Problem 7 - University Physics with Modern Physics

Question

A sinusoidal electromagnetic wave of frequency \(6.10 \times 10^{14} \mathrm{Hz}\) travels in vacuum in the \(+z\) -direction. The \(\overrightarrow{\boldsymbol{B}}\) -field is parallel to the \(y\) -axis and has amplitude \(5.80 \times 10^{-4}\) T. Write the vector equations for \(\overrightarrow{\boldsymbol{E}}(z, t)\) and \(\overrightarrow{\boldsymbol{B}}(z, t) .\)

Step-by-Step Solution

Answer

The electric field is \(\overrightarrow{\boldsymbol{E}}(z, t) = (1.74 \times 10^{5}) \sin(1.28 \times 10^{7}z - 3.83 \times 10^{15}t) \hat{i}\) and the magnetic field is \(\overrightarrow{\boldsymbol{B}}(z, t) = (5.80 \times 10^{-4}) \sin(1.28 \times 10^{7}z - 3.83 \times 10^{15}t) \hat{j}\)."

Step 1: Identify the Wave Characteristics

The electromagnetic wave is sinusoidal and traveling in the +z-direction. We know that the magnetic field (\(\overrightarrow{\boldsymbol{B}}\)) is parallel to the y-axis, and it has an amplitude of \(5.80 \times 10^{-4}\) T. The frequency of the wave is given as \(6.10 \times 10^{14}\, \mathrm{Hz}.\) From this, we need to find the expressions for both the electric field (\(\overrightarrow{\boldsymbol{E}}(z, t)\)) and the magnetic field (\(\overrightarrow{\boldsymbol{B}}(z, t)\)).

Step 2: Find the Angular Frequency and Wavelength

The angular frequency \(\omega\) is calculated using the formula \(\omega = 2\pi f\), where \(f = 6.10 \times 10^{14}\, \mathrm{Hz}.\) Thus, \(\omega = 2\pi \times 6.10 \times 10^{14}\, \mathrm{Hz} = 3.83 \times 10^{15}\, \mathrm{rad/s}.\) The wavelength \(\lambda\) is found using the speed of light in vacuum, \(c = 3.00 \times 10^{8}\, \mathrm{m/s}.\) The relation is \(\lambda = \frac{c}{f}\). Substituting the values, \(\lambda = \frac{3.00 \times 10^{8}\, \mathrm{m/s}}{6.10 \times 10^{14}\, \mathrm{Hz}} = 4.92 \times 10^{-7}\, \mathrm{m}.\)

Step 3: Write the Magnetic Field Equation

The magnetic field vector \(\overrightarrow{\boldsymbol{B}}(z, t)\) is parallel to the y-axis and can be expressed as: \[\overrightarrow{\boldsymbol{B}}(z, t) = B_0 \sin(kz - \omega t + \phi_B) \hat{j}\]where \(B_0 = 5.80 \times 10^{-4}\, \mathrm{T}\) is the amplitude, \(k = \frac{2\pi}{\lambda} = \frac{2\pi}{4.92 \times 10^{-7}\, \mathrm{m}} = 1.28 \times 10^{7}\, \mathrm{m}^{-1}\) is the wave number, and assuming zero phase constant \(\phi_B = 0\). Thus, \[\overrightarrow{\boldsymbol{B}}(z, t) = (5.80 \times 10^{-4}) \sin(1.28 \times 10^{7}z - 3.83 \times 10^{15}t) \hat{j}\]

Step 4: Use the Relationship Between E and B to Find the Electric Field

In a vacuum, the magnitudes of \(\overrightarrow{\boldsymbol{E}}\) and \(\overrightarrow{\boldsymbol{B}}\) are related by \(c = \frac{E_0}{B_0}\). Therefore, \[E_0 = c \cdot B_0 = (3.00 \times 10^{8}\, \mathrm{m/s}) \cdot (5.80 \times 10^{-4}\, \mathrm{T}) = 1.74 \times 10^{5}\, \mathrm{V/m}.\] The direction of \(\overrightarrow{\boldsymbol{E}}\) is perpendicular to the direction of \(\overrightarrow{\boldsymbol{B}}\) and the direction of wave propagation. Thus, \(\overrightarrow{\boldsymbol{E}}\) is parallel to the x-axis.

