Field lines in a parallel plate capacitor with border effect

Question

I am drawing qualitatively the field lines bending near the edges of a parallel plate capacitor with tikz. The code is:

\documentclass[margin=10pt]{standalone}
\usepackage{tikz}
\usepackage{bm}
%\usepackage{pgfmath}
\usetikzlibrary {positioning}
\usetikzlibrary{calc,fadings,decorations.pathreplacing, arrows}
\usetikzlibrary{decorations.pathmorphing,patterns}
\usetikzlibrary{decorations.markings}
\begin{document}
\begin {tikzpicture}[thick, scale=0.9, every node/.style={transform
shape}, decoration={
    markings, mark=at position 0.5 with {\arrow{latex}}}]
\def\Lx{5.0}
\def\xi{0.25}
\def\dx{0.5}
\def\dy{0.35}
\def\hh{2.0}
% top plate
\filldraw[opacity=0.2] (0, 0) -- (\Lx, 0) -- (\Lx, \dy) -- (0, \dy);
\draw (\Lx/2, \hh+2.*\dy) node[] {$\bm -Q$};
% bottom plate
\filldraw[opacity=0.2] (0, \hh) -- (\Lx, \hh) -- (\Lx, \hh+\dy) -- (0, \hh+\dy);
\draw (\Lx/2, -\dy) node[] {$\bm +Q$};
% left curved lines
\draw[postaction={decorate}] (\xi, \dy) node[below= -0.14, red] {$\bm +$} to
[bend left=15]  (\xi, \hh) node[above=-0.24, blue] {$\bm -$};
\draw[postaction={decorate}] (0, \dy) .. controls (-\dx, {0.35*(\hh+\dy)}) and (-\dx,{0.65*(\hh+\dy)}) ..  (0, \hh);
\draw[postaction={decorate}] (0, 0.7*\dy) .. controls (-3*\dx, {0.1*(\hh+\dy)})
and (-3*\dx,{0.9*(\hh+\dy)}) .. (0, \hh+0.3*\dy);
\draw[postaction={decorate}] (0, 0.2*\dy) .. controls (-6*\dx, {0.0*(\hh+\dy)})
and (-6*\dx,{(\hh+\dy)}) ..  (0, \hh+0.8*\dy);
% Middle lines
\foreach \nL in {1, 2, ..., 8}{
    \draw[red] (\xi+\nL*\dx, 0.65*\dy) node[] {$\bm +$};
    \draw[blue] (\xi+\nL*\dx, \hh+0.1*\dy) node[] {$\bm -$};
    \draw[postaction={decorate}] (\xi+\nL*\dx, \dy) --++ (0, \hh-\dy);
}
% right curved lines
\draw[postaction={decorate}] (\xi+9*\dx, \dy) node[below= -0.14, red] {$\bm +$}
to [bend right=15]  (\xi+9*\dx, \hh) node[above=-0.24, blue] {$\bm -$};
\draw[postaction={decorate}] (\Lx, \dy) .. controls (\Lx+\dx, {0.35*(\hh+\dy)}) and (\Lx+\dx,{0.65*(\hh+\dy)}) .. (\Lx, \hh);
\draw[postaction={decorate}] (\Lx, 0.7*\dy) .. controls (\Lx+3*\dx,
{0.1*(\hh+\dy)}) and (\Lx+3*\dx,{0.90*(\hh+\dy)}) ..  (\Lx, \hh+0.3*\dy);
\draw[postaction={decorate}] (\Lx, 0.2*\dy) .. controls (\Lx+6*\dx,
{0.0*(\hh+\dy)}) and (\Lx+6*\dx,{(\hh+\dy)}) .. (\Lx, \hh+0.8*\dy);
\end{tikzpicture}
\end{document}

The image I get is this:

I would like to improve the alignment of the arrow tips on the curved lines. The arrow bases are off the lines. They should be aligned more symmetrically. How could I fix/improve this? Perhaps just changing the arrow types will do it. The stealth style improves, but still a bit off.

Thanks in advance!

I've been puttng a lot of thought into computing field lines. First, field lines are the gradient of the potential (scalar) field. Second, conductors form equipotential volumes. The charge distribution is not even, but will migrate until all the potentials even out. Consequently, the equipotential lines are parallel between the plates all the way to the edge. — John Kormylo, May 02 '19 at 21:01
This is just qualitative. I didn't mean to be accurate. It's just to show that there will be field lines outside the parallel plates. If the problem is 2d, one can use conformal mapping. See here: http://demonstrations.wolfram.com/ElectrostaticFieldsUsingConformalMapping/. Actually, only the middle equipotential will be truly planar. — minmax, May 08 '19 at 17:31
And the interiors of the conductors. A thicker conductor (I suspect the linked example was infinitely thin) would produce a different result. I just couldn't imagine a field that could be constant over such a large volume without being constant in between. — John Kormylo, May 08 '19 at 22:13

score 6 · Accepted Answer · answered May 02 '19 at 15:36

Welcome to TeX-SE! The issue is that you attach a straight arrow to a curved line. So the first step is to bend the arrow. But then the decorations do not know a priori the curvature at a given point, which can be fixed by recording some coordinates along the path and then draw a curved arrow through these coordinates.

