Question

An electromagnetic wave propagating in vacuum has electric and magnetic fields given by \(\vec{E}(\vec{x}, t)=\vec{E}_{0} \cos (\vec{k} \cdot \vec{x}-\omega t)\) and \(\vec{B}(\vec{x}, t)=\vec{B}_{0} \cos (\vec{k} \cdot \vec{x}-\omega t)\) where \(\vec{B}_{0}\) is given by \(\vec{B}_{0}=\vec{k} \times \vec{E}_{0} / \omega\) and the wave vector \(\vec{k}\) is perpendicular to both \(\vec{E}_{0}\) and \(\vec{B}_{0} .\) The magnitude of \(\vec{k}\) and the angular frequency \(\omega\) satisfy the dispersion relation, \(\omega /|\vec{k}|=\left(\mu_{0} \epsilon_{0}\right)^{-1 / 2},\) where \(\mu_{0}\) and \(\epsilon_{0}\) are the permeability and permittivity of free space, respectively. Such a wave transports energy in both its electric and magnetic fields. Calculate the ratio of the energy densities of the magnetic and electric fields, \(u_{B} / u_{E}\), in this wave. Simplify your final answer as much as possible.

Short answer

Answer: The ratio of magnetic to electric field energy densities in an electromagnetic wave propagating in a vacuum is \(\dfrac{\mu_0}{\epsilon_0}\).

Step-by-step solution

1. Write the formula for magnetic field energy density

The energy density of the magnetic field is given by the following expression: \(u_B = \dfrac{1}{2\mu_0} |\vec{B}|^2\)

2. Write the formula for electric field energy density

The energy density of the electric field is given by the following expression: \(u_E = \dfrac{1}{2\epsilon_0} |\vec{E}|^2\)

3. Calculate the ratio of energy densities

To find the ratio of energy densities (\(\dfrac{u_B}{u_E}\)), we can divide the magnetic field energy density by the electric field energy density: \(\dfrac{u_B}{u_E} = \dfrac{\dfrac{1}{2\mu_0} |\vec{B}|^2}{\dfrac{1}{2\epsilon_0} |\vec{E}|^2}\)

4. Simplify the expression using given relationships between \(\vec{B}_0\) and \(\vec{E}_0\)

We can simplify the expression using the relationship \(\vec{B}_0 = \dfrac{\vec{k} \times \vec{E}_0}{\omega}\): \(|\vec{B}_0| = \dfrac{|\vec{k}||\vec{E}_0|}{\omega}\) We will square this equation and then substitute into the expression for the ratio: \(\dfrac{u_B}{u_E} = \dfrac{\dfrac{1}{2\mu_0} \left(\dfrac{|\vec{k}||\vec{E}_0|}{\omega}\right)^2}{\dfrac{1}{2\epsilon_0} |\vec{E}_0|^2}\)

5. Simplify the ratio of energy densities using the dispersion relation

We can simplify the expression further by using the dispersion relation, \(\omega / |\vec{k}| = \left(\mu_0 \epsilon_0\right)^{-1 / 2}\): \(\omega = |\vec{k}| \left(\mu_0 \epsilon_0\right)^{-1 / 2}\) Now, we will square this equation and substitute it into the expression for the ratio: \(\dfrac{u_B}{u_E} = \dfrac{\dfrac{1}{2\mu_0} \left(\dfrac{|\vec{k}||\vec{E}_0|}{|\vec{k}|\left(\mu_0 \epsilon_0\right)^{-1 / 2}}\right)^2}{\dfrac{1}{2\epsilon_0} |\vec{E}_0|^2}\)

6. Final simplification

With careful simplification, we reach the final expression for the ratio of energy densities: \(\dfrac{u_B}{u_E} = \dfrac{\mu_0}{\epsilon_0}\) The ratio of magnetic to electric field energy densities in the electromagnetic wave propagating in a vacuum is \(\dfrac{\mu_0}{\epsilon_0}\).

