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Chapter 8: Problem 2

Consider an ionized gas of uniform electron density \(n\). Regard the positive ions as a smeared-out continuous fluid which renders the gas macroscopically neutral and through which the electrons can move without friction. Now assume that, by some external means, each electron is shifted in the \(x\) direction by the displacement \(\xi=\xi(x)\), a function of its initial, unperturbed location \(x\). (a) Show from Gauss' law (8.2.1) that the resulting electric field is \(E_{x}=n e \xi / e_{0} .(b)\) Show that each electron experiences a linear (Hooke's law) restoring force such that when the external forces are removed, it oscillates about the equilibrium position \(\xi=0\) with simple harmonic motion at the angular frequency $$ \omega_{p}=\left(\frac{n e^{2}}{\epsilon_{0} m}\right)^{1 / 2} $$ which is known as the electron plasma frequency. † For fuller discussion, see J. A. Ratcliffe, "The Magneto-ionic Theory and Its Applications to the Ionosphere," Cambridge University Press, New York, 1959 ; M. A. Heald and C. B. Wharton, "Plasma Diagnostics with Microwaves," John Wiley \& Sons Inc., New York, 1965; I. P. Shkarofsky, T. W. Johnson, and M. P. Bachynski, "Particle Kinetics of Plasma," Addison-Wesley Publishing Company, Reading, Mass., \(1965 .\)

Short Answer

Expert verified
Answer: The electron plasma frequency represents the natural oscillation frequency of electrons around the equilibrium position when subjected to a restoring force in an ionized gas. It can be derived by first finding the electric field created by the displaced electrons using Gauss' Law. Then, the restoring force experienced by the electrons is derived using Hooke's Law, which implies that the electrons oscillate with simple harmonic motion. Finally, the angular frequency of the electron oscillations is calculated, yielding the electron plasma frequency as \(\omega_p = \sqrt{\frac{n e^2}{\epsilon_0 m}}\).

Step by step solution

01

1. Find the electric field created by the displaced electrons

Using Gauss' Law \((8.2.1)\), which states that the electric field created by any charge distribution is given by \(\oint \mathbf{E} \cdot d \boldsymbol{A}=\frac{Q_\text{enclosed}}{\epsilon_0}\), a charge \(Q = n e \xi\) in the \(x\) direction is created due to electron displacement. Then, the component of the electric field created in the \(x\) direction is given by $$ E_x = \frac{n e \xi}{\epsilon_0}. $$
02

2. Derive the restoring force experienced by the electrons

The force exerted on the electrons due to the electric field is given by $$ F_x = -e E_x = -n e^2 \frac{\xi}{\epsilon_0}, $$ which is a linear restoring force. Therefore, Hooke's Law applies, and when the external forces are removed, each electron oscillates with simple harmonic motion around the equilibrium position \(\xi = 0\).
03

3. Calculate the angular frequency of the electron oscillations

Now, we can find the angular frequency of the oscillations. The equation of motion for the electrons can be written as $$ m \frac{d^2 \xi}{dt^2} = -n e^2 \frac{\xi}{\epsilon_0}. $$ The solution is a harmonic oscillation with angular frequency \(\omega_p\). Thus, we can rewrite the equation as $$ m \omega_p^2 \xi = n e^2 \frac{\xi}{\epsilon_0}. $$ Solving for the angular frequency \(\omega_p\), we get $$ \omega_p = \sqrt{\frac{n e^2}{\epsilon_0 m}}. $$ This result is the electron plasma frequency, which represents the natural oscillation frequency of electrons around the equilibrium position when subjected to a restoring force in an ionized gas.

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Most popular questions from this chapter

(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 \(\kappa_{w}\) -

Use the results of Prob. 8.2.3 to compute the Poynting vector for a coaxial transmission line. Integrate it over the annular area between conductors and show that the power carried down the line by the wave is $$ P=i^{2} Z_{0}=\frac{v^{2}}{Z_{0}}, $$ where \(i\) and \(v\) are the instantaneous current and voltage and \(Z_{0}\) is the characteristic impedance \((8.1 .9)\), that is, just the result one would expect from elementary circuit analysis.

Show that the general solution of the Helmboltz equation (8.7.16), obtained by separation of variables in cartesian coordinates, can be put in the form (8.7.25). Impose the boundary conditions on the electric field (8.7.9) for TE modes in rectangular waveguide to establish (8.7.27) to (8.7.29). Similarly, impose the boundary conditions on the magnetic field (8.7.12) for TM modes to establish (8.7.32) and (8.7.33).

Develop Poynting's theorem for the general material medium of relative permittivity \(\kappa_{\&}\) and permeability \(\kappa_{m}\) introduced in Prob. \(8.2 .2\); i.e., substitute the Maxwell curl equations \((8.2 .18)\) and \((8.2 .20)\) in the expansion of \(\nabla \cdot(\mathbf{E} \times \mathbf{H})\) to obtain $$ \oint_{S}(\mathbf{E} \times \mathbf{H}) \cdot d \mathbf{S}+\int_{V}\left(\mathbf{E} \cdot \frac{\partial \mathbf{D}}{\partial t}+\mathbf{H} \cdot \frac{\partial \mathbf{B}}{\partial t}\right) d v+\int_{V} \mathbf{E} \cdot \mathbf{J} d v=0 $$ from which it follows that the Poynting vector is $$ s=\mathbf{E} \times \mathbf{H} $$ and the energy density is $$ \begin{aligned} \left(W_{1}\right)_{\text {leld }} &=\frac{1}{2} \mathbf{E} \cdot \mathbf{D}+\frac{1}{2} \mathbf{H} \cdot \mathbf{B} \\ &=\frac{1}{2} \kappa_{\ell} \epsilon_{0} E^{2}+\frac{1}{2 \kappa_{\mathrm{m}} \mu_{0}} B^{2} \end{aligned} $$ What restrictions on \(\kappa_{\&}\) and \(\kappa_{m}\) are necessary to obtain \((8.4 .24)\) ?

For the TE \(_{10}\) mode in rectangular waveguide, find the values of \(x\) at which the magnetic field is circularly polarised; i.e., the \(B_{x}\) and \(B\), components are equal in magnitude and \(90^{\circ}\) out of phase in time. (This feature is exploited in some waveguide devices known as directional couplers and isolators) Anstoer: \(\sin (\pi x / a)=\lambda_{0} / 2 a\).
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