File:VFPt metal balls plusminus potential+contour.svg
Original file (SVG file, nominally 800 × 600 pixels, file size: 200 KB)
Captions
Summary
[edit]DescriptionVFPt metal balls plusminus potential+contour.svg |
English: Electric field around two identical conducting spheres at opposite electric potential. The shape of the field lines is computed exactly, using the method of image charges with an infinite series of charges inside the two spheres. Field lines are always orthogonal to the surface of each sphere. In reality, the field is created by a continuous charge distribution at the surface of each sphere, indicated by small plus and minus signs. The electric potential is depicted as background color with yellow at 0V and uniformely spaced equipotential lines are shown. |
Date | |
Source | Own work |
Author | Geek3 |
Other versions |
|
SVG development InfoField | This plot was created with VectorFieldPlot. |
Source code InfoField | Python code# paste this code at the end of VectorFieldPlot 3.3
# https://commons.wikimedia.org/wiki/User:Geek3/VectorFieldPlot
u = 100.0
doc = FieldplotDocument('VFPt_metal_balls_plusminus_potential+contour',
commons=True, width=800, height=600, center=[400, 300], unit=u)
# define two spheres with position, radius and charge
s1 = {'p':array([-1.5, 0.]), 'r':1.0}
s2 = {'p':array([1.5, 0.]), 'r':1.0}
spheres = s1, s2
# make charge proportional to capacitance, which is proportional to radius.
s1['Q'] = s1['r']
s2['Q'] = -s2['r']
d = vabs(s2['p'] - s1['p'])
v12 = vnorm(s2['p'] - s1['p'])
# compute series of charges https://dx.doi.org/10.2174/1874183500902010032
charges = [{'x':s1['p'][0], 'y':s1['p'][1], 'Q':s1['Q']},
{'x':s2['p'][0], 'y':s2['p'][1], 'Q':s2['Q']}]
r1 = r2 = 0.
q1, q2 = s1['Q'], s2['Q']
q0 = max(fabs(q1), fabs(q2))
for i in range(10):
q1, q2 = -s1['r'] * q2 / (d - r2), -s2['r'] * q1 / (d - r1),
r1, r2 = s1['r']**2 / (d - r2), s2['r']**2 / (d - r1)
p1, p2 = s1['p'] + r1 * v12, s2['p'] - r2 * v12
charges.append({'x':p1[0], 'y':p1[1], 'Q':q1})
charges.append({'x':p2[0], 'y':p2[1], 'Q':q2})
if max(fabs(q1), fabs(q2)) < 1e-3 * q0:
break
field = Field([ ['monopole', c] for c in charges ])
U1 = field.V(s1['p'] - s1['r'] * array([1., 0.]))
U2 = field.V(s2['p'] + s2['r'] * array([1., 0.]))
doc.draw_scalar_field(func=field.V, cmap=doc.cmap_AqYlFs, vmin=U2, vmax=U1)
doc.draw_contours(func=field.V, linewidth=1, linecolor='#111111',
levels=sc.linspace(U2, U1, 11)[1:-1], attributes={'opacity':'0.7'})
# draw symbols
for c in charges:
doc.draw_charges(Field([ ['monopole', c] ]), scale=0.6*sqrt(fabs(c['Q'])))
gradr = doc.draw_object('linearGradient', {'id':'rod_shade', 'x1':0, 'x2':0,
'y1':0, 'y2':1, 'gradientUnits':'objectBoundingBox'}, group=doc.defs)
for col, of in (('#666', 0), ('#ddd', 0.6), ('#fff', 0.7), ('#ccc', 0.75),
('#888', 1)):
doc.draw_object('stop', {'offset':of, 'stop-color':col}, group=gradr)
gradb = doc.draw_object('radialGradient', {'id':'metal_spot', 'cx':'0.53',
'cy':'0.54', 'r':'0.55', 'fx':'0.65', 'fy':'0.7',
'gradientUnits':'objectBoundingBox'}, group=doc.defs)
for col, of in (('#fff', 0), ('#e7e7e7', 0.15), ('#ddd', 0.25),
('#aaa', 0.7), ('#888', 0.9), ('#666', 1)):
doc.draw_object('stop', {'offset':of, 'stop-color':col}, group=gradb)
for si, s in enumerate(spheres):
ball = doc.draw_object('g', {'id':'metal_ball{:}'.format(si+1),
'transform':'translate({:.3f},{:.3f})'.format(*(s['p'])),
'style':'fill:none; stroke:#000;stroke-linecap:square', 'opacity':1})
# draw metal balls
doc.draw_object('circle', {'cx':0, 'cy':0, 'r':s['r'],
'style':'fill:url(#metal_spot); stroke-width:0.02'}, group=ball)
s['cgroup'] = doc.draw_object('g', {'style':'stroke-width:0.02'}, group=ball)
# find start positions of field lines
def start(t):
phi = 2. * pi * t
return array(s1['p']) + 1.5 * array([cos(phi), sin(phi)])
# draw the field lines
nlines = 24
p0_list = Startpath(field, start).npoints(nlines)
for iline in range(nlines):
line = FieldLine(field, p0_list[iline], directions='both', maxr=1e4)
# draw little charge signs near the surface
path_minus = 'M {0:.5f},0 h {1:.5f}'.format(-2 / u, 4 / u)
path_plus = 'M {0:.5f},0 h {1:.5f} M 0,{0:.5f} v {1:.5f}'.format(-2 / u, 4 / u)
for si, s in enumerate(spheres):
# check if fieldline ends inside the sphere
for ci in range(2):
if vabs(line.get_position(ci) - s['p']) < s['r']:
# find the point where the field line cuts the surface
t = optimize.brentq(lambda t: vabs(line.get_position(t)
- s['p']) - s['r'], 0., 1.)
pr = line.get_position(t) - s['p']
cpos = 0.9 * s['r'] * vnorm(pr)
doc.draw_object('path', {'stroke':'black', 'd':
[path_plus, path_minus][ci],
'transform':'translate({:.5f},{:.5f})'.format(
round(u*cpos[0])/u, round(u*cpos[1])/u)},
group=s['cgroup'])
pot = []
if iline != 6 and iline != nlines - 1 - 6:
pot.append(0.)
if iline >= 6 and iline < nlines - 6:
pot += [0.27 * U1 + 0.73 * U2, 0.73 * U1 + 0.27 * U2]
doc.draw_line(line, linewidth=2.4, arrows_style={'at_potentials':pot})
doc.write()
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- Under the following conditions:
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Date/Time | Thumbnail | Dimensions | User | Comment | |
---|---|---|---|---|---|
current | 12:26, 16 December 2020 | 800 × 600 (200 KB) | Geek3 (talk | contribs) | Uploaded own work with UploadWizard |
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Short title | VFPt_metal_balls_plusminus_potential+contour |
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Image title | VFPt_metal_balls_plusminus_potential+contour
created with VectorFieldPlot 3.3 https://commons.wikimedia.org/wiki/User:Geek3/VectorFieldPlot about: https://commons.wikimedia.org/wiki/File:VFPt_metal_balls_plusminus_potential+contour.svg rights: Creative Commons Attribution ShareAlike 4.0 |
Width | 800 |
Height | 600 |