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Summary
DescriptionVFPt metal balls plusminus potential.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. |
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Date | |||
Source | Own work | ||
Author | Geek3 | ||
Other versions |
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SVG development InfoField | This W3C-invalid vector image was created with Inkscape, or with something else.
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Source code InfoField | SVG code# paste this code at the end of VectorFieldPlot 1.10
# https://commons.wikimedia.org/wiki/User:Geek3/VectorFieldPlot
u = 100.0
doc = FieldplotDocument('VFPt_metal_balls_plusminus_potential',
commons=True, width=800, height=600, center=[400, 300], unit=u)
# define two spheres with position, radius and charge
s1 = {'p':sc.array([-1.5, 0.]), 'r':1.0, 'q':1.}
s2 = {'p':sc.array([1.5, 0.]), 'r':1.0, 'q':-1.}
# 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 = (s2['p'] - s1['p']) / d
# compute series of charges https://dx.doi.org/10.2174/1874183500902010032
charges = [[s1['p'][0], s1['p'][1], s1['q']], [s2['p'][0], s2['p'][1], 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([p1[0], p1[1], q1])
charges.append([p2[0], p2[1], q2])
if max(fabs(q1), fabs(q2)) < 1e-3 * q0:
break
field = Field({'monopoles':charges})
# draw potential in background
p_array = sc.array([c[:2] for c in charges])
q_array = sc.array([c[2] for c in charges])
def potential(xy):
return sc.dot(q_array, 1. / sc.linalg.norm(xy - p_array, axis=1))
from matplotlib import colors
# colormap from aqua through yellow to fuchsia
cmap = colors.ListedColormap([sc.clip((2*x, 2*(1-x), 4*(x-0.5)**2), 0, 1)
for x in sc.linspace(0., 1., 2048)])
doc.draw_scalar_field(func=potential, cmap=cmap,
vmin=potential(s2['p'] + s2['r'] * sc.array([1., 0.])),
vmax=potential(s1['p'] + s1['r'] * sc.array([-1., 0.])))
# draw symbols
for c in charges:
doc.draw_charges(Field({'monopoles':[c]}), scale=0.6*sqrt(fabs(c[2])))
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)
ball_charges = []
for ib in range(2):
ball = doc.draw_object('g', {'id':'metal_ball{:}'.format(ib+1),
'transform':'translate({:.3f},{:.3f})'.format(*([s1, s2][ib]['p'])),
'style':'fill:none; stroke:#000;stroke-linecap:square', 'opacity':1})
# draw metal balls
doc.draw_object('circle', {'cx':0, 'cy':0, 'r':[s1, s2][ib]['r'],
'style':'fill:url(#metal_spot); stroke-width:0.02'}, group=ball)
ball_charges.append(doc.draw_object('g',
{'style':'stroke-width:0.02'}, group=ball))
# find start positions of field lines
def startpath(t):
phi = 2. * pi * t
return (sc.array(s1['p']) + 1.5 * sc.array([cos(phi), sin(phi)]))
def dstartpath(t):
return (startpath(t+1e-6) - startpath(t-1e-6)) / 2e-6
def FieldSum(t0, t1):
return ig.quad(lambda t:
sc.cross(field.F(startpath(t)), dstartpath(t)), t0, t1)[0]
Ftotal = FieldSum(0, 1)
def startpos(s):
t = op.brentq(lambda t: FieldSum(0, t) / Ftotal - s, 0, 1)
return startpath(t)
# draw the field lines
nlines = 24
for i in range(nlines):
p0 = startpos((0.5 + i) / nlines)
line = FieldLine(field, p0, 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 in range(2):
sphere = [s1, s2][si]
# check if fieldline ends inside the sphere
for ci in range(2):
if vabs(line.get_position(ci) - sphere['p']) < sphere['r']:
# find the point where the field line cuts the surface
t = op.brentq(lambda t: vabs(line.get_position(t)
- sphere['p']) - sphere['r'], 0., 1.)
pr = line.get_position(t) - sphere['p']
cpos = 0.9 * sphere['r'] * pr / vabs(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=ball_charges[si])
arrow_d = 2.0
of = [0.5 + s1['r'] / arrow_d, 0.5, 0.5, 0.5 + s2['r'] / arrow_d]
doc.draw_line(line, arrows_style={'dist':arrow_d, 'offsets':of})
doc.write()
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- 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.
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Date/Time | Thumbnail | Dimensions | User | Comment | |
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current | 20:05, 30 December 2018 | 800 × 600 (170 KB) | Geek3 | User created page with UploadWizard |
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Short title | VFPt_metal_balls_plusminus_potential |
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Image title | VFPt_metal_balls_plusminus_potential
created with VectorFieldPlot 1.10 https://commons.wikimedia.org/wiki/User:Geek3/VectorFieldPlot about: https://commons.wikimedia.org/wiki/File:VFPt_metal_balls_plusminus_potential.svg rights: Creative Commons Attribution ShareAlike 4.0 |
Width | 800 |
Height | 600 |