Equipotential lines

December 18, 2011 | en http://www.gnuplotting.org/equipotential-lines/

In physics equipotential lines describe lines in space which are at the same potential, for example of the electric field.

Fig. 1 Equipotential lines of a plate with electric charges (code to produce this figure)

In Fig. 1 equipotential lines for the electric field of six charges equally spaced on a plate are shown. In order to get these lines we need the function of the potential v(x,y) and make a contour plot with splot to a file to save the positions of the lines.

# calculate and save equipotential lines
set view map
unset surface
set contour base
# distance between contour lines according to 1/r
# => equal distance between lines
set cntrparam levels discrete 4,5,6.67,10
set isosam 31,31
set table 'equipotential_lines.txt'
splot v(x,y) w l ls 1
unset table

plot 'equipotential_lines.txt' u 1:2 w l ls 1

The positions of the lines are given by the cntrparam levels which are chosen in a way, to get equally spaced lines in the far field. The set table command stores the contour lines to a file, and finally the last command plots the stored lines.

In addition to the equipotential lines the value of the contour is stored as a third column in the equipotential_lines.txt file and can be plotted on the graph, too. This is shown in Fig. 2.

Fig. 2 Equipotential lines of a plate with electric charges with labels (code to produce this figure)

To get the label of the contour we have to choose a x-position which is given by lx0 in the following. The labels(x,y) function sets a string to the value of the third column, if the right x-position is given and we are above the plate. The function f(x,y) checks if we are near the point where a label should be drawn and undefines the contour line around this point. The size of this area is given by eps.

lx0 = 1.14899
eps = 0.15
labels(x,y) = (x==lx0 && y>y0) ? stringcolumn(3) : ""
f(x,y) = (abs(lx0-x)<eps && y>y0) ? 1/0 : y