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Diffstat (limited to 'source/know/concept/debye-length/index.md')
| -rw-r--r-- | source/know/concept/debye-length/index.md | 30 |
1 files changed, 16 insertions, 14 deletions
diff --git a/source/know/concept/debye-length/index.md b/source/know/concept/debye-length/index.md index 5961c4f..063e308 100644 --- a/source/know/concept/debye-length/index.md +++ b/source/know/concept/debye-length/index.md @@ -12,8 +12,7 @@ If a charged object is put in a plasma, it repels like charges and attracts opposite charges, leading to a **Debye sheath** around the object's surface with a net opposite charge. -This has the effect of **shielding** the object's presence -from the rest of the plasma. +This has the effect of **shielding** the rest of the plasma from the object's presence. We start from [Gauss' law](/know/concept/maxwells-equations/) for the [electric field](/know/concept/electric-field/) $$\vb{E}$$, @@ -23,12 +22,12 @@ and splitting the charge density into ions $$n_i$$ and electrons $$n_e$$: $$\begin{aligned} \nabla^2 \phi(\vb{r}) - = - \frac{1}{\varepsilon_0} \Big( q_i n_i(\vb{r}) + q_e n_e(\vb{r}) + q_t \delta(\vb{r}) \Big) + = - \frac{1}{\varepsilon_0} \Big( q_i n_i(\vb{r}) + q_e n_e(\vb{r}) + Q \delta(\vb{r}) \Big) \end{aligned}$$ The last term represents a *test particle*, which will be shielded. -This particle is a point charge $$q_t$$, +This particle is a point charge $$Q$$, whose density is simply a [Dirac delta function](/know/concept/dirac-delta-function/) $$\delta(\vb{r})$$, and is not included in $$n_i$$ or $$n_e$$. @@ -63,10 +62,10 @@ where we have assumed quasi-neutrality such that $$q_i n_{i0} = q_e n_{e0}$$: $$\begin{aligned} \nabla^2 \phi &= - \frac{1}{\varepsilon_0} - \bigg( q_i n_{i0} - n_{i0} \frac{q_i^2 \phi}{k_B T_i} + q_e n_{e0} - n_{e0} \frac{q_e^2 \phi}{k_B T_e} + q_t \delta(\vb{r}) \bigg) + \bigg( q_i n_{i0} - n_{i0} \frac{q_i^2 \phi}{k_B T_i} + q_e n_{e0} - n_{e0} \frac{q_e^2 \phi}{k_B T_e} + Q \delta(\vb{r}) \bigg) \\ &= \bigg( \frac{n_{i0} q_i^2}{\varepsilon_0 k_B T_i} + \frac{n_{e0} q_e^2}{\varepsilon_0 k_B T_e} \bigg) \phi - - \frac{q_t}{\varepsilon_0} \delta(\vb{r}) + - \frac{Q}{\varepsilon_0} \delta(\vb{r}) \end{aligned}$$ We now define the **ion** and **electron Debye lengths** @@ -101,24 +100,27 @@ suggesting exponential decay: $$\begin{aligned} \nabla^2 \phi(\vb{r}) &= \frac{1}{\lambda_D^2} \phi(\vb{r}) - - \frac{q_t}{\varepsilon_0} \delta(\vb{r}) + - \frac{Q}{\varepsilon_0} \delta(\vb{r}) \end{aligned}$$ -This has the following solution, -known as the **Yukawa potential**, -which decays exponentially, -representing the plasma's **self-shielding** -over a characteristic distance $$\lambda_D$$: +This has the solution below, known as the **Yukawa potential**, +which looks like Coulomb's law but with an extra exponential factor, +representing the plasma's **self-shielding**: $$\begin{aligned} \boxed{ \phi(r) - = \frac{q_t}{4 \pi \varepsilon_0 r} \exp\!\Big( \!-\!\frac{r}{\lambda_D} \Big) + = \frac{Q}{4 \pi \varepsilon_0 r} \exp\!\Big( \!-\!\frac{r}{\lambda_D} \Big) } \end{aligned}$$ +We call it *self*-shielding because in reality +$$Q$$ is simply an electron or ion of the plasma. +This explains why plasmas are macroscopically neutral, +despite consisting of charged particles. + Note that $$r$$ is a scalar, -i.e. the potential depends only on the radial distance to $$q_t$$. +i.e. the potential depends only on the radial distance to $$Q$$. This treatment only makes sense if the plasma is sufficiently dense, such that there is a large number of particles |