91Ó°ÊÓ

Electromagnetic waves move at the speed of light in a vacuum. The speed of light, denoted as \( c \), is a well-known constant value of \( 3 \times 10^8 \ ext{m/s} \). This speed is crucial when you're converting between wavelength and frequency, two essential properties of waves. The formula connecting wavelength \( \lambda \) and frequency \( f \) is:

• \( f = \frac{c}{\lambda} \)

In this equation, if you know the wavelength of the wave, you can easily find its frequency by dividing the speed of light by the wavelength. Remember, the units are important here. Wavelength often comes in nanometers (nm) for these equations, so be sure to convert to meters (m) by multiplying by \( 10^{-9} \).
In our example, with a wavelength of 435 nm, the conversion goes as follows:

• \( \lambda = 435 \times 10^{-9} \ ext{m} \)
• \( f = \frac{3 \times 10^8}{435 \times 10^{-9}} \approx 6.90 \times 10^{14} \ ext{Hz} \)

This relationship helps us understand that as the wavelength decreases, the frequency increases—a fundamental concept in wave physics.

Electromagnetic waves consist of oscillating electric and magnetic fields that are perpendicular to each other and to the direction of wave propagation. The relationship between the electric field \( E \) and the magnetic field \( B \) is direct and can be given by the formula:

• \( E = c \cdot B \)

Here, \( c \) is again the speed of light. The electric field's amplitude \( E_0 \) can be used to find the magnetic field's amplitude \( B_0 \) using:

• \( B_0 = \frac{E_0}{c} \)

In practice, if you know the amplitude of the electric field, finding the magnetic field's amplitude is straightforward. For our scenario, with an electric field amplitude of \( 2.70 \times 10^{-3} \ ext{V/m} \):

• \( B_0 = \frac{2.70 \times 10^{-3}}{3 \times 10^8} = 9.00 \times 10^{-12} \ ext{T} \)

This illustrates how intimately tied these two fields are in electromagnetic waves. They sustain each other as the wave propagates through space.

Understanding wave equations is essential for describing electromagnetic wave behavior. The vector wave equations for the electric field \( \overrightarrow{E} \) and the magnetic field \( \overrightarrow{B} \) involve amplitude, wave number \( k \), and angular frequency \( \omega \).For an electric field propagating in the \(-z\)-direction and oscillating along the \(x\)-axis, the equation is:

• \( \overrightarrow{E}(z, t) = E_0 \cos(kz + \omega t) \hat{i} \)

Here:

• \( E_0 = 2.70 \times 10^{-3} \ ext{V/m} \)
• \( k = \frac{2\pi}{\lambda} \approx 1.45 \times 10^7 \ ext{m}^{-1} \)
• \( \omega = 2\pi f \approx 4.33 \times 10^{15} \ ext{rad/s} \)

The corresponding magnetic field, perpendicular and propagating in the same direction, can be expressed as:

• \( \overrightarrow{B}(z, t) = B_0 \cos(kz + \omega t) \hat{j} \)
• \( B_0 = 9.00 \times 10^{-12} \ ext{T} \)

Wave equations show the spatial and temporal dependence of the fields, depicting how they vary over time and space. These relationships govern how electromagnetic waves travel through a vacuum, maintaining the interconnected nature of their fields.