File:VFPt metal balls largesmall transparent.svg

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Summary

Description
English: Electric field around a large and a small conducting sphere 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, shown in red and blue. In reality, the field is created by a continuous charge distribution at the surface of each sphere and the field lines inside the sphere don't exist. Field lines are always orthogonal to the surface of each sphere.
Date
Source Own work
Author Geek3
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Python 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_largesmall_transparent',
    commons=True, width=800, height=600, center=[400, 300], unit=u)

# define two spheres with position, radius and charge
s1 = {'p':sc.array([-1.0, 0.]), 'r':1.5}
s2 = {'p':sc.array([2.0, 0.]), 'r':0.5}

# 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 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':0.5})
    
    # draw rods
    if ib == 0:
        x1, x2 = -4.1 - s1['p'][0], -0.9 * s1['r']
    else:
        x1, x2 = 0.9 * s2['r'], 4.1 - s2['p'][0]
    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

startp = []
def startpath1(t):
    phi = 2. * pi * t
    return (sc.array(s2['p']) + 1.5 * sc.array([cos(phi), sin(phi)]))
start_func1 = get_startpoint_function(startpath1, field)
nlines1 = 16
for i in range(nlines1):
    startp.append(start_func1((0.5 + i) / nlines1))

def startpath2(t):
    phi = 2. * pi * (0.195 + 0.61 * t)
    return (sc.array(s1['p']) + 1.5 * sc.array([cos(phi), -sin(phi)]))
start_func2 = get_startpoint_function(startpath2, field)
nlines2 = 14
for i in range(nlines2):
    startp.append(start_func2((0.5 + i) / nlines2))

# draw the field lines
for p0 in startp:
    line = FieldLine(field, p0, directions='both', maxr=7.)
    
    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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current20:05, 30 December 2018Thumbnail for version as of 20:05, 30 December 2018800 × 600 (41 KB)wikimediacommons>Geek3User created page with UploadWizard

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