[플라즈마 물리][Plasma Physics]Debye Shielding 디바이 차폐

Example Problem: Debye Shielding
Consider a positive point charge immersed in a plasma as we discussed in the class. Show that the net charge in the Debye shielding cloud exactly cancels the test charge. Assume that the ions are fixed and that $e\phi << KT_e$. Note that we assumed that ion distribution is similar to that of the electrons in the class.

Answer:

While deriving, we assume potential to be

$$\begin{aligned} \frac{e\phi}{KT_e} << 1 \quad \quad \quad n_i \simeq n_e\\ \lambda_D = \sqrt{\frac{\epsilon KT}{ne^2}}\\ \phi(r) = \frac{e}{4\pi\epsilon_0 r }e^{-\frac{r}{\lambda_D}}\end{aligned}$$

Assume $\frac{m_i}{m_e} \rightarrow \infty$ so that ions are fixed; $n_i(r)=n$. Further assume electrons obey the Boltzmann distribution.

$$\begin{aligned} f(u) &= A e^{\frac{\frac{-1}{2}mv^2 + q\phi}{KT_e}}\\ n_e(r) &= ne^{\frac{e\phi(r)}{K_B T}}\end{aligned}$$

$$\begin{aligned} \rho &= -e(n_i - n_e) \\ &=-e \left(n - ne^{\frac{e\phi(r)}{K_BT}} \right)\\ &= ne \left(e^{\frac{e\phi(r)}{K_BT}} - 1 \right)\\\end{aligned}$$

for $\frac{e\phi}{KT_e} << 1$,

$$\begin{aligned} e^{\frac{e\phi(r)}{K_B T}} = 1 + \frac{1}{2}\frac{e\phi(r)}{K_B T} + ...\end{aligned}$$

$$\begin{aligned} \rho &= ne(1+\frac{e\phi(r)}{K_B T} - 1) \\ &=\frac{ne^2 \phi(r)}{K_B T} \\ &= \frac{\epsilon_0}{\lambda_D^2}\phi(r)\end{aligned}$$

$$\begin{aligned} \int \rho dv &= Q \\ &= \int_V \frac{\epsilon_0}{\lambda_D^2}\frac{e}{4\pi\epsilon_0 r }e^{-\frac{r}{\lambda_D}} r^2 \sin\theta dr d\theta d\phi\\ &= \int \frac{4\pi \epsilon_0}{\lambda_D^2}\frac{e}{4\pi \epsilon_0} r e^{-\frac{r}{\lambda_D}}dr \\ &= \frac{e}{\lambda_D^2} \int_{0}^{\infty} r e^{-\frac{r}{\lambda_D}}dr \\ &= \frac{e}{\lambda_D^2} (\lambda_D^2 e^{-\frac{r}{\lambda_D}}-\lambda_D r e^{-\frac{r}{\lambda_D}}) =0\end{aligned}$$

On the last step, when $r = \lambda_D$, charge becomes zero. Hence, the Debye sheilding cloud exactly cancels the test charge.

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