File:VFPt horseshoe-magnet absB.svg
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. 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 |
| SVG development | Category:Valid SVG created with VectorFieldPlot code#VFPt%20horseshoe-magnet%20absB.svg  This file uses embedded text. |
| Source code | 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 np.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 + np.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=np.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 = np.array([0, 0.07, 0.25, 0.6, 1])
grad_opa = np.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()
|
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