Torus and Arc with tikz and addplot3

Question

I'm new in using tikz/pgfplots to create graphics. It is quite powerful but therefore also complex.

My specific case deals with combining a single torus with an arc segment. The image below should give an impression

What I managed to do so far is this MWE

\documentclass[a4paper,10pt]{article}
\usepackage[utf8]{inputenc}
\usepackage[dvipsnames]{xcolor}
\usepackage{pgfplots}

\begin{document}

\begin{tikzpicture}[thkline/.style={thick, blue, <-, >=stealth}]
    \coordinate(O) at (-2.75,-1.35);
    \coordinate(A) at (1.25,-0.35);
    \begin{axis}[anchor=origin, xmax=15, ymax=15, zmax=6, axis lines = none, 
                 colormap={red}{color=(lightgray) color=(lightgray)}]
        \addplot3[domain=0:360,y domain=0:360, samples=25, surf, z buffer=sort]
            (
            {3 * sin(x)},
            {(4 + 3 * cos(x)) * sin(y)},
            {-(4 + 3 * cos(x)) * cos(y)});
    \end{axis}
%     \draw[thkline, rotate = 3] (O)  arc (-90:60:3 and 1.5) node [below right] {$e^-$} ;
    \draw[thkline, rotate = 0] (O)  arc (-90:90:3 and 1.) node [below right] {$e^-$} ;
\end{tikzpicture} 


\end{document}

I was hoping, an expert could point me in the right direction. Thanks in advance!

Edit I

According to the answer given below, I reworked the code using the first solution and came across a rather interesting issue, since the objects are misaligned:

Could that behavior related to the pgf version? Since we're using \pgfplotsset{compat=1.16}

Edit II

The strange shift remains after adding \begin{scope}[canvas is xy plane at z=0,>=stealth, shift={(axis cs: 0,0,0)}] to both scopes. It seems to apply to the top line (annotated with 'acceleration'), the torus itself and part of the second scope (the short arc with arrow head):

Answer 1

Welcome! You were already close. In order to draw the stuff in the xy plane, it might be a good idea to load the TikZ library 3d.

\documentclass[a4paper,10pt]{article}
\usepackage[utf8]{inputenc}
\usepackage[dvipsnames]{xcolor}
\usepackage{pgfplots}
\pgfplotsset{compat=1.16}
\usetikzlibrary{3d}
\begin{document}

\begin{tikzpicture}[thkline/.style={thick, blue, >=stealth},font=\sffamily]
    \begin{axis}[anchor=origin, xmax=15, ymax=15, zmax=6, axis lines = none, 
                 colormap={red}{color=(lightgray) color=(lightgray)},
                 clip=false]
        %background stuff        
        \draw[ultra thick] (0,0,0) coordinate(O) -- (-25,0,0)  
            node[pos=2/3,above,sloped]{acceleration};
        \begin{scope}[canvas is xy plane at z=0,>=stealth]
         \draw[ultra thick,->]  (0,0) -- (-25,0);
         \draw[very thick,->] (0,30) -- (0,0);
         \draw[->] (0,0) coordinate(O) -- (0,-7);
         \draw[very thick] (0,0) arc(0:-30:50);
        \end{scope}      
        % torus
        \addplot3[domain=0:360,y domain=0:360, samples=25, surf, z buffer=sort,
            opacity=0.6]
            (
            {3 * sin(x)},
            {(4 + 3 * cos(x)) * sin(y)},
            {-(4 + 3 * cos(x)) * cos(y)});
        % foreground    
        \begin{scope}[canvas is xy plane at z=0,>=stealth]
         \draw[->] (0,-7) -- (0,-30);
         \draw[very thick,->,overlay] (-50,0)+(-8:50) arc(-8:-20:50);
         \draw (0,-10) arc(-90:-180:10) node[midway,left]{$\mathsf{90}^\circ$}; 
        \end{scope}      
        \end{axis}
\end{tikzpicture} 
\end{document}

Unfortunately there is no real 3d engine coming with TikZ/pgfplots, so one has to draw the things in the appropriate order. I indicate this in the code. There is an angle of 8 which I estimated.

It is also possible to project the arrows on the planes with \pgflowlevelsynccm. However, the transformations pgfplots performs prevent me from adding this to the scopes inside the axis. Therefore, one may export the view and scaling from the axis and add the scopes outside, with the stuff behind the torus on the background layer.

\documentclass[a4paper,10pt]{article}
\usepackage[utf8]{inputenc}
\usepackage[dvipsnames]{xcolor}
\usepackage{pgfplots}
\pgfplotsset{compat=1.16}
\usetikzlibrary{3d,backgrounds,calc}
\begin{document}

\begin{tikzpicture}[thkline/.style={thick, blue, >=stealth},font=\sffamily]
    \begin{axis}[anchor=origin, xmax=15, ymax=15, zmax=6, axis lines = none, 
                 colormap={red}{color=(lightgray) color=(lightgray)},
                 clip=false]
        %background stuff        
        \draw[ultra thick] (0,0,0) coordinate(O) -- (-25,0,0)  
            node[pos=2/3,above,sloped]{acceleration};
        \path let \p1=($(1,0,0)-(0,0,0)$),\p2=($(0,1,0)-(0,0,0)$),
            \p3=($(0,0,1)-(0,0,0)$) in
            \pgfextra{\xdef\myxx{\x1}\xdef\myxy{\y1}
            \xdef\myyx{\x2}\xdef\myyy{\y2}
            \xdef\myzx{\x3}\xdef\myzy{\y3}};
        % torus
        \addplot3[domain=0:360,y domain=0:360, samples=25, surf, z buffer=sort,
            opacity=0.6]
            (
            {3 * sin(x)},
            {(4 + 3 * cos(x)) * sin(y)},
            {-(4 + 3 * cos(x)) * cos(y)});
        % foreground    
        \begin{scope}[canvas is xy plane at z=0,>=stealth]
         \draw (0,-10) arc(-90:-180:10) node[midway,left]{$\mathsf{90}^\circ$}; 
        \end{scope} 
        \end{axis}
        \begin{scope}[x={(\myxx,\myxy)},y={(\myyx,\myyy)},z={(\myzx,\myzy)},
            canvas is xy plane at z=0,>=stealth,on background layer]
         \pgflowlevelsynccm 
         \draw[line width=0.7cm,->]  (0,0) -- (-10,0);
         \draw[line width=0.5cm] (0,30) -- (0,0);
         \draw[line width=0.5cm,->] (0,30) -- (0,20);
         \draw[line width=0.1cm,->] (0,0) coordinate(O) -- (0,-7);
         \draw[line width=0.5cm] (0,0) arc(0:-30:50);
        \end{scope}      
        % foreground    
        \begin{scope}[x={(\myxx,\myxy)},y={(\myyx,\myyy)},z={(\myzx,\myzy)},
            canvas is xy plane at z=0,>=stealth]
         \pgflowlevelsynccm     
         \draw[line width=0.1cm,->] (0,-7) -- (0,-30);
         \draw[line width=0.5cm,->,overlay] (-50,0)+(-8:50) arc(-8:-20:50);
        \end{scope}      
\end{tikzpicture} 
\end{document}