\documentclass[margin=10pt]{standalone}
\usepackage{tikz}
\usepackage{bm}
%\usepackage{pgfmath}
\usetikzlibrary{positioning}
\usetikzlibrary{calc,fadings,decorations.pathreplacing, arrows.meta,bending}
\usetikzlibrary{decorations.pathmorphing,patterns}
\usetikzlibrary{decorations.markings}
\begin{document}
% from https://tex.stackexchange.com/a/430239/121799
\tikzset{% inspired by https://tex.stackexchange.com/a/316050/121799
    arc arrow/.style args={%
    to pos #1 with length #2}{
    decoration={
        markings,
         mark=at position 0 with {\pgfextra{%
         \pgfmathsetmacro{\tmpArrowTime}{#2/(\pgfdecoratedpathlength)}
         \xdef\tmpArrowTime{\tmpArrowTime}}},
        mark=at position {#1-\tmpArrowTime} with {\coordinate(@1);},
        mark=at position {#1-2*\tmpArrowTime/3} with {\coordinate(@2);},
        mark=at position {#1-\tmpArrowTime/3} with {\coordinate(@3);},
        mark=at position {#1} with {\coordinate(@4);
        \draw[-{Latex[length=#2,bend]}]       
        (@1) .. controls (@2) and (@3) .. (@4);},
        },
     }
}

\begin{tikzpicture}[thick, scale=0.9, every node/.style={transform
shape}, arc arrow=to pos 0.525 with length 2mm]
\def\Lx{5.0}
\def\xi{0.25}
\def\dx{0.5}
\def\dy{0.35}
\def\hh{2.0}
% top plate
\filldraw[opacity=0.2] (0, 0) -- (\Lx, 0) -- (\Lx, \dy) -- (0, \dy);
\draw (\Lx/2, \hh+2.*\dy) node[] {$\bm -Q$};
% bottom plate
\filldraw[opacity=0.2] (0, \hh) -- (\Lx, \hh) -- (\Lx, \hh+\dy) -- (0, \hh+\dy);
\draw (\Lx/2, -\dy) node[] {$\bm +Q$};
% left curved lines
\draw[postaction={decorate}] (\xi, \dy) node[below= -0.14, red] {$\bm +$} to
[bend left=15]  (\xi, \hh) node[above=-0.24, blue] {$\bm -$};
\draw[postaction={decorate}] (0, \dy) .. controls (-\dx, {0.35*(\hh+\dy)}) and (-\dx,{0.65*(\hh+\dy)}) ..  (0, \hh);
\draw[postaction={decorate}] (0, 0.7*\dy) .. controls (-3*\dx, {0.1*(\hh+\dy)})
and (-3*\dx,{0.9*(\hh+\dy)}) .. (0, \hh+0.3*\dy);
\draw[postaction={decorate}] (0, 0.2*\dy) .. controls (-6*\dx, {0.0*(\hh+\dy)})
and (-6*\dx,{(\hh+\dy)}) ..  (0, \hh+0.8*\dy);
% Middle lines
\foreach \nL in {1, 2, ..., 8}{
    \draw[red] (\xi+\nL*\dx, 0.65*\dy) node[] {$\bm +$};
    \draw[blue] (\xi+\nL*\dx, \hh+0.1*\dy) node[] {$\bm -$};
    \draw[postaction={decorate}] (\xi+\nL*\dx, \dy) --++ (0, \hh-\dy);
}
% right curved lines
\draw[postaction={decorate}] (\xi+9*\dx, \dy) node[below= -0.14, red] {$\bm +$}
to [bend right=15]  (\xi+9*\dx, \hh) node[above=-0.24, blue] {$\bm -$};
\draw[postaction={decorate}] (\Lx, \dy) .. controls (\Lx+\dx, {0.35*(\hh+\dy)}) and (\Lx+\dx,{0.65*(\hh+\dy)}) .. (\Lx, \hh);
\draw[postaction={decorate}] (\Lx, 0.7*\dy) .. controls (\Lx+3*\dx,
{0.1*(\hh+\dy)}) and (\Lx+3*\dx,{0.90*(\hh+\dy)}) ..  (\Lx, \hh+0.3*\dy);
\draw[postaction={decorate}] (\Lx, 0.2*\dy) .. controls (\Lx+6*\dx,
{0.0*(\hh+\dy)}) and (\Lx+6*\dx,{(\hh+\dy)}) .. (\Lx, \hh+0.8*\dy);

\end{tikzpicture}
\end{document}

Worked very well. Very nice piece of tikz witchcraft! Thanks! — minmax, May 02 '19 at 19:58

Field lines in a parallel plate capacitor with border effect

1 Answers1