Pstricks: Magnetic Field Lines of a Bar Magnet

Question

As the title suggests I am looking for a pstricks way to draw the magnetic field lines of a bar magnet.

There are drawings online, for example: hyperphysics.phy-astr.gsu.edu/hbase/magnetic/imgmag/bar.gif, but I want to use pstricks.

Some of the plots online are mathematically not correct, which I consider unfortunate. Wolfram online, however, has some math to do this, Wolfram Demonstration: Magnetic Field of a Cylindrical Bar Magnet.

I realize there are some solutions for Tikz, compare TikZ - Magnetic field lines images method and TikZ - Magnetic field.

I am also aware of the general approach to vector fields, for example Drawing vector field using PSTricks, but have not been able to do apply this. Also, I would prefer a solution that does not include this process.

There is a pstricks package pst-magneticfield, but it is for drawing fields of solenoids (basic coils).

EDIT

I have tried the suggested solution

\documentclass{standalone}
\usepackage[dvipsnames]{pstricks}
\usepackage{pst-magneticfield}
\usepackage{graphicx}
\begin{document}
\psset{unit=0.75cm}
\begin{pspicture*}(-5,-7)(5,7)
\psBarMagnet[showField](0,0)
\end{pspicture*}
\end{document}

together with the recent pst-magneticfield.tex, but somehow end up with changing linewidths:

It’s not every day some asks a question about magnetic fields AND TeX, my two favourite things :-) Note that the magnetic field from an ideal cylindrical magnet is identical to an ideal solenoid of the same dimensions. — Will Robertson, Jan 19 '19 at 01:26

Circumscribe · Answer 1 · 2019-01-19T01:26:01.977

I should preface this answer by saying that I know pretty much nothing about PSTricks and that I had never heard of the pst-magneticfield package (which is amazing by the way) before today. I do know some physics though. Also, I'm assuming a cylindrical magnet counts as a "bar magnet".

The magnetic field generated by a uniformly magnetised cylindrical magnet is identical to that produced by a uniform current along the shell of a cylinder of the same dimensions. This can be approximated by a solenoid of the same length and radius with a large number of loops.

The pst-magneticfield package can draw this if you choose the parameters properly. My starting point was the code from Raaja's answer, which is not quite accurate.

\documentclass{standalone}
\usepackage{pst-magneticfield}
\begin{document}

\psset{unit=0.5}
\begin{pspicture*}(-20,-16)(20,16)
  \psframe[linecolor=black, fillstyle=solid,fillcolor=gray](-2,-6)(2,6)
  \psmagneticfield[N=128,R=2,L=11.95,
      nL=7,pointsB=4000,
      nS=1,numSpires=13,pointsS=8000,
      linecolor=black,drawSelf=false](-20,-16)(20,16)
  \rput(0,-5.2){\textcolor{white}{S}}
  \rput(0,5.2){\textcolor{white}{N}}
\end{pspicture*}

\end{document}

Producing the following image takes about 110 second on my system if I use XeLaTeX. Running latex, dvips and then ps2pdf (not pstopdf) takes about 93 seconds (which is mostly ps2pdfs running time).

Note that the field lines coming out the sides don't do so at right angles (which is correct). I've only drawn one such line for three reasons: (1) the field there is weak, (2) the field inside should be close to uniform and (3) technical reasons.

The technical reason is that the second field line would almost loop back on itself and produce another field line I didn't ask for. This doesn't happen if pointsS is reduced, but it doesn't seem to be possible to choose different values for different coils. It also appears that nothing is drawn if nL=0, even if nS is positive. I'm not sure why.

Parameters:

I've set the length L=11.95 to just a teensy bit less than the length of the magnet and the radius R=2 to its actual radius.
I've set the number of coils to N=128, which was kind of arbitrary. I'm sure some smaller number would also work.
nL=7 is the number of field lines to draw that come out of the ends of the solenoid.
If you set nS=0 (number of field lines to draw around each individual coils), only the field lines coming out of the ends of the magnet are shown and if you set nS=1 things go a little crazy (I would assume, I didn't wait for it). I've set nS=1 and numSpires=13 to only draw one field line coming out the side for the 13th coil. The number 13 was chosen so that the field lines inside are still roughly uniform, and it was obtained by trial and error.
The maximum number of points to use for the field lines coming out the ends (pointsB=4000) and the same thing for the field lines coming out the sides (pointsS=8000) were also obtained by trial end error.

