TikZ - Magnetic field

Question

Here is what I want to do:

And here is what I've done so far

\begin{tikzpicture}[scale=1]
\draw[line width=1pt,->] (-4,0) -- (4,0);
\draw[line width=1pt,->] (0,-4) -- (0,4);

\foreach \u in {0,...,8}
\draw[color=red,domain=-3.14:3.14,samples=200,smooth] plot (canvas polar cs:angle=\x r,radius={\u*sin(\x r)*\u*sin(\x r)*5}); 
\end{tikzpicture}

As you can see, there is a problem: the function exceeds the axes.

How can we change that please?

(By the way I am French so if my explanations are not clear let me know)

Answer 1

If you like you can adapt this drawing of mine --- the field lines are not exact, they're just qualitative: (notice that you can rotate the magnet changing \angle at the start of the picture).

\documentclass[border=10pt]{standalone}
\usepackage{tikz}
\usetikzlibrary{shadings, calc, decorations.markings}
\tikzset{->-/.style={decoration={
  markings,
  mark=at position #1 with {\arrow{>}}},postaction={decorate}},
  ->-/.default=0.5,
  }
\begin{document}
\begin{tikzpicture}
    \def\angle{50}% change this to rotate the magnet
    \path [use as bounding box] (-3.2,-3.2) rectangle (3.2,3.2);
    %\draw[very thin, gray!50] (-3,-3) grid (3,3);
    \draw[->](-3,0) -- (3,0) node [above] {$x$};
    \draw[->](0,-3) -- (0,3) node [right] {$z$};
    \draw[thick, color=blue] (0, 0) circle (2);
    \draw [semithick, ->] (0:1.2) arc (0:\angle:1.2);
    \node at (0.5*\angle:1.4) {$\theta$};
    \begin{scope}[gray,text=black, rotate=\angle]
        \node[draw, left color=red!60,right color=blue!60,  middle color=white, 
        rotate=\angle, shading angle=90+\angle, minimum width=1cm,
        ]
        at (0,0) (I) {};
        \draw[->-] (I.north east) .. controls (1.5,1) and (-1.5,1) ..  (I.north west);
        \coordinate (n1) at ($ (I.north east)!0.5!(I.east) $);
        \coordinate (s1) at ($ (I.north west)!0.5!(I.west) $);
        \draw[->-] (n1) .. controls (3, 2) and (-3, 2) .. (s1);
        \draw[->-] (I.east) .. controls (6,3) and (-6,3) .. (I.west);
        \draw[->-] (I.south east) .. controls (1.5,-1) and (-1.5,-1) ..  (I.south west);
        \coordinate (n2) at ($ (I.south east)!0.5!(I.east) $);
        \coordinate (s2) at ($ (I.south west)!0.5!(I.west) $);
        \draw[->-] (n2) .. controls (3, -2) and (-3, -2) .. (s2);
        \draw[->-] (I.east) .. controls (6,-3) and (-6,-3) .. (I.west);
    \end{scope}
\end{tikzpicture}
\end{document}

Answer 2

This is the nearest thing that I got.

1. Used arcs to draw the field lines and cropped using \clip.
2. Used decorations.markings to mark arrows on field lines.

3. The green arrows has to be improved further.

   \documentclass{standalone}
   \usepackage{tikz}
   \usetikzlibrary{calc}
   \usetikzlibrary{decorations.markings}
   
   \tikzstyle directed=[postaction={decorate,decoration={markings, % arrows on the field lines
     mark=at position .1 with {\arrowreversed[scale=1.5]{stealth}},
     mark=at position .9 with {\arrowreversed[scale=1.5]{stealth}}}}]
   \tikzstyle tangent=[postaction={decorate,decoration={markings, % Tangent to the field line
     mark=at position .7 with {\draw[ultra thick,stealth-,green!60!black,solid](-12pt,0)--(12pt,0)node[above]{$\vec{B}$};}}}]
   \tikzstyle fLines=[thick,dashed,directed,tangent]
   
   \begin{document}
   \begin{tikzpicture}
   \def\lmag{1.8}  % length of magnet
   \def\wmag{0.4}  % thickness of magnet
   \def\nc{5}      % no. of lines = 2*\nc+1
   
   \begin{scope}
   \coordinate (A) at (-\lmag/2,\wmag/2);
   \coordinate (B) at (\lmag/2,-\wmag/2);
   \draw[fill, color=blue](A) rectangle ++(\lmag/2,-\wmag)node[white,midway]{S};
   \draw[fill, color=red](0,-\wmag/2) rectangle ++(\lmag/2,\wmag)node[white,midway]{N};
   
   \clip (-5,-3) rectangle (5,3);
   \foreach \r in {1,...,\nc}{
   \draw[fLines]($(A)-(0,0.5*\r*\wmag/\nc)$) arc(({270-asin(\lmag/(2*\r))}):({-90+asin(\lmag/(2*\r))}):\r);
   \draw[fLines]($(B)+(0,0.5*\r*\wmag/\nc)$) arc(({90-asin(\lmag/(2*\r))}):({-270+asin(\lmag/(2*\r))}):\r); }
   \draw[fLines] (-\lmag/2,0) -- ++(-6,0);
   \draw[fLines] (\lmag/2,0) ++(6,0)--(\lmag/2,0);
   \end{scope}
   \end{tikzpicture}
   \end{document}
   

Answer 3

6 years seems to be the right time. I'm not a huge fan of your example since it seems to be just a couple of cropped circles. The streamlines of a magnetic dipole are defined by $$y^2=x^{\frac{4}{3}}C-x^2$$ You can easily find informations about this online.
Now here is the code I used to illustrate the earth's magnetic field. The code is dirty but the output is nice and physically accurate.
Edit: I modified the little red arrows so they are now tangent to the streamlines.

