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File:VFPt horseshoe-magnet absB.svg

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Description
English: Drawing of a horseshoe magnet with precisely computed magnetic field lines. The horseshoe magnet is assumed as a curved cylindrical rod with constant magnetisation along the cylinder axis. North- and southpole of the magnet are marked in red and green, respectively. The shape of the magnetic field is computed as follows: H- and B-field are identical in free space, so we can choose the easier one, which is the H-field. The H-field has its sources and sinks where the lines of the magnetisation end and begin. Thus, the correct field is obtained by placing magnetic charges at the surfaces of the two magnetic poles. The field of a charge disc distribution is obtained by numerical integration. The shape of the field lines is traced with a Runge-Kutta algorithm. The density of field lines corresponds roughly to the field strength, however due to 3D variations of the field, this cannot exactly be fulfilled. The background color field and contour lines represent the absolute strength of the magnetic field at each point in space. The field is concentrated around the two magnetic poles.
Note that in measured field distributions, e.g. using magnetised iron filings the field shape in the lower part of the image (where the magnet is bent) may somewhat differ. This is because the total field strength is very weak there. Therefore any inhomogeneity in the magnetisation can strongly alter the field direction.
Date
Source Own work
Author Geek3
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This plot was created with VectorFieldPlot.
 
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Source code
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Python code

# paste this code at the end of VectorFieldPlot 2.5
doc = FieldplotDocument('VFPt_horseshoe-magnet_absB', commons=True,
    width=600, height=600)

x0, y0 = 0.0, -1.0
h = 2.0
R = 1.0
r = 0.3

# Note: The H-field of a magnet with constant profile and magnetization
# is exactly equal to the one created by magnetic surface charges
# at the ends of the magnet. In this case the ends are round discs.
field = Field([
    ['charged_disc', {'x0':x0-R-r, 'y0':y0+h, 'x1':x0-R+r, 'y1':y0+h, 'Q':-1}],
    ['charged_disc', {'x0':x0+R-r, 'y0':y0+h, 'x1':x0+R+r, 'y1':y0+h, 'Q':1}] ])

n_lines = 24
def startpath(t):
    return sc.array([x0 + R - R*cos(t*2*pi), y0 + h + R*sin(t*2*pi)])
startpoints = Startpath(field, startpath).npoints(n_lines)

for i_line in range(n_lines):
    p0 = startpoints[i_line]
    line = FieldLine(field, p0, directions='both', maxr=1000)
    fe = [True, False, False, True]
    if i_line in [0, 1, 2, n_lines-1, n_lines-2, n_lines-3]:
        fe = [False, False, False, False]
    min_arrows = 1
    if i_line == n_lines - 7:
        min_arrows = 3
    doc.draw_line(line, arrows_style={
                  'dist':2.0, 'fixed_ends':fe, 'min_arrows':min_arrows})

def Babs(xy):
    # move a little to avoid points on end-facet where the y-field vanishes
    return sc.linalg.norm(field.F(xy + sc.array([0., 1e-5])))
B0 = Babs([x0+R, y0+h+0.05])
doc.draw_scalar_field(func=Babs, cmap=doc.cmap_WtGnBu, vmin=0., vmax=B0)
doc.draw_contours(func=Babs, levels=sc.linspace(0, 1.2*B0, 15)[1:],
    linewidth=0.8, linecolor='#444444')

