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| author | Prefetch | 2022-10-27 20:40:09 +0200 |
|---|---|---|
| committer | Prefetch | 2022-10-27 20:40:09 +0200 |
| commit | 6e70f28ccbd5afc1506f71f013278a9d157ef03a (patch) | |
| tree | a8ca7113917f3e0040d6e5b446e4e41291fd9d3a /source/know/concept/guiding-center-theory | |
| parent | bcae81336764eb6c4cdf0f91e2fe632b625dd8b2 (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.md | 22 |
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: |