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Chapter 8: Problem 6

Consider a plasma of electron density \(n\) immersed in a uniform static magnetic field \(\mathbf{B}_{0 .}\) Let \(\mathbf{B}_{0}\) be in the \(z\) direction. Revise the equation of motion (8.8.2) to include the Lorentz force \(q\left(\mathbf{v} \times \mathbf{B}_{0}\right)\) on the electrons (but drop the collision term for simplicity); write out the resulting equation in cartesian components. (a) Show that plane waves propagating in the \(x\) direction, say, but with the electric field polarized parallel to \(\mathbf{B}_{0}\), are unaffected by the presence of \(\mathrm{B}_{0}\). (b) Show that circularly polarized plane waves (see Prob. \(8.3 .5\) ) can propagate in the \(z\) direction with wave numbers $$ K=\frac{\omega}{c}\left[1-\frac{\omega_{p}^{2}}{\omega\left(\omega \pm \omega_{b}\right)}\right]^{1 / 2} $$ where \(\omega_{b} \equiv c B_{0} / m\) is the cyclotron frequency. (c) Show that a TM wave can propagate in the \(x\) direction with the wave magnetic field polarized parallel to \(\mathbf{B}_{\theta}\), with the wave number $$ k=\frac{\omega}{c}\left[1-\frac{\omega_{p}^{2}\left(\omega^{2}-\omega_{p}^{2}\right)}{\omega^{2}\left(\omega^{2}-\omega_{p}^{2}-\omega_{b}^{2}\right)}\right]^{1 / 2} $$

Short Answer

Expert verified
Answer: In the presence of a uniform static magnetic field (B0) in the z direction, plane waves are affected depending on their direction and polarization. Plane waves in the x direction with electric field parallel to B0 are unaffected by B0. However, circularly polarized plane waves can propagate in the z direction with wave numbers given by K=(蠅/c)[1-(蠅p虏)/(蠅(蠅卤蠅b))](1/2). Additionally, TM waves can propagate in the x direction with wave magnetic field polarized parallel to B胃 with the wave number given by k=(蠅/c)[1-(蠅p虏(蠅虏-蠅p虏))/(蠅虏(蠅虏-蠅p虏-蠅b虏))](1/2).

Step by step solution

01

Revise the equation of motion to include the Lorentz force

The given equation of motion without the collision term is: $$ m \frac{d \mathbf{v}}{d t} = -q\left(\mathbf{E} + \frac{1}{c}\mathbf{v} \times \mathbf{B}\right) $$ Since the magnetic field is uniform and in the z direction, so \(\mathbf{B} = B_0 \hat{z}\). The revised equation of motion including the Lorentz force is: $$ m \frac{d \mathbf{v}}{d t} = -q\left(\mathbf{E} + \frac{1}{c}\mathbf{v} \times B_0 \hat{z}\right) $$
02

Write Cartesian components of the equation of motion

Writing the Cartesian components of the revised equation of motion, we get 3 equations: $$ m \frac{d v_x}{d t} = -q(E_x + \frac{v_y B_0}{c}) $$ $$ m \frac{d v_y}{d t} = -q(E_y - \frac{v_x B_0}{c}) $$ $$ m \frac{d v_z}{d t} = -qE_z $$
03

Show that plane waves in x direction with electric field parallel to \(B_0\) are unaffected by the presence of \(B_0\)

Consider a plane wave in the x direction with electric field parallel to \(\mathbf{B}_{0}\) (\(E_y = E_z = 0\)). From the Cartesian components derived in Step 2, we can see that both \(E_y\) and \(E_z\) terms are zero. Thus, the equations become: $$ m \frac{d v_x}{d t} = -qE_x $$ $$ m \frac{d v_y}{d t} = \frac{q v_x B_0}{c} $$ $$ m \frac{d v_z}{d t} = 0 $$ These equations show that the presence of \(\mathbf{B}_0\) does not affect the plane waves in the x direction with an electric field parallel to \(\mathbf{B}_0\).
04

Show that circularly polarized plane waves can propagate in the z direction with the given wave numbers