\usepackage{tikz}
\usetikzlibrary{arrows.meta,positioning}
\begin{document}
\def\R{8}
\def\Rp{7.8}
\def\CC{10}
\centering
\begin{tikzpicture}[line cap=round,line join=round,x=1.0cm,y=1.0cm]
\clip(-10,-10) rectangle (10,10);
\drawcolor=gray, line width=3pt,dash pattern=on 7pt off 7pt --(0,-10);
\begin{scope}[rotate=11]
\foreach \C in {2,3,4}{ 
\draw[line width=2.pt,smooth,samples=200,domain=0:{\C^(3/2)}] plot(\x,{sqrt( abs((\x)^(4/3)\C-(\x)^(2)))});
\draw[line width=2.pt,smooth,samples=200,domain=0:{\C^(3/2)}] plot(-\x,{sqrt(abs((\x)^(4/3)\C-(\x)^(2)))});
\draw[line width=2.pt,smooth,samples=200,domain=0:{\C^(3/2)}] plot(\x,{-sqrt(abs((\x)^(4/3)\C-(\x)^(2)))});
\draw[line width=2.pt,smooth,samples=200,domain=0:{\C^(3/2)}] plot(-\x,{-sqrt(abs((\x)^(4/3)\C-(\x)^(2)))});
\draw[-{Stealth[scale=3]},draw opacity=0] ({\C^(3/2)},0) -- ({\C^(3/2)},0.25);
\draw[-{Stealth[scale=3]},draw opacity=0] ({-\C^(3/2)},0) -- ({-\C^(3/2)},0.25);}
\foreach \C in {6,10,25}{ 
\draw[line width=2.pt,smooth,samples=200,domain=0:{\C^(3/2)}] plot(\x,{sqrt( abs((\x)^(4/3)\C-(\x)^(2)))});
\draw[line width=2.pt,smooth,samples=200,domain=0:{\C^(3/2)}] plot(-\x,{sqrt(abs((\x)^(4/3)\C-(\x)^(2)))});
\draw[line width=2.pt,smooth,samples=200,domain=0:{\C^(3/2)}] plot(\x,{-sqrt(abs((\x)^(4/3)\C-(\x)^(2)))});
\draw[line width=2.pt,smooth,samples=200,domain=0:{\C^(3/2)}] plot(-\x,{-sqrt(abs((\x)^(4/3)\C-(\x)^(2)))});%}
\draw[-{Stealth[scale=3]},draw opacity=0] ({\R^(3/2)\C^(-3/4)},{\Rsqrt(1-\R\C^(-3/2))})--
({\Rp^(3/2)\C^(-3/4)},{\Rpsqrt(1-\Rp\C^(-3/2))});
\draw[-{Stealth[scale=3]},draw opacity=0] ({-\R^(3/2)\C^(-3/4)},{\Rsqrt(1-\R\C^(-3/2))})--
({-\Rp^(3/2)\C^(-3/4)},{\Rpsqrt(1-\Rp\C^(-3/2))});
\draw[-{Stealth[scale=3]},draw opacity=0] ({\Rp^(3/2)\C^(-3/4)},{-\Rpsqrt(1-\Rp\C^(-3/2))})--
 ({\R^(3/2)\C^(-3/4)},{-\Rsqrt(1-\R\C^(-3/2))});
\draw[-{Stealth[scale=3]},draw opacity=0]({-\Rp^(3/2)\C^(-3/4)},{-\Rpsqrt(1-\Rp\C^(-3/2))})--
 ({-\R^(3/2)\C^(-3/4)},{-\Rsqrt(1-\R\C^(-3/2))});
}
\draw[line width=2pt] (0,-10) -- (0,10);

\draw-{Stealth[scale=3]},draw opacity=0--(0,\Rp);
\draw-{Stealth[scale=3]},draw opacity=0--(0,-\R);
\drawcolor=red, fill=redrectangle (1,3);
\draw [color=white] (0,1.5) node{\Large S} ;
\drawcolor=blue, fill=bluerectangle (1,-3);
\draw[color=white] (0,-1.5) node{\Large N} ;
\end{scope}
\drawcolor=blue, line width=3pt circle (4);

\drawcolor=blue, line width=3pt,dash pattern=on 7pt off 7pt --(4,0);
\begin{scope}[rotate=11]
\draw-Stealth,color=red,line width=2pt--(4.1,0.5);
\draw-Stealth,color=red,line width=2pt--(0,4);
\draw-Stealth,color=red,line width=2pt--(0,-5);
\draw[-Stealth,color=red,line width=2pt] (4.48081,1.68121)
--(3.50953,1.91916);
\draw-Stealth,color=red,line width=2pt
-- (4.48081,-1.68121);
\draw-Stealth,color=red,line width=2pt
-- (2.08677,3.41253);
\draw-Stealth,color=red,line width=2pt
--(2.83512,-4.07584);
\end{scope}

\end{tikzpicture}
\end{document}