Wavefunctions of EM Waves: Videos & Practice Problems

Electromagnetic waves are transverse waves characterized by oscillating electric (E) and magnetic (B) fields, which are always in phase. Their wave functions can be expressed using sinusoidal functions, with the maximum values denoted as E_max and B_max. The wave number (k) is calculated as $(2\pi /\lambda )$, and angular frequency (ω) can be derived from the speed of light (c) and k. Understanding these concepts is crucial for solving problems related to electromagnetic radiation.

Wavefunctions of EM Waves

Electromagnetic waves are a type of transverse wave, characterized by oscillating electric (E) and magnetic (B) fields that are perpendicular to each other and to the direction of wave propagation. Understanding how to write the wave functions for these fields is essential for solving related problems. The wave functions for electromagnetic waves share a similar structure to those of mechanical waves, but with some key differences due to the nature of the fields involved.

The wave function for electromagnetic waves can be expressed using sinusoidal functions, either sine or cosine, depending on the initial conditions of the wave. The general form of the wave function for the electric field can be written as:

\[E(y, t) = E_{\text{max}} \sin(kx - \omega t)\]

For the magnetic field, the wave function is similarly structured:

\[B(z, t) = B_{\text{max}} \sin(kx - \omega t)\]

In these equations, E_max and B_max represent the maximum electric and magnetic field strengths, respectively. The term k is the wave number, defined as:

\[k = \frac{2\pi}{\lambda}\]

where λ is the wavelength of the wave. The angular frequency ω can be calculated using:

\[\omega = 2\pi f\]

or, for electromagnetic waves, it can also be expressed as:

\[\omega = v \cdot k\]

where v is the speed of light (approximately \(3 \times 10^8 \, \text{m/s}\)). It is important to note that the electric and magnetic fields are always in phase, meaning they reach their maximum and minimum values simultaneously, which simplifies the process of determining their wave functions.

To illustrate this, consider an infrared laser emitting a wavelength of 10 micrometers. The electric field oscillates parallel to the y-axis with a maximum value of 1.5, while the magnetic field oscillates parallel to the z-axis. The wave functions can be derived by first calculating the wave number and angular frequency:

1. Calculate the wave number:

\[k = \frac{2\pi}{10 \times 10^{-6}} \approx 6.28 \times 10^5 \, \text{m}^{-1}\]

2. Calculate the angular frequency:

\[\omega = c \cdot k = (3 \times 10^8) \cdot (6.28 \times 10^5) \approx 1.88 \times 10^{14} \, \text{s}^{-1}\]

3. Determine B_max using the relationship between electric and magnetic fields:

\[B_{\text{max}} = \frac{E_{\text{max}}}{c} = \frac{1.5 \times 10^6}{3 \times 10^8} \approx 5 \times 10^{-3} \, \text{T}\]

With all variables determined, the final wave functions can be expressed as:

\[E(y, t) = 1.5 \times 10^6 \sin(6.28 \times 10^5 x - 1.88 \times 10^{14} t)\]

\[B(z, t) = 5 \times 10^{-3} \sin(6.28 \times 10^5 x - 1.88 \times 10^{14} t)\]

This demonstrates how to derive the wave functions for electromagnetic waves, highlighting the similarities and differences compared to mechanical waves while emphasizing the importance of understanding the relationships between the various parameters involved.

Electromagnetic waves produced byX-raymachines typically haveafrequencyof approximately$3.5\times10^{16}Hz$ .What is the wave numberof thesewaves?

• A. $2.20\times10^{17}m^{-1}$
• B. $1.80\times10^{-16}m^{-1}$
• C. $1.17\times10^8m^{-1}$
• D. $7.33\times10^8m^{-1}$

Answer: (D)

Example 1

In this example, we explore the wave function of an electromagnetic wave represented by the equation \( E = 54 \sin(2.2 \times 10^7 x - \omega t) \). Here, the term \( 2.2 \times 10^7 \) corresponds to the wave number \( k \), while the angular frequency \( \omega \) remains unknown. The amplitude of the electric field, denoted as \( E_{\text{max}} \), is given as 54.

To find the amplitude of the magnetic field, \( B_{\text{max}} \), we can use the relationship between the electric field and the magnetic field in electromagnetic waves, expressed as:

\( E_{\text{max}} = c \cdot B_{\text{max}} \)

Where \( c \) is the speed of light, approximately \( 3 \times 10^8 \) m/s. Rearranging this formula allows us to solve for \( B_{\text{max}} \):

\( B_{\text{max}} = \frac{E_{\text{max}}}{c} \)

Substituting the known values, we calculate:

\( B_{\text{max}} = \frac{54}{3 \times 10^8} = 1.8 \times 10^{-7} \) T (Teslas).