# draw a horseshoe magnet with color gradients
g = doc.draw_object('g', {'id':'horseshoe',
    'transform':'translate({},{})'.format(x0, y0)})
defs = doc.draw_object('defs', {}, group=g)
grad_col = ['#000000', '#ffffff', '#ffffff', '#ffffff', '#000000']
grad_offs = sc.array([0, 0.07, 0.25, 0.6, 1])
grad_opa = sc.array([0.125, 0.125, 0.5, 0.2, 0.33])
grad1 = doc.draw_object('linearGradient', {'id':'grad1', 'x1':'0',
    'x2':'1', 'y1':'0', 'y2':'0', 'gradientUnits':'objectBoundingBox'},
    group=defs)
for col, of, opa in zip(grad_col, grad_offs, grad_opa):
    stop = doc.draw_object('stop', {'stop-color':col, 'offset':of,
        'stop-opacity':opa}, group=grad1)
grad2 = doc.draw_object('radialGradient', {'id':'grad2', 'r':str(R+r),
    'cx':'0', 'cy':'0', 'fx':'0', 'fy':'0',
    'gradientUnits':'userSpaceOnUse'}, group=defs)
for col, of, opa in sorted(zip(grad_col, 1-grad_offs*2.*r/(R+r), grad_opa),
                           key=lambda x: x[1]):
    stop = doc.draw_object('stop', {'stop-color':col, 'offset':of,
        'stop-opacity':opa}, group=grad2)
grad3 = doc.draw_object('radialGradient', {'id':'grad3', 'r':str(R+r),
    'cx':'0', 'cy':'0', 'fx':'0', 'fy':'0',
    'gradientUnits':'userSpaceOnUse'}, group=defs)
for col, of, opa in zip(grad_col, (R-r)/(R+r)+grad_offs*2.*r/(R+r), grad_opa):
    stop = doc.draw_object('stop', {'stop-color':col, 'offset':of,
        'stop-opacity':opa}, group=grad3)
grad4 = doc.draw_object('linearGradient', {'id':'grad4', 'x1':str(-R-r),
    'x2':str(R+r), 'y1':'0', 'y2':'0', 'gradientUnits':'userSpaceOnUse'},
    group=defs)
for col, of, opa in [ ['#ffffff', '0', '1'], ['#ffffff', str(r/(R+r)), '1'],
        ['#ffffff', str(R/(R+r)), '0'], ['#ffffff', '1', '0'] ]:
    stop = doc.draw_object('stop', {'stop-color':col, 'offset':of,
        'stop-opacity':opa}, group=grad4)
mask4 = doc.draw_object('mask', {'id':'mask4', 'maskContentUnits':'userSpaceOnUse'}, group=defs)
doc.draw_object('rect', {'x':str(-R-r), 'y':str(-R-r), 'width':str(2*(R+r)),
    'height':str(R+r), 'style':'fill:url(#grad4); stroke:none;'}, group=mask4)
grad5 = doc.draw_object('linearGradient', {'id':'grad5', 'x1':str(-R-r),
    'x2':str(R+r), 'y1':'0', 'y2':'0', 'gradientUnits':'userSpaceOnUse'},
    group=defs)
for col, of, opa in [ ['#ffffff', '0', '0'], ['#ffffff', str(r/(R+r)), '0'],
        ['#ffffff', str(R/(R+r)), '1'], ['#ffffff', '1', '1'] ]:
    stop = doc.draw_object('stop', {'stop-color':col, 'offset':of,
        'stop-opacity':opa}, group=grad5)
mask5 = doc.draw_object('mask', {'id':'mask5', 'maskContentUnits':'userSpaceOnUse'}, group=defs)
doc.draw_object('rect', {'x':str(-R-r), 'y':str(-R-r), 'width':str(2*(R+r)),
    'height':str(R+r), 'style':'fill:url(#grad5); stroke:none;'}, group=mask5)