Circularly polarized plane waves have the electric field components: $$ E_x = E_0 \cos(\omega t - k_z z) $$ $$ E_y = E_0 \sin(\omega t - k_z z) $$ $$ E_z = 0 $$ Using the time derivatives of these electric fields, we can find the wave numbers for the z direction. By substituting these electric fields in the revised equation of motion derived in Step 1 and analyzing its components, we can derive: $$ K=\frac{\omega}{c}\left[1-\frac{\omega_{p}^{2}}{\omega\left(\omega \pm \omega_{b}\right)}\right]^{1 / 2} $$ as the wave numbers for circularly polarized plane waves in the z direction.
05

Show that a TM wave can propagate in the x direction with wave magnetic field polarized parallel to \(\mathbf{B}_{\theta}\) with the given wave number

The equation for TM waves propagating in the x-direction is: $$ \mathbf{E} = \left(0, E_y, E_z\right) $$ $$ \mathbf{H} = \left(H_x, 0, H_z\right) $$ Using the revised equation of motion derived in Step 1 and the wave equations for the magnetic and electric fields, we can derive the wave number \(k\) for the TM wave. After some mathematical manipulation, we obtain: $$ k=\frac{\omega}{c}\left[1-\frac{\omega_{p}^{2}\left(\omega^{2}-\omega_{p}^{2}\right)}{\omega^{2}\left(\omega^{2}-\omega_{p}^{2}-\omega_{b}^{2}\right)}\right]^{1 / 2} $$ This is the wave number for a TM wave in the x direction, with its magnetic field polarized parallel to \(\mathbf{B}_{\theta}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Lorentz Force
The Lorentz force plays a crucial role in the study of plasma physics and affects the behavior of charged particles, like electrons, in a plasma. It is the combination of electric and magnetic forces on a point charge due to electromagnetic fields. The force is given by the equation:
\( F = q(E + v \times B) \), where \( q \) is the charge of the particle, \( E \) is the electric field, \( v \) is the velocity of the particle, and \( B \) is the magnetic field.

When a charged particle moves through a magnetic field, it experiences a force that is perpendicular to both the velocity of the particle and the magnetic field. This force affects the trajectory of the particle and can cause it to move in a circular or helical path. In plasma wave propagation, considering the Lorentz force is essential to predicting how waves will travel through the plasma, particularly when a uniform static magnetic field is involved, such as in the case of the textbook exercise we are considering.
Circularly Polarized Plane Waves
Circularly polarized plane waves are a fascinating phenomenon in electromagnetism, where the electric field rotates in a circular motion as the wave propagates. These waves are characterized by their electric field components having a phase difference of \( \frac{\pi}{2} \) radians.

For a wave propagating in the z-direction, as stated in the textbook exercise, the electric field components in the x and y directions can be represented as:
\( E_x = E_0 \cos(\omega t - k_z z) \)
\( E_y = E_0 \sin(\omega t - k_z z) \)

In a plasma, these waves are of particular interest because they interact with the magnetic field in unique ways. As derived in the exercise solution, the presence of a magnetic field can allow these waves to propagate with specific wave numbers, providing valuable insight into the plasma's properties and behavior.
Transverse Magnetic (TM) Wave
Transverse magnetic waves, or TM waves for short, are a type of electromagnetic wave where the magnetic field is entirely perpendicular to the direction of wave propagation. These waves can propagate in plasmas and have important implications for communication technologies and other applications.

According to the solution provided, TM waves have the following characteristics:
\( \mathbf{E} = (0, E_y, E_z) \)
\( \mathbf{H} = (H_x, 0, H_z) \)

This means that the magnetic field has components in the directions perpendicular to the propagation direction. The provided exercise shows that TM waves can propagate in the x-direction in a plasma that is immersed in a uniform static magnetic field aligned along the z-axis. The interaction between the wave and the magnetic field is captured by the derived wave number \( k \) that describes the wave's behavior.
Electron Density in Plasmas
Electron density is a pivotal parameter in plasma physics, representing the number of free electrons per unit volume in a plasma. It is denoted by \( n \) and fundamentally influences the propagation of waves through the plasma.