Addendum

I just remembered (thanks in part to a comment by KJO) that the H-field of a uniformly cylindrical (or another shape) magnetised bar is identical to that of two disc-shaped (or other-shaped) monopoles at the ends. The magnetic field is equal to the H-field outside the magnet.

The following is roughly what the magnetic field of two very flat bar magnets should look like in several configurations. I've drawn it using the pst-electricfield package. It will likely be qualitatively correct for cylindrical/bar shaped magnets. It does look a little less polished than the other picture.

\documentclass{standalone}
\usepackage{pst-electricfield}

\begin{document}

\psset{unit=0.5}

\let\oppositeends\empty
\multido{\rA=-2+0.2}{21}{%
  \xdef\oppositeends{\oppositeends[1 \rA\space 6][-1 \rA\space -6][-1 \rA\space 10][1 \rA\space 22]}
}
\begin{pspicture*}(-6,-10)(6,26)
  \psElectricfield[Q={\oppositeends},linecolor=black,N=5,points=1000,radius=0]
  \psframe[linecolor=black, fillstyle=solid,fillcolor=gray=12](-2,-6)(2,6)
  \psframe[linecolor=black, fillstyle=solid,fillcolor=gray=12](-2,10)(2,22)
  \rput(0,-5.2){\textcolor{white}{S}}
  \rput(0,5.2){\textcolor{white}{N}}
  \rput(0,10.8){\textcolor{white}{S}}
  \rput(0,21.2){\textcolor{white}{N}}
\end{pspicture*}

\let\sameends\empty
\multido{\rA=-2+0.2}{21}{%
  \xdef\sameends{\sameends[1 \rA\space 6][-1 \rA\space -6][1 \rA\space 10][-1 \rA\space 22]}
}
\begin{pspicture*}(-6,-10)(6,26)
  \psElectricfield[Q={\sameends},linecolor=black,N=5,points=1000,radius=0]
  \psframe[linecolor=black, fillstyle=solid,fillcolor=gray=12](-2,-6)(2,6)
  \psframe[linecolor=black, fillstyle=solid,fillcolor=gray=12](-2,10)(2,22)
  \rput(0,-5.2){\textcolor{white}{S}}
  \rput(0,5.2){\textcolor{white}{N}}
  \rput(0,10.8){\textcolor{white}{N}}
  \rput(0,21.2){\textcolor{white}{S}}
\end{pspicture*}

\let\sidebyside\empty
\multido{\rA=-2+0.2,\rB=6+0.2}{21}{%
  \xdef\sidebyside{\sidebyside[1 \rA\space 6][-1 \rA\space -6][1 \rB\space 6][-1 \rB\space -6]}
}
\begin{pspicture*}(-6,-10)(14,10)
  \psElectricfield[Q={\sidebyside},linecolor=black,N=5,points=1000,radius=0]
  \psframe[linecolor=black, fillstyle=solid,fillcolor=gray=12](-2,-6)(2,6)
  \psframe[linecolor=black, fillstyle=solid,fillcolor=gray=12](6,-6)(10,6)
  \rput(0,-5.2){\textcolor{white}{S}}
  \rput(0,5.2){\textcolor{white}{N}}
  \rput(8,-5.2){\textcolor{white}{S}}
  \rput(8,5.2){\textcolor{white}{N}}
\end{pspicture*}

\let\reverseside\empty
\multido{\rA=-2+0.2,\rB=6+0.2}{21}{%
  \xdef\reverseside{\reverseside[1 \rA\space 6][-1 \rA\space -6][-1 \rB\space 6][1 \rB\space -6]}
}
\begin{pspicture*}(-6,-10)(14,10)
  \psElectricfield[Q={\reverseside},linecolor=black,N=5,points=1000,radius=0]
  \psframe[linecolor=black, fillstyle=solid,fillcolor=gray=12](-2,-6)(2,6)
  \psframe[linecolor=black, fillstyle=solid,fillcolor=gray=12](6,-6)(10,6)
  \rput(0,-5.2){\textcolor{white}{S}}
  \rput(0,5.2){\textcolor{white}{N}}
  \rput(8,-5.2){\textcolor{white}{N}}
  \rput(8,5.2){\textcolor{white}{S}}
\end{pspicture*}

\end{document}

(Drawing all four of these took about 220 second with latex+dvips+ps2pdf by the way.)

