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authorPrefetch2022-10-27 20:40:09 +0200
committerPrefetch2022-10-27 20:40:09 +0200
commit6e70f28ccbd5afc1506f71f013278a9d157ef03a (patch)
treea8ca7113917f3e0040d6e5b446e4e41291fd9d3a /source/know/concept/guiding-center-theory
parentbcae81336764eb6c4cdf0f91e2fe632b625dd8b2 (diff)
Optimize last images, add proof template, improve CSS
Diffstat (limited to 'source/know/concept/guiding-center-theory')
-rw-r--r--source/know/concept/guiding-center-theory/index.md22
1 files changed, 8 insertions, 14 deletions
diff --git a/source/know/concept/guiding-center-theory/index.md b/source/know/concept/guiding-center-theory/index.md
index 5368966..412c88b 100644
--- a/source/know/concept/guiding-center-theory/index.md
+++ b/source/know/concept/guiding-center-theory/index.md
@@ -72,6 +72,7 @@ we can use this average to approximately remove the finer dynamics,
and focus only on the guiding center.
+
## Uniform electric and magnetic field
Consider the case where $$\vb{E}$$ and $$\vb{B}$$ are both uniform,
@@ -149,6 +150,7 @@ $$\begin{aligned}
\end{aligned}$$
+
## Non-uniform magnetic field
Next, consider a more general case, where $$\vb{B}$$ is non-uniform,
@@ -193,11 +195,8 @@ $$\begin{aligned}
\approx - \frac{u_L^2}{2 \omega_c} \nabla B
\end{aligned}$$
-<div class="accordion">
-<input type="checkbox" id="proof-nonuniform-B-averages"/>
-<label for="proof-nonuniform-B-averages">Proof</label>
-<div class="hidden" markdown="1">
-<label for="proof-nonuniform-B-averages">Proof.</label>
+
+{% include proof/start.html id="proof-averages" -%}
We know what $$\vb{x}_L$$ is,
so we can write out $$(\vb{x}_L \cdot \nabla) \vb{B}$$
for $$\vb{B} = (B_x, B_y, B_z)$$:
@@ -290,9 +289,8 @@ $$\begin{aligned}
\end{pmatrix}
= - \frac{u_L^2}{2 \omega_c} \nabla B
\end{aligned}$$
+{% include proof/end.html id="proof-averages" %}
-</div>
-</div>
With this, the guiding center's equation of motion
is reduced to the following:
@@ -332,11 +330,8 @@ $$\begin{aligned}
\approx - u_{gc\parallel} \frac{\vb{R}_c}{R_c^2}
\end{aligned}$$
-<div class="accordion">
-<input type="checkbox" id="proof-nonuniform-B-curvature"/>
-<label for="proof-nonuniform-B-curvature">Proof</label>
-<div class="hidden" markdown="1">
-<label for="proof-nonuniform-B-curvature">Proof.</label>
+
+{% include proof/start.html id="proof-curvature" -%}
Assuming that $$\vu{b}$$ does not explicitly depend on time,
i.e. $$\ipdv{\vu{b}}{t} = 0$$,
we can rewrite the derivative using the chain rule:
@@ -381,9 +376,8 @@ $$\begin{aligned}
= - \frac{\vu{R}_c}{R_c}
= - \frac{\vb{R}_c}{R_c^2}
\end{aligned}$$
+{% include proof/end.html id="proof-curvature" %}
-</div>
-</div>
With this, we arrive at the following equation of motion
for the guiding center: