnetic waves, both electric (\( E \)) and magnetic (\( B \)) field components oscillate perpendicular to each other, as well as perpendicular to the direction of wave propagation.
If \( B_x \) is aligned along the x-axis, the electric field must be aligned along the z-axis for the wave to propagate along the y-axis, based on the right-hand rule and the nature of electromagnetic wave behavior.

• The plane containing the oscillations is thus the xz-plane.
• The wave travels along the y-axis.
• This configuration is typical for polarized light traveling in one specified direction.

Polarization is an essential feature of electromagnetic waves that has significant practical applications across physics and engineering.

Frequency is a fundamental property of waves that tells us how many cycles pass a point per second. We need to determine the frequency of the light wave in this problem.
The relationship between frequency \( f \), wave number \( k \), and the speed of light \( c \) is given by the formula \( f = \frac{c \cdot k}{2\pi} \).
Here, the speed of light \( c \) is \( 3.00 \times 10^8 \text{ m/s} \) and the wave number \( k \) is \( 1.57 \times 10^7 \text{ m}^{-1} \).

Substituting these values into the formula, we get:

• \( f = \frac{3.00 \times 10^8 \times 1.57 \times 10^7}{2\pi} \)
• Calculating this gives a frequency of approximately \( 7.50 \times 10^{14} \text{ Hz} \).

This high frequency is typical for visible light, falling within the optical range of electromagnetic spectrum, revealing much about the energy and behavior of the wave.

Wave intensity is a measure of the energy passing through a surface area in a given time. For electromagnetic waves, intensity is calculated using the formula \( I = \frac{1}{2} c \varepsilon_0 E^2_{\text{max}} \).
Here, \( \varepsilon_0 \) is the electric constant, \( 8.85 \times 10^{-12} \text{ F/m} \), and \( E_{\text{max}} \) is the maximum electric field strength.
The maximum electric field is calculated from the maximum magnetic field, using \( E_{\text{max}} = c \cdot B_{\text{max}} \), where \( B_{\text{max}} = 4.0 \times 10^{-6} \text{ T} \).

• The maximum electric field, \( E_{\text{max}} = 3.00 \times 10^8 \times 4.0 \times 10^{-6} = 1.2 \times 10^3 \text{ V/m} \).
• The intensity calculation becomes: \( I = \frac{1}{2} \times 3.00 \times 10^8 \times 8.85 \times 10^{-12} \times (1.2 \times 10^3)^2 \)
• This calculates to an intensity of approximately \( 1.90 \times 10^3 \text{ W/m}^2 \).

Intensity is crucial for understanding how much energy is transferred by a wave, impacting everything from how we perceive light to how signals are transmitted in communication systems.

The wave number is a concept often used to express the spatial frequency of a wave, detailing how many wavelengths fit into a unit distance.
It is denoted by \( k \) and defined as \( k = \frac{2\pi}{\lambda} \), where \( \lambda \) is the wavelength of the wave.

In the problem, you are provided with a wave number \( k = 1.57 \times 10^7 \text{ m}^{-1} \), which tells you the number of wave cycles per meter. This is a way to quantify how quickly the wave oscillates in space, which complements the concept of frequency that describes oscillations in time.

• Higher wave numbers mean shorter wavelengths and are typical for visible light and beyond, like ultraviolet or X-rays.
• Knowing the wave number helps to understand the scale and properties of electromagnetic waves in space.

The concept of the wave number is pivotal in physics as it assists in bridging the wave properties in space and time dynamics, providing a complete view of wave behavior.