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File:VFPt horseshoe-magnet.svg

Original file ‎(SVG file, nominally 600 × 600 pixels, file size: 40 KB)

Summary

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.
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
SVG development
InfoField
 
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This plot was created with VectorFieldPlot.
 
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Source code
InfoField

Python code

# paste this code at the end of VectorFieldPlot 3.0
doc = FieldplotDocument('VFPt_horseshoe-magnet', 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}] ])

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

for iline, p0 in enumerate(startpoints):
    line = FieldLine(field, p0, directions='both', maxr=1000)
    fe = {'start':True, 'leave_image':False, 'enter_image':False, 'end':True}
    if iline in [0, 1, 2, nlines-1, nlines-2, nlines-3]:
        fe['start'] = fe['end'] = False
    min_arrows = 1
    if iline == nlines - 7:
        min_arrows = 3
    doc.draw_line(line, arrows_style={
                  'dist':2.0, 'fixed_ends':fe, 'min_arrows':min_arrows})

# 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

I, the copyright holder of this work, hereby publish it under the following license:
w:en:Creative Commons
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horseshoe magnet

inception

7 July 2018

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Date/TimeThumbnailDimensionsUserComment
current18:03, 7 July 2018Thumbnail for version as of 18:03, 7 July 2018600 × 600 (40 KB)Geek3User created page with UploadWizard

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