Question

Show that \(E_{x s}=A e^{j\left(k_{0} z+\phi\right)}\) is a solution of the vector Helmholtz equation, Eq. (30), for \(k_{0}=\omega \sqrt{\mu_{0} \epsilon_{0}}\) and any \(\phi\) and \(A\).

Short answer

Answer: Yes, the given electric field satisfies the vector Helmholtz equation for any values of \(\phi\) and \(A\).

Step-by-step solution

Understand the vector Helmholtz equation

The vector Helmholtz equation is given by: \begin{equation} \nabla^2 \vec{E} + k_0^2 \vec{E} = 0 \end{equation} where \(\nabla^2\) is the Laplacian operator, \(\vec{E}\) is the electric field, and \(k_0\) is the wave number.

Substitute the given electric field in the vector Helmholtz equation

We are given the electric field by: \begin{equation} \vec{E}_{xs} = A e^{j(k_0z + \phi)} \end{equation} To show that this electric field is a solution of the vector Helmholtz equation, we need to substitute it in the equation and check if it satisfies it.

Calculate the Laplacian of the electric field

The Laplacian operator, when applied to the electric field, results in the second derivative with respect to all three components (x, y, and z). In our given electric field, only the z-component is variable, therefore we only need to take the second derivative with respect to z: \begin{equation} \nabla^2 \vec{E}_{xs} = \frac{\partial^2 \vec{E}_{xs}}{\partial z^2} = \frac{\partial^2}{\partial z^2}\left(A e^{j(k_0z + \phi)}\right) \end{equation} Now we can calculate the second derivative: \begin{equation} \frac{\partial^2 \vec{E}_{xs}}{\partial z^2} = A e^{j(k_0z + \phi)}(-k_0^2). \end{equation}

Substitute the Laplacian and electric field in the vector Helmholtz equation

Now we can substitute the Laplacian of the electric field and the electric field itself in the vector Helmholtz equation: \begin{equation} \nabla^2 \vec{E}_{xs} + k_0^2 \vec{E}_{xs} = A e^{j(k_0z + \phi)}(-k_0^2) + k_0^2 A e^{j(k_0z + \phi)} = 0. \end{equation} As we can see, this equation is true for any value of \(\phi\) and \(A\), so the given electric field is a solution of the vector Helmholtz equation.

Check the condition for \(k_0\)

We are given that \(k_0 = \omega \sqrt{\mu_0 \epsilon_0}\). As the given electric field is a solution of the vector Helmholtz equation, this condition is consistent with the problem statement. In conclusion, we have shown that the given electric field is a solution of the vector Helmholtz equation for any values of \(\phi\) and \(A\).

Key concepts

Electromagnetic Field

The electromagnetic field is a fundamental concept in physics that describes the physical field produced by electrically charged objects. It affects the behavior of charged objects in the vicinity of the field. An electromagnetic field consists of two parts, the electric field \textbf{E} and the magnetic field \textbf{B}, which are interconnected and interact with each other. The behavior of electromagnetic fields is governed by Maxwell's equations, which describe how electric charges and currents produce electric and magnetic fields, and how those fields interact with each other.

When we consider waves or disturbances in an electromagnetic field, we typically describe them using a mathematical function that represents the spatial and temporal variations of the field. In the context of the vector Helmholtz equation, we specifically look at wave solutions, which typically represent the propagation of electromagnetic waves through space. The function given in the exercise, \(E_{x s}=A e^{j\left(k_{0} z+\phi\right)}\), represents a wave traveling in the z-direction, with \(A\) being the amplitude, \(\phi\) the phase, and \(e^{j(k_{0} z + \phi)}\) depicting the oscillatory nature of the wave in the spatial and temporal domain.

Laplacian Operator

The Laplacian operator is a second-order differential operator that plays a key role in various areas of physics and mathematics, particularly in the study of field theory and potential theory. It is denoted by \(abla^2\) and is defined as the divergence of the gradient of a function. In three-dimensional Cartesian coordinates, the Laplacian of a scalar function \(f\) is given by \(abla^2 f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}\).

When applied to vector fields, such as an electromagnetic field, the Laplacian operator affects each component of the vector field separately. In the context of electromagnetic theory, solving the vector Helmholtz equation involving the Laplacian operator helps us find the behavior and propagation of electromagnetic waves in free space or within certain materials. When we deal with wave-like solutions, such as in the example \(\vec{E}_{xs}\), calculating the Laplacian helps determine if the proposed function is a valid representation of the physical phenomenon under study, such as the propagation of an electromagnetic wave.

Wave Number

The wave number is an important property of waves that describes the number of wave cycles that exist over a certain unit of distance. It is typically represented by \(k\) and is mathematically defined as \(k = \frac{2\pi}{\lambda}\), where \(\lambda\) is the wavelength, or the distance over which the wave's shape repeats. The wave number is directly proportional to the frequency \(u\) and inversely proportional to the wavelength.

The term \(k_0\) in the problem is referred to as the angular wave number or propagation constant and is related to the angular frequency \(\omega\) by \(k_{0}=\omega \sqrt{\mu_{0} \epsilon_{0}}\), where \(\mu_0\) is the permeability of free space, and \(\epsilon_0\) is the permittivity of free space. This relationship connects the wave number with the speed of light in vacuum and with the properties of the medium through which the wave is propagating. A wave's wave number affects its phase velocity and is crucial in the analysis of wave phenomena, including the solution of differential equations like the vector Helmholtz equation.

Partial Differential Equations

Partial differential equations (PDEs) are equations that involve the partial derivatives of a multivariable function. They are fundamental to the study of various physical phenomena, such as heat conduction, fluid flow, and the behavior of electromagnetic fields. PDEs can be used to describe the propagation, interaction, and evolution of waves and fields over time and space.

The vector Helmholtz equation is a specific type of PDE that arises in the study of wave phenomena, such as electromagnetic field propagation. It is used not just in electromagnetic theory, but also in acoustics, quantum mechanics, and other disciplines. This equation typically takes the form \(abla^2 \vec{E} + k_0^2 \vec{E} = 0\), expressing a relationship between the spatial distribution of an electromagnetic field and its rate of change. Solving such an equation often involves finding functions that satisfy the equation's constraints, as was done in the textbook exercise where we demonstrated that a given electric field expression was indeed a solution to the vector Helmholtz equation.