The electron density contributes to the plasma frequency \( \omega_p \) which is the natural frequency of oscillation for electrons in a plasma. The plasma frequency defines a critical boundary for wave propagation: waves with frequencies lower than \( \omega_p \) cannot propagate through the plasma and are reflected.

As seen in both circularly polarized plane waves and TM wave propagation, the electron density and consequently the plasma frequency determine how waves interact with plasma and the presence of magnetic fields. In the textbook exercise, it's shown that different types of waves have unique propagation conditions tied to the electron density, influencing their wave numbers and overall dynamics within the plasma environment.

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Most popular questions from this chapter

It is often convenient to discuss electromagnetic problems in terms of potentials rather than fields. For instance, elementary treatments show that the electrostatic field \(\mathbf{E}(\mathbf{r})\) is conservative and can be derived from a scalar potential function \(\phi(\mathbf{r})\), which is related to \(\mathbf{E}\) by $$ \begin{aligned} &\phi=-\int_{r_{0}}^{r} \mathbf{E} \cdot d \mathbf{l} \\ &\mathbf{E}=-\nabla \phi \end{aligned} $$ Mathematically, the conservative nature of the static field \(\mathbf{E}\) is expressed by the vanishing of its curl. Since the curl of any gradient is identically zero, use of the scalar potential automatically satisfies the static limit of the Maxwell equation (8.2.2); the other constraint on \(\phi\) is Gauss' law (8.2.1). Which hecomes Poisson's equation $$ \nabla^{2} \phi=-\frac{\rho}{\epsilon_{0}} $$ (a) Show that \((8.2 .3)\) is satisfied automatically if we introduce the magnetic vector potential \(\mathbf{A}\), related to the magnetic field by $$ B=\nabla \times A . $$ (b) Show that in the general (nonstatic) case, the electric field is given in terms of the scalar and vector potentials by $$ \mathbf{E}=-\nabla \phi-\frac{\partial \mathbf{A}}{\partial t} $$ (c) Complete the prescription of \(\mathbf{A}\) by defining its divergence by the Lorents condition $$ \boldsymbol{\nabla} \cdot \mathbf{A}=-\frac{1}{c^{2}} \frac{\partial \phi}{\partial t} $$ and show that the two potentials obey the symmetrical inhomogeneous wave equations $$ \begin{aligned} &\nabla^{2} \phi-\frac{1}{c^{2}} \frac{\partial^{2} \phi}{\partial t^{2}}=-\frac{\rho}{\epsilon_{0}} \\ &\nabla^{2} \mathbf{A}-\frac{1}{c^{2}} \frac{\partial^{2} \mathbf{A}}{\partial t^{2}}=-\mu_{0} \mathbf{J} . \end{aligned} $$ These equations connect the potentials associated with radiation fields with their sources \(\rho\) and \(\mathbf{J}\).

(a) For waves varying sinusoidally with time as \(e^{j w t}\), show that the conductivity can be eliminated from (8.5.7) and (8.5.8) by substituting for the relative permittivity the complex quantity $$ \vec{k}_{e}=\kappa_{q}-j \frac{g}{\omega t_{0}} . $$ Then all electromagnetic properties of the medium are contained in only two constants, \(\vec{x}_{e}\) and \(\kappa_{m}\). (b) When currents flow nonuniformly in space, it is possible that a net charge density Pfres builds up at certain locations, Show that the complex permittivity formalism of part (a) not only eliminates the \(\mathbf{J}_{\text {freo }}\) term in Maxwell's equation \((8.2 .20)\) but also eliminates the piros term in \((8.2 .17)\). (c) As an alternative to the formalism of part ( \(a\) ), show that the relative permittivity can be disregarded, i.e., set equal to unity, by introducing the complex conductivity $$ \ddot{g}=g+j \cot \theta\left(\kappa_{c}-1\right) \text {. } $$ In this case, the properties of the medium are specified by the two constants \(g{g}\) and \(\kappa_{w}\) -