I've drawn the magnets over the field lines because the lines inside the magnet are field lines for the H-field rather than for the B-field. For completeness, here is a version where I'm not covering them up:

It's really beautiful, I admire it! – AndréC Jan 20 '19 at 12:26 — AndréC, Jan 20 '19 at 12:26

score 11 · Accepted Answer · answered Jan 17 '19 at 11:09

11

\documentclass{standalone}
\usepackage[dvipsnames]{pstricks}
\usepackage{pst-magneticfield}
\usepackage{graphicx}
\begin{document}

\psset{unit=0.4}
\begin{pspicture*}(-10,-12)(10,12)
\psmagneticfield[linecolor=black,N=2,R=1,L=1,PasB=0.4,nS=0,nL=7,
  pointsB=1000](-10,-12)(10,12)
\psframe*[linecolor=Green](-1,0)(1,-3)
\psframe*[linecolor=BrickRed](-1,0)(1,3)
\rput(0,-2){\bfseries\textcolor{white}{S}}
\rput(0,2){\bfseries\textcolor{white}{N}}
\end{pspicture*}

%% or with latest pat-magneticfield http://archiv.dante.de/~herbert/TeXnik/tex/generic/pst-magneticfield/pst-magneticfield.tex
\psset{unit=0.75cm}
\begin{pspicture*}(-5,-7)(5,7)
\psBarMagnet[showField](0,0)
\end{pspicture*}

\end{document}

answered Jan 17 '19 at 11:09

1

What happens if there are more than one magnet bar placed relatively closed each other? – Display Name Jan 17 '19 at 22:15
1

I think the first picture your code produces pretty much the same picture as Raaja's code (see my comments there). I think the second picture rougly shows the field lines for a slightly shorter and thinner magnet and the lines you manually added at the side are probably not correct. – Circumscribe Jan 18 '19 at 10:00
@Herbert: I have tried your first solution, but agree with Circumscribe that it's "pretty much the same picture as Raaja's code". My problem is with those cusps that seem wrong tbh – fborchers Jan 18 '19 at 15:46
I like your second solution though. Your packages are gold, pure gold – fborchers Jan 18 '19 at 15:49
The linewidths do however change in a strange way. I have tried and failed to get them all the same – fborchers Jan 18 '19 at 16:21
@fborchers: give a hort example and I'll have a look at it. – Jan 18 '19 at 16:43
@Herbert I have added a picture of how it looks on my machine. I use a standard latex, dvips and ps2pdf routine – fborchers Jan 20 '19 at 11:46
I see, will fix it ... – Jan 20 '19 at 17:31
And uploaded a new version to CTAN – Jan 21 '19 at 05:48
Thanks, but somehow this did not solve my problem. This might by due to old TeX files on my machine (deb repo) though. I have hacked your code and use multido with linewidth=0.4\pslinewidth now, which works for me. – fborchers Jan 21 '19 at 18:32

Raaja_is_at_topanswers.xyz · Answer 3 · 2019-01-18T08:49:56.500

As a first try, you can tweak the <options> of pst-magneticfield as in

%&pdflatex
% !TeX TXS-program:compile = txs:///pdflatex/[--shell-escape]
\documentclass{standalone}
\usepackage{pst-magneticfield}
\usepackage{graphicx}
\usepackage{pstricks-add, auto-pst-pdf}
\begin{document}

 \psset{unit=0.5}
 \begin{pspicture*}(-10,-12)(10,12)
 \psframe[linecolor=black, fillstyle=solid,fillcolor=gray](-2,-3)(2,3)
 \psmagneticfield[linecolor=black,N=2,R=2,L=1,PasB=0.4,nS=0,nL=7,pointsB
=1000](-10,-12)(10,12)
 \rput(0,-2.6){\textcolor{white}{S}}
 \rput(0,2.6){\textcolor{white}{N}}
 \end{pspicture*}
\end{document}

to achieve something closer to your requirement as in

However, this is not the exact output you would desire. Plese note that I still am figuring out how to remove the red-lines that is appearing over the magnet. Also, dont forgot to escape the shell, if you are compiling with pdflatex which is necessary due to the usage of auto-pst-pdf. ** Update 1** By making drawSelf = false, you can remove the unwanted coils atop your magnets.

\psmagneticfield[linecolor=black,N=2,R=2,L=1,PasB=0.4,nS=0,nL=7,pointsB=1000, drawSelf = false]

This gives:

Addendum 1:

Apparently scripting this from scratch yields much more better results and makes our life easier ;). However, I am not sure of the technical accuracy! So, what do we need: A few ellipses, a box, a straight line and finally a few texts.

Note: This script can be optimised much better than it is currently presented.

