File:VFPt metal balls neutral potential.svg
![File:VFPt metal balls neutral potential.svg](https://upload.wikimedia.org/wikipedia/commons/thumb/d/d9/VFPt_metal_balls_neutral_potential.svg/800px-VFPt_metal_balls_neutral_potential.svg.png?20181230200512)
Original file (SVG file, nominally 800 × 600 pixels, file size: 133 KB)
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Summary
[edit]DescriptionVFPt metal balls neutral potential.svg |
English: Electric field around a positively charged and a neutral conducting sphere. 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 the maximum positive potential in pink. |
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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_neutral_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':0.34}
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)])
U0 = potential(s1['p'] + s1['r'] * sc.array([-1., 0.]))
Ucorner = potential(sc.array([4., 3.]))
doc.draw_scalar_field(func=potential, cmap=cmap, vmin=2*Ucorner-U0, vmax=U0)
# 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 rods
if ib == 0:
x1, x2 = -4.1 - s1['p'][0], -0.9 * s1['r']
doc.draw_object('rect', {'x':x1, 'width':x2-x1,
'y':-0.1/1.2+0.01, 'height':0.2/1.2-0.02,
'style':'fill:url(#rod_shade); stroke-width:0.02'}, group=ball)
# 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 well-distributed start positions of field lines
def get_startpoint_function(startpath, field):
'''
Given a vector function startpath(t), this will return a new
function such that the scalar parameter t in [0,1] progresses
indirectly proportional to the orthogonal field strength.
'''
def dstartpath(t):
return (startpath(t+1e-6) - startpath(t-1e-6)) / 2e-6
def FieldSum(t0, t1):
return ig.quad(lambda t: sc.absolute(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)
return startpos
def startpath1(t):
phi = 2. * pi * t
return (s1['p'] + s1['r'] * sc.array([cos(phi), sin(phi)]))
start_func1 = get_startpoint_function(startpath1, field)
startp = []
nlines1 = 24
for i in range(nlines1):
startp.append(start_func1((0.5 + i) / nlines1))
def startpath(t):
phi = 2. * pi * t
return (s2['p'] + s2['r'] * sc.array([-cos(phi), sin(phi)]))
def dstartpath(t):
return (startpath(t+1e-6) - startpath(t-1e-6)) / 2e-6
t0 = op.brentq(lambda t: sc.cross(field.F(startpath(t)), dstartpath(t)), 0, 0.5)
def startpath2(t):
phi = 2. * pi * (t0 + t * (1. - 2. * t0))
return (s2['p'] + 1.0 * sc.array([-cos(phi), sin(phi)]))
start_func2 = get_startpoint_function(startpath2, field)
nlines = 4
for i in range(nlines):
startp.append(start_func2((0.5 + i) / nlines))
for phi in sc.linspace(-0.35, 0.35, 4):
startp.append(s1['p'] + 0.05 * sc.array([cos(phi), sin(phi)]))
for phi in sc.linspace(-1.4, 1.4, 4):
startp.append(s2['p'] + 0.05 * sc.array([-cos(phi), sin(phi)]))
for ip, p0 in enumerate(startp):
line = FieldLine(field, p0, directions='both', maxr=10.)
# 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'] and
vabs(line.get_position(1-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]
ma = 1
if ip >= len(startp) - 8:
ma = 0
doc.draw_line(line, arrows_style={'dist':arrow_d, 'offsets':of,
'min_arrows':ma})
doc.write()
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Date/Time | Thumbnail | Dimensions | User | Comment | |
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current | 20:05, 30 December 2018 | ![]() | 800 × 600 (133 KB) | Geek3 (talk | contribs) | User created page with UploadWizard |
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Short title | VFPt_metal_balls_neutral_potential |
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Image title | VFPt_metal_balls_neutral_potential
created with VectorFieldPlot 1.10 https://commons.wikimedia.org/wiki/User:Geek3/VectorFieldPlot about: https://commons.wikimedia.org/wiki/File:VFPt_metal_balls_neutral_potential.svg rights: Creative Commons Attribution ShareAlike 4.0 |
Width | 800 |
Height | 600 |