Next, we determine the frequency \( f \) of the electromagnetic wave. Although we do not have \( \omega \), we can relate \( k \) and \( f \) using the equation:

\( 2 \pi f = c \cdot k \)

Rearranging this gives us:

\( f = \frac{c \cdot k}{2 \pi} \)

Substituting the known values, we find:

\( f = \frac{(3 \times 10^8) \cdot (2 \times 10^7)}{2 \pi} \approx 9.55 \times 10^{14} \) Hz.

In summary, we have calculated the amplitude of the magnetic field as \( 1.8 \times 10^{-7} \) T and the frequency of the electromagnetic wave as \( 9.55 \times 10^{14} \) Hz. Understanding these relationships between electric and magnetic fields is crucial in the study of electromagnetic waves.

The magnetic field of a travelling electromagnetic wave is described by the wave function $B\left(z,t\right)=1.0\times10^{-3}\sin\left(kz-1.27\times10^{12}t\right)$, where k is the wave number. Write the complete wave function for the electric field of this wave.

• A. $E(z,t)=1.0\times {10}^3\sin (3.81\times {10}^{20}z-1.27\times {10}^{12}t)$
• B. $E(z,t)=3.0\times {10}^{11}\sin (3.81\times {10}^{20}z-1.27\times {10}^{12}t)$
• C. $E(z,t)=3.0\times {10}^5\sin (4.23\times {10}^{3}z-1.27\times {10}^{12}t)$
• D. $E(z,t)=1.0\times {10}^3\sin (4.23\times {10}^{3}z-1.27\times {10}^{12}t)$

Answer: (C)

Here’s what students ask on this topic:

What is the wave function of an electromagnetic wave?

The wave function of an electromagnetic wave describes the oscillating electric (E) and magnetic (B) fields. These functions can be expressed using sinusoidal functions. For the electric field, the wave function is:

$E(x,t)=E{\max }_{}sin(\mathrm{kx}-\mathrm{\omega t})$

For the magnetic field, the wave function is:

$B(x,t)=B{\max }_{}sin(\mathrm{kx}-\mathrm{\omega t})$

Here, $k$ is the wave number, $\omega$ is the angular frequency, and $E{\max }_{}$ and $B{\max }_{}$ are the maximum values of the electric and magnetic fields, respectively.

How do you calculate the wave number (k) for an electromagnetic wave?

The wave number $k$ for an electromagnetic wave is calculated using the wavelength $\lambda$. The formula is:

$k=\frac{2\pi }{\lambda }$

Here, $\lambda$ is the wavelength of the electromagnetic wave. This relationship shows that the wave number is inversely proportional to the wavelength.

What is the relationship between the electric field (E) and magnetic field (B) in an electromagnetic wave?

In an electromagnetic wave, the electric field (E) and magnetic field (B) are always in phase and perpendicular to each other. They oscillate sinusoidally and reach their maximum, minimum, and zero values simultaneously. The relationship between their magnitudes is given by:

$E=cB$

Here, $c$ is the speed of light in a vacuum. This equation shows that the electric field strength is proportional to the magnetic field strength, with the speed of light as the proportionality constant.

How do you determine the angular frequency (ω) of an electromagnetic wave?

The angular frequency $\omega$ of an electromagnetic wave can be determined using the wave number $k$ and the speed of light $c$. The formula is:

$\omega =ck$

Alternatively, it can also be calculated using the frequency $f$:

$\omega =2\pi f$

These relationships show that the angular frequency is directly proportional to both the wave number and the frequency of the wave.

What does it mean for the electric and magnetic fields to be 'in phase' in an electromagnetic wave?

When the electric (E) and magnetic (B) fields are 'in phase' in an electromagnetic wave, it means that they reach their maximum, minimum, and zero values simultaneously. Mathematically, this is represented by having the same $\mathrm{kx}-\mathrm{\omega t}$ term in their wave functions:

$E(x,t)=E{\max }_{}sin(\mathrm{kx}-\mathrm{\omega t})$

$B(x,t)=B{\max }_{}sin(\mathrm{kx}-\mathrm{\omega t})$

This phase relationship ensures that the fields are synchronized in their oscillations, which is a fundamental characteristic of electromagnetic waves.