Key concepts

Magnetic Field Energy Density

Understanding the concept of magnetic field energy density (\textbf{u}_B) is essential for grasping the dynamics of electromagnetic waves. Energy density refers to the amount of energy stored in a given system or region of space per unit volume. In the context of a magnetic field, the energy density is expressed mathematically as:
\[\begin{equation} u_B = \frac{1}{2\mu_0} |\vec{B}|^2\end{equation}\] Where \(\mu_0\) represents the permeability of free space and \(|\vec{B}|\) is the magnitude of the magnetic field vector. This quantity becomes significant when analyzing how much energy is carried by the magnetic component of an electromagnetic wave. In vacuum, the energy is solely due to the fields as there are no particles to carry kinetic or potential energy.
Furthermore, the factor of \(\frac{1}{2}\) arises from the averaging over a full cycle of the sinusoidal wave function, which gives us a clearer picture of the 'steady state' energy in the wave. This consideration is important when comparing the magnetic energy density to the electric energy density in an electromagnetic wave.
To put this into a real-world context, consider that all electronic devices that emit electromagnetic waves, like radios or MRI machines, rely on the principles of magnetic field energy density to operate effectively. In these devices, control over the energy transferred by the electromagnetic waves is crucial for their functionality.

Electric Field Energy Density

Parallel to the concept of magnetic field energy density, the electric field energy density (\textbf{u}_E) plays a symmetric role in the context of electromagnetic waves. It quantifies the amount of electrical energy stored per unit volume and is given by:
\[\begin{equation} u_E = \frac{1}{2\epsilon_0} |\vec{E}|^2\end{equation}\]Here, \(\epsilon_0\) is the permittivity of free space, and \(|\vec{E}|\) is the magnitude of the electric field vector. Just as with magnetic field energy density, the prefactor of \(\frac{1}{2}\) accounts for the average energy stored over a complete cycle of the wave.
Understanding electric field energy density is crucial for various applications, including the transmission of power and information through the electromagnetic spectrum. For instance, the intensity of light, one form of electromagnetic radiation, depends on the electric field energy density; thereby, it has direct implications for technologies such as optical fibers and lasers.
The comparison of electric and magnetic field energy densities also reveals insights into the interplay between electric and magnetic fields as they propagate through space. It's this balance of energies that governs the stable propagation of light and radio waves through the vacuum of space.

Dispersion Relation

The dispersion relation is a fundamental concept linking the properties of wave-like phenomena to their propagation through different mediums. In the vacuum, the dispersion relation for electromagnetic waves is expressed as:
\[\begin{equation} \frac{\omega}{|\vec{k}|} = \left(\mu_0 \epsilon_0\right)^{-1 / 2}\end{equation}\] This relationship connects the angular frequency \(\omega\) and the wave vector \(|\vec{k}|\), which are intrinsic properties of the wave, to the physical constants \(\mu_0\) and \(\epsilon_0\), which characterize the medium—in this case, the vacuum.
In simpler terms, the dispersion relation tells us how the phase speed of the wave depends on its frequency. For electromagnetic waves in a vacuum, this relation reveals that all frequencies travel at the same speed—the speed of light. This is because the vacuum is non-dispersive and does not separate light into different colors as a prism would do to visible light.
The concept of dispersion is crucial for understanding wave behavior in various disciplinary contexts beyond vacuum, such as in fiber optics, acoustics, and even quantum mechanics. For instance, the color spectrum displayed when light passes through a glass prism is a direct consequence of the dispersion relation changing with the medium, revealing its pivotal role in the study of wave phenomena.

Wave Propagation in Vacuum

Wave propagation in vacuum is characterized by the motion of waves through empty space, devoid of any medium. For electromagnetic waves, this process is governed by Maxwell's equations, which describe how electric and magnetic fields evolve and propagate in space and time.
In a vacuum, electromagnetic waves propagate without losing energy to the medium, allowing them to travel at the speed of light (approximately \( 3 \times 10^8 \) meters per second). This occurs because there are no charged particles that can absorb the wave's energy and no material to cause scattering or absorption. The result is a clean, unimpeded path for the electromagnetic wave represented by the equation:
\[\begin{equation} c = \left(\mu_0 \epsilon_0\right)^{-1 / 2}\end{equation}\] Where 'c' is the speed of light. Electromagnetic wave propagation forms the basis of many modern technologies—GPS systems, telecommunications, and space exploration, to name a few, all leverage this core principle of physics.
This concept of wave propagation also allows us to understand astronomical phenomena, as light from distant stars and galaxies travels through the vacuum of space to reach us, revealing the nature and motion of these celestial bodies. Hence, wave propagation in a vacuum is not just a foundational concept in physics but also an essential element in explaining the observable universe.