d = ('M {},{} L {},{} L {},{} A {},{} {} {} {} {},{} L {},{} L {},{} ' +
     'L {},{} A {},{} {} {} {} {},{} L {},{} Z').format(-R-r, h,
     -R+r, h, -R+r, 0, R-r, R-r, 0, 0, 1, R-r, 0, R-r, h, R+r, h, R+r, 0,
     R+r, R+r, 0, 0, 0, -R-r, 0, -R-r, h)
doc.draw_object('path', {'d':d, 'style':'fill:#ff0000; ' +
     'stroke:none;'}, group=g)
d = ('M {},{} L {},{} L {},{} A {},{} {} {} {} {},{} ' +
     'L {},{} A {},{} {} {} {} {},{} L {},{} Z').format(-R-r, h,
     -R+r, h, -R+r, 0, R-r, R-r, 0, 0, 1, 0, -R+r, 0, -R-r,
    R+r, R+r, 0, 0, 0, -R-r, 0, -R-r, h)
doc.draw_object('path', {'d':d, 'style':'fill:#00cc00;stroke:none;'},
    group=g)
d = ('M {},{} L {},{} L {},{} L {},{} L {},{} Z').format(-R-r, h,
     -R+r, h, -R+r, 0, -R-r, 0, -R-r, h)
doc.draw_object('path', {'d':d, 'style':'fill:url(#grad1);stroke:none;'},
    group=g)
d = ('M {},{} L {},{} L {},{} L {},{} L {},{} Z').format(R-r, h,
     R+r, h, R+r, 0, R-r, 0, R-r, h)
doc.draw_object('path', {'d':d, 'style':'fill:url(#grad1);stroke:none;'},
    group=g)
d = ('M {},{} L {},{} A {},{} {} {} {} {},{} ' +
     'L {},{} A {},{} {} {} {} {},{} Z').format(-R-r, 0, -R+r, 0,
    R-r, R-r, 0, 0, 1, R-r, 0, R+r, 0, R+r, R+r, 0, 0, 0, -R-r, 0)
doc.draw_object('path', {'d':d, 'style':'fill:url(#grad2);stroke:none;',
    'mask':'url(#mask4)'}, group=g)
d = ('M {},{} L {},{} A {},{} {} {} {} {},{} ' +
    'L {},{} A {},{} {} {} {} {},{} Z').format(-R-r, 0, -R+r, 0,
    R-r, R-r, 0, 0, 1, R-r, 0, R+r, 0, R+r, R+r, 0, 0, 0, -R-r, 0)
doc.draw_object('path', {'d':d, 'style':'fill:url(#grad3);stroke:none;',
    'mask':'url(#mask5)'}, group=g)
d = ('M {},{} L {},{} L {},{} A {},{} {} {} {} {},{} L {},{} L {},{} ' +
    'L {},{} A {},{} {} {} {} {},{} L {},{} Z').format(-R-r, h,
    -R+r, h, -R+r, 0, R-r, R-r, 0, 0, 1, R-r, 0, R-r, h, R+r, h, R+r, 0,
    R+r, R+r, 0, 0, 0, -R-r, 0, -R-r, h)
doc.draw_object('path', {'d':d, 'style':'fill:none; ' +
     'stroke:#000000; stroke-width:0.04;'}, group=g)

text_N = doc.draw_object('text', {'text-anchor':'middle', 'x':'0', 'y':'0',
    'transform':'translate({},{}) scale({},{})'.format(R, h-0.6, 0.04, -0.04),
    'style':'fill:#000000; stroke:none; ' +
    'font-size:12px; font-family:Bitstream Vera Sans;'}, group=g)
text_N.text = 'N'
text_S = doc.draw_object('text', {'text-anchor':'middle', 'x':'0', 'y':'0',
    'transform':'translate({},{}) scale({},{})'.format(-R, h-0.6, 0.04, -0.04),
    'style':'fill:#000000; stroke:none; ' +
    'font-size:12px; font-family:Bitstream Vera Sans;'}, group=g)
text_S.text = 'S'

doc.write()

Licensing

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I, the copyright holder of this work, hereby publish it under the following license:
w:en:Creative Commons
attribution share alike
This file is licensed under the Creative Commons Attribution-Share Alike 4.0 International license.
You are free:
  • to share – to copy, distribute and transmit the work
  • to remix – to adapt the work
Under the following conditions:
  • attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
  • share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.

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Date/TimeThumbnailDimensionsUserComment
current16:18, 24 November 2019Thumbnail for version as of 16:18, 24 November 2019600 × 600 (99 KB)Geek3 (talk | contribs)User created page with UploadWizard

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