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# 物理代写|电动力学代考Electrodynamics代写|Cold plasma: general dispersion equation

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## 物理代写|电动力学代考Electrodynamics代写|Cold plasma: general dispersion equation

We first consider the classical example of a wave in a cold plasma neglecting the ion motion. The dispersion relations conventionally used for metamaterials have common features with ones for plasma, hence we include the wave theory for such materials in this chapter, as well in section 18.3.3.

To explain the idea of the method consider the 1D perturbation, when the electromagnetic field propagates along the $x$-axis. Let the longitudinal component of electric field be formed as a wave-train, localized in a range $t>0$, the dimensionless parameter $\beta \ll 1$ is introduced to characterize the scale of slow-varied amplitude $A$ to be convenient in a perturbation theory development

$$E_x=A(\beta x, \beta t) \exp i(k x-\omega t)+c . c .$$
Equation (13.14) simplifies as
$$\chi_t+\frac{p}{m} \chi_x=F(x, t)$$
it is directly integrated by the method of characteristics
$$\chi=\int_0^t F\left(x-\frac{p}{m}(t-\tau), \tau\right) d \tau .$$
where $m$ is the electron mass and $p$ is the $x$-component of its momentum. Plugging equation (13.17) into expression for $F$ from equation (13.14) and, next, to equation (13.19), we get in the zeroth order (in the small parameter $\beta$, tearing the slow varying amplitude $A$ out from the integrand) approximation
$$\begin{gathered} \chi=e \frac{\partial f_0}{\partial p} A(\beta x, \beta t) \int_0^t \exp i\left[k\left(x-\frac{p}{m} t\right)+\left(\frac{k p}{m}-\omega\right) \tau\right] d \tau+c . c . \ =e \frac{\partial f_0}{\partial p} A(\beta x, \beta t) \frac{\exp i[k x-\omega t]}{i\left(\omega-k \frac{p}{m}\right)}+c . c . \end{gathered}$$

## 物理代写|电动力学代考Electrodynamics代写|Maxwell distribution background: Langmuir waves

For the Maxwell distribution $f_0$ it has the form
$$\omega^2=\omega_{L e}^2+3 \kappa_B T_e \frac{k^2}{m}$$
where
$$\omega_{L e}=2 e \sqrt{\frac{\pi n}{m}}$$
is the Langmuir frequency of the electrons, $\kappa_B$-Boltzmann constant, and $T_e$ is the electron temperature.
If the electron density
$$\rho=\rho_0 \exp (-i \omega t)+c . c .$$
and $\frac{\left\langle p^2\right\rangle^{1 / 2} k}{m \omega} \ll 1$ and the same condition is valid for ions, then equation (13.20) with ion terms account gives
$$\chi=\sum_a e_a \frac{\partial f_{0 a}}{\partial p_a} A \frac{\exp i[k x-\omega t]}{i\left(\omega-k \frac{p_a}{m_a}\right)}+c . c .,$$
where the $x$-component of ion ‘ $\mathrm{a}$ ‘ momentum is denoted as $p_a$ so as the ion mass is $m_a$

Approximate equality occurs if there is no great difference between the concentrations $n_a$. For example, this is true for a single-ion quasineutral plasma. Obviously the Langmuir frequency is basic for longitudinal plasma oscillations in the long wavelength range.

# 电动力学代写

## 物理代写|电动力学代考Electrodynamics代写|Cold plasma: general dispersion equation

$$E_x=A(\beta x, \beta t) \exp i(k x-\omega t)+c . c .$$

$$\chi_t+\frac{p}{m} \chi_x=F(x, t)$$

$$\chi=\int_0^t F\left(x-\frac{p}{m}(t-\tau), \tau\right) d \tau$$

$$\chi=e \frac{\partial f_0}{\partial p} A(\beta x, \beta t) \int_0^t \exp i\left[k\left(x-\frac{p}{m} t\right)+\left(\frac{k p}{m}-\omega\right) \tau\right] d \tau+c . c .=e \frac{\partial f_0}{\partial p} A(\beta x, \beta t) \frac{\exp i[k x-\omega t]}{i\left(\omega-k \frac{p}{m}\right)}+c . c .$$

## 物理代写|电动力学代考Electrodynamics代写|Maxwell distribution background: Langmuir waves

$$\omega^2=\omega_{L e}^2+3 \kappa_B T_e \frac{k^2}{m}$$

$$\omega_{L e}=2 e \sqrt{\frac{\pi n}{m}}$$

$$\rho=\rho_0 \exp (-i \omega t)+c . c .$$

$$\chi=\sum_a e_a \frac{\partial f_{0 a}}{\partial p_a} A \frac{\exp i[k x-\omega t]}{i\left(\omega-k \frac{p_a}{m_a}\right)}+c . c .,$$

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## MATLAB代写

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