%&pdflatex
% !TeX TXS-program:compile = txs:///pdflatex/[--shell-escape]
\documentclass[a4paper, pdf, x11names]{standalone}
\usepackage{pstricks}
\usepackage{graphicx}
\usepackage{pstricks-add, auto-pst-pdf}
\begin{document}
    \psset{unit = 6mm}
    \begin{pspicture}(-5,-4)(2,4)
    % magnet
    \psframe[linecolor=black, fillstyle=solid,fillcolor=gray](-3,-2.5)(0,2.5)
    %right side
    \psellipticarc[rot=0, linecolor = red]{->}(0.4,0)(1,2){0}{360}
    \psellipticarc[rot=0, linecolor = red]{->}(0.4,0)(1.2,2.7){0}{360}
        \psellipticarc[rot=0, linecolor = red]{->}(0.4,0)(1.4,3.4){0}{360}
        %closing the gaps
            \psellipticarc[rot=0, linecolor = red](0.4,0)(1,2){350}{10}
        \psellipticarc[rot=0, linecolor = red](0.4,0)(1.2,2.7){350}{10}
        \psellipticarc[rot=0, linecolor = red](0.4,0)(1.4,3.4){350}{10}
                %left side and mirror
                \psellipticarc[rot=0, linecolor = red]{<-}(-3.4,0)(1,2){-180}{0}
                \psellipticarc[rot=0, linecolor = red]{<-}(-3.4,0)(1.2,2.7){-180}{0}
                \psellipticarc[rot=0, linecolor = red]{<-}(-3.4,0)(1.4,3.4){-180}{0}
                \psellipticarc[rot=0, linecolor = red](-3.4,0)(1,2){0}{190}
                \psellipticarc[rot=0, linecolor = red](-3.4,0)(1.2,2.7){0}{190}
                \psellipticarc[rot=0, linecolor = red](-3.4,0)(1.4,3.4){0}{190}
                %interesting stuff
                \psellipticarc[rot=0, linecolor = red]{-<}(-3.4,0)(1.6,4.1){-90}{80}
                \psellipticarc[rot=0, linecolor = red](-3.4,0)(1.6,4.1){70}{90}
                \psellipticarc[rot=0, linecolor = red]{>-}(0.4,0)(1.6,4.1){100}{-90}
                \psellipticarc[rot=0, linecolor = red](0.4,0)(1.6,4.1){90}{110}
                % allied extras
                \psellipticarc[rot=0, linecolor = red]{-<}(-3.4,0)(1.8,4.8){-90}{80}
                \psellipticarc[rot=0, linecolor = red](-3.4,0)(1.8,4.8){70}{90}
                \psellipticarc[rot=0, linecolor = red]{>-}(0.4,0)(1.8,4.8){100}{-90}
                \psellipticarc[rot=0, linecolor = red](0.4,0)(1.8,4.8){90}{110}
                %the straight strip
                \psline[linecolor=red]{>-}(-1.5,4)(-1.5,-4)
                \psline[linecolor=red](-1.5,4)(-1.5,3.9)
                %which direction is my magnetic field going huh?
                \rput(-1.5,-2.1){N}
                \rput(-1.5,2.1){S}
                % now you know
    \end{pspicture}
\end{document}

This gives,

Addendum 2: Based on @GodMustBeCrazy's comment on What if there is more than one magnet, we can achieve close to desired results by abusing the pst-electricfield package as in

%&pdflatex
% !TeX TXS-program:compile = txs:///pdflatex/[--shell-escape]
\documentclass[a4paper, pdf, x11names]{standalone}
\usepackage{pstricks}
\usepackage{pst-electricfield}
\usepackage{pstricks-add, auto-pst-pdf}
%https://tex.stackexchange.com/questions/308036/how-to-draw-a-circle-with-black-border-with-pstricks
\begin{document}
    \psset{unit = 6mm}
    \begin{pspicture*}(-6,-6)(6,6)
\psframe*[linecolor=white!50](-6,-6)(6,6)
\psElectricfield[Q={[-1 -2 2][1 2 2][-1 2 -2][1 -2 -2]},linecolor=red, radius=0.1]
% fitting the magnets
\psframe[linecolor=black, fillstyle=solid,fillcolor=gray](-2.3,-2.5)(-1.7,-1.5)
\psframe[linecolor=black, fillstyle=solid,fillcolor=gray](-2.3,2.5)(-1.7,1.5)
\psframe[linecolor=black, fillstyle=solid,fillcolor=gray](2.3,-2.5)(1.7,-1.5)
\psframe[linecolor=black, fillstyle=solid,fillcolor=gray](2.3,2.5)(1.7,1.5)
% drawing N-S
 \rput(-2,-2.3){\textcolor{white}{\tiny S}}
\rput(-2,-1.7){\textcolor{white}{\tiny N}}
 \rput(-2,2.3){\textcolor{white}{\tiny N}}
\rput(-2,1.7){\textcolor{white}{\tiny S}}
 \rput(2,-2.3){\textcolor{white}{\tiny N}}
\rput(2,-1.7){\textcolor{white}{\tiny S}}
 \rput(2,2.3){\textcolor{white}{\tiny S}}
\rput(2,1.7){\textcolor{white}{\tiny N}}
\end{pspicture*}
\end{document}