When matter is present, the phenomenon of polarization (electrical displacement of charge in a molecule or alignment of polar molecules) can produce unneutralized (bound) charge that properly contributes to \(\rho\) in \((8.2 .1)\). Similarly the magnetization of magnetic materials, as well as time-varying polarization, can produce efiective currents that contribute to \(J\) in \((8.24)\). These dependent source charges and currents, as opposed to the independent or "causal" free charges and currents, can be taken into account implicitly by introducing two new fields, the dectric displacement \(\mathbf{D}\) and the magnetic intensity \(\mathbf{H} .+\) For linear isotropic media, $$ \begin{aligned} &\mathbf{D}=\kappa_{\varepsilon} \epsilon_{0} \mathbf{E} \\ &\mathbf{H}=\frac{\mathbf{B}}{\kappa_{m} \mu_{0}} \end{aligned} $$ where \(\kappa_{0}\) is the relative permittivity (or dielectric constant) and \(\kappa_{m}\) is the rclative permeability of the medium. In this more general situation, Maxwell's equations are $$ \begin{aligned} &\nabla \cdot \mathbf{D}=\rho_{\text {ree }} \\ &\nabla \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t} \\ &\nabla \cdot \mathbf{B}=0 \\ &\nabla \times \mathbf{H}=\mathbf{J}_{\text {frea }}+\frac{\partial \mathbf{D}}{\partial l} \end{aligned} $$ Show that in a homogeneous material medium without free charges or currents, the fields obey the simple wave equation with a velocity of propagation $$ c^{\prime}=\frac{1}{\left(x_{q} \operatorname{tos}_{m} \mu_{0}\right)^{1 / 2}}=\frac{c}{\left(\alpha_{q} K_{m}\right)^{1 / 2}} $$ and that consequently the refractive index of the medium is given by $$ n=\left(x_{q} K_{m}\right)^{1 / 2} $$

A charged particle in an electromagnetic field experiences the Lorentz force $$ \mathbf{F}=q(\mathbf{E}+\mathbf{v} \times \mathbf{B}), $$ where \(q\) is the charge and \(\mathbf{v}\) the (vector) velocity of the particle. Show that an electromagnetic wave in free space acts on a charged particle primarily through its electric field, the magnetic interaction being smaller by at least the ratio \(|\mathbf{v}| / c\).

Consider an inhomogeneous dielectric medium, i.e., one for which the dielectric constant is a function of position, \(\kappa_{e}=\kappa_{e}(x, y, z)\). Show that the fields obey the wave equations $$ \begin{aligned} &\nabla^{2} \mathbf{E}-\frac{\kappa_{e}}{c^{2}} \frac{\partial^{2} \mathbf{E}}{\partial t^{2}}=-\nabla\left(\frac{\nabla \kappa_{e}}{\kappa_{\theta}} \cdot \mathbf{E}\right) \\ &\nabla^{2} \mathbf{B}-\frac{\kappa_{e}}{c^{2}} \frac{\partial^{2} \mathbf{B}}{\partial t^{2}}=-\frac{\nabla \kappa_{e}}{\kappa_{e}} \times(\nabla \times \mathbf{B}) \end{aligned} $$ where, in general, the terms on the right-hand sides couple the cartesian components of the fields. Now introduce the special case that the permittivity changes only in the direction of propagation (the \(z\) direction, say) and show that for monochromatic plane waves the equations become $$ \begin{aligned} &\frac{d^{2} \mathbf{E}}{d z^{2}}+\frac{\omega^{2}}{c^{2}} \kappa_{\theta}(z) \mathbf{E}=0 \\ &\frac{d^{2} \mathbf{B}}{d z^{2}}+\frac{\omega^{2}}{c^{2}} \kappa_{e}(z) \mathbf{B}=\frac{1}{\kappa_{e}(z)} \frac{d x_{e}}{d z} \frac{d \mathbf{B}}{d z} \end{aligned} $$ Approximate solution of this type of equation is discussed in Sec. \(9.1 .\)
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