Improve knowledge baseHEADmaster

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