Step 5: Write the Electric Field Equation

The electric field vector \(\overrightarrow{\boldsymbol{E}}(z, t)\) can be expressed as: \[\overrightarrow{\boldsymbol{E}}(z, t) = E_0 \sin(kz - \omega t + \phi_E) \hat{i}\]where \(E_0 = 1.74 \times 10^{5}\, \mathrm{V/m}\) is the amplitude, the wave number \(k\) and angular frequency \(\omega\) are the same as those for \(\overrightarrow{\boldsymbol{B}}(z, t)\). Assuming zero phase constant \(\phi_E = 0\), \[\overrightarrow{\boldsymbol{E}}(z, t) = (1.74 \times 10^{5}) \sin(1.28 \times 10^{7}z - 3.83 \times 10^{15}t) \hat{i}\]

Key Concepts

Sinusoidal Waves

A sinusoidal wave is a type of wave that oscillates in intensity and is represented by the mathematical function sinus or cosine. In the electromagnetic spectrum, these waves can be found in light, radio waves, and more.
Sinusoidal waves are characterized by their smooth, repetitive oscillations, which are similar to the motion of a sine curve. These waves have peaks and troughs, where the peak is the maximum displacement from its equilibrium position and the trough is the lowest.

• Think of a sinusoidal wave in terms of its amplitude, frequency, and wavelength, each critical in defining its behavior.
• The frequency refers to how many cycles occur in a second, usually measured in Hertz (Hz).
• The wavelength is the distance between consecutive peaks or troughs.
• Amplitude is the wave's height from the center line to a peak. In electromagnetic waves, amplitude is related to the strength of the electric or magnetic field.

Sinusoidal waves are central in understanding the propagation of electromagnetic waves because they neatly illustrate how fields vary in space and time.

Wave Number

The wave number is an essential concept when studying waves, particularly electromagnetic ones. Denoted as \( k \), the wave number is a measure of how many wave cycles fit into a unit distance. It describes the spatial frequency of a wave, basically how `crammed' the waves are into a given space.
Wave number \( k \) is calculated using the formula \( k = \frac{2\pi}{\lambda} \), where \( \lambda \) is the wavelength of the wave. The wave number gives insight into the wave's spatial variation and is expressed in radians per meter (rad/m).

• Smaller wavelengths result in larger wave numbers, meaning more cycles fit into a given distance.
• The wave number helps in determining how rapidly the wave oscillates in space—which is opposite to frequency, that describes time oscillations.
• In electromagnetic waves, knowing the wave number is crucial to understand how the wave interacts with different materials.

The wave number serves an important role, especially when creating equations for wave propagation through different mediums.

Angular Frequency

Angular frequency is another vital aspect of wave characteristics, denoted as \( \omega \). It describes how quickly the wave oscillates with time.
Given in radians per second, it provides an understanding of how fast these oscillations occur. To calculate angular frequency, use the formula \( \omega = 2\pi f \), where \( f \) is the frequency in Hertz.

• Higher angular frequencies indicate that the wave oscillates more rapidly.
• It represents the rate of rotation in a circular motion if that circular path was the wave.
• Angular frequency is particularly important in electromagnetism because it also relates to the energy of photons in a given wave.

Understanding angular frequency is key to grasping wave dynamics and energy distribution in electromagnetic waves.

Electric and Magnetic Fields Equations

In electromagnetic waves, electric and magnetic fields are interlinked and form sinusoidal wave patterns.
The electromagnetic equations for these fields are used to predict wave behavior, and they reflect how fields vary over time and distance.

• The electric field \( \overrightarrow{E}(z, t) \) and magnetic field \( \overrightarrow{B}(z, t) \) are expressed in sinusoidal forms such as \[ \overrightarrow{E}(z, t) = E_0 \sin(kz - \omega t) \hat{i} \]
• Correspondingly, the magnetic field is \[ \overrightarrow{B}(z, t) = B_0 \sin(kz - \omega t) \hat{j} \]
• Here, \( E_0 \) and \( B_0 \) are the amplitudes, and \( kz - \omega t \) describes wave displacement.

In these equations, the wave number \( k \) and angular frequency \( \omega \) identify how these fields oscillate in space and time. The electric and magnetic fields are always perpendicular to each other and the direction of wave propagation, following the right-hand rule.