to get:

This is just a try for fun, I am really not sure about the technical accuracy of the results (been a little rusty in electromagnetics, been a long since I looked into them). Thanks to @Herbert for a nice answer that he made in the past ;)

Good idea. You can put the bar magnet _ over_ the field lines. See first example in my answer. — , Jan 17 '19 at 11:08
@Herbert I specifically did not do that because in OP's example, the field lines were visible atop the magnet :D — Raaja_is_at_topanswers.xyz, Jan 17 '19 at 11:41
@Herbert I tried that one with lualatex threw some errors, but after I reach home, I will check and update the answer ;) thanks for info! — Raaja_is_at_topanswers.xyz, Jan 17 '19 at 17:56
I don't think these pictures are correct. You're approximating a bar magnet by just two coils somewhere near its center instead of a coil with an infinite number of loops along its entire length. Inside, the field lines inside should be mostly parallel along the entire length and the few coming out the side should bend backwards. They're definitely not ellipses (and they do not come close together far away from the magnet). — Circumscribe, Jan 18 '19 at 09:55
@Circumscribe I really dont know the technical accuracy which, I would whole heartedly agree :). However, this is just "an approach" which, someone with the background can play with. — Raaja_is_at_topanswers.xyz, Jan 18 '19 at 09:58
@Raaja: Fair enough. I just wanted to point this out since mathematical/physical accuracy was stressed in the question. — Circumscribe, Jan 18 '19 at 10:03
@Circumscribe I completely agree :), that's why after the addendum 2, I posted a remark explicitly stating that this is for fun :D — Raaja_is_at_topanswers.xyz, Jan 18 '19 at 10:05
It'll probably be mostly accurate if you set L to the actual length (L=6 in this case) and N to some large number (let's say N=65). I think the package doesn't draw the lines coming out the sides by design (though there shouldn't be many of them since the field there is weak), but you can draw two by adding e.g. nS=1,numSpires=8 33 (doesn't look good). — Circumscribe, Jan 18 '19 at 11:00
@Circumscribe I will try your suggestions, however I am little bit vary about the expected outcome per se. — Raaja_is_at_topanswers.xyz, Jan 18 '19 at 11:52
@Circumscribe you can post that as an answer, I don't mind at all :) — Raaja_is_at_topanswers.xyz, Jan 18 '19 at 17:28
Raaja I too can be forgetting my old lab teachers experiments with iron filings however I think the last example is based on 2 magnets rather than 4 the flow (flux) is traditionally N to S thus either a top bar S to N above a bottom bar N to S alternatively left hand bar S down to N with RH bar from N down to S — , Jan 18 '19 at 19:45
@KJO: The last image shows (a) the electric field lines for four electrically charged particles, (b) the magnetic field lines for four magnetic monopoles (which probably don't exist) or (c) the magnetic field lines of two infinitely thin permanent magnets with endpoints at these four spots. — Circumscribe, Jan 18 '19 at 21:19
@Circumscribe If I recollect (probably badly without checking) my reading of elctro package was the convergence of two dipole antenna (e.g. similar to two double ended objects) — , Jan 18 '19 at 21:28
@Circumscribe saw it, seems to match some of my expectation though without some nails and a couple of magnets I cant tell how accurate the fields follow reality :-) — , Jan 19 '19 at 01:28
This last picture cannot be correct as the magnetic field lines cannot cross. If the field is zero in the center (by symmetry) it would be necessary to omit those lines. — fborchers, Jan 21 '19 at 18:26
@fborchers As I have stated in my answer, the last part is just for fun. It is already mentioned that it is technically inaccurate :) — Raaja_is_at_topanswers.xyz, Jan 21 '19 at 22:42

Pstricks: Magnetic Field Lines of a Bar Magnet

EDIT

3 Answers3

Parameters:

Addendum