File:VFPt metal balls largesmall2 potential+contour.svg

# paste this code at the end of VectorFieldPlot 3.1
# https://commons.wikimedia.org/wiki/User:Geek3/VectorFieldPlot
u = 100.0
doc = FieldplotDocument('VFPt_metal_balls_largesmall2_potential+contour',
    commons=True, width=800, height=600, unit=u)

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

def U_sphere(sphere, charges):
    f = Field([ ['monopole', {'x':c['p'][0], 'y':c['p'][1], 'Q':c['Q']}] for c in charges])
    return sc.mean([f.V(sphere['c'] + sphere['r'] * array((cos(phi), sin(phi))))
        for phi in sc.linspace(0, 2*pi, 64, endpoint=False)])

def Q_sphere(isphere, charges):
    return sum([c['Q'] for c in charges if c['i'] == isphere])

# compute series of charges https://dx.doi.org/10.2174/1874183500902010032
def mirrored_charges(p, Q, isphere, spheres, Qmin):
    '''
    Recursive function. Returns list of mirrored charges for n spheres
    '''
    if fabs(Q) < Qmin:
        return []
    charges = [{'p':p, 'Q':Q, 'i':isphere}]
    for i, s in enumerate(spheres):
        if i != isphere:
            pnew = s['c'] + (p - s['c']) * (s['r'] / vabs(p - s['c']))**2
            Qnew = -Q * s['r'] / vabs(p - s['c'])
            charges += mirrored_charges(pnew, Qnew, i, spheres, Qmin)
    return charges

charges_raw = [mirrored_charges(s['c'], 1., si, spheres, 1e-4) for si,s in enumerate(spheres)]
# Use charge normalization from paper above
# Here one can also solve for charge conditions such as neutrality
matrixU = [ [U_sphere(s, cs) for cs in charges_raw] for s in spheres]
matrixQ = [ [Q_sphere(si, cs) for cs in charges_raw] for si in range(len(spheres))]
Q0, Q1 = 1., -1
charge_factors = sc.linalg.solve(matrixQ, [Q0, Q1])
for il in range(len(charges_raw)):
    for ic in range(len(charges_raw[il])):
        charges_raw[il][ic]['Q'] *= charge_factors[il]

charges = [c for cl in charges_raw for c in cl]
charges = sorted(charges, key=lambda x: -fabs(x['Q']))
for si, s in enumerate(spheres):
    s['U'] = U_sphere(s, charges)
    s['Q'] = Q_sphere(si, charges)
    #print('sphere', si, s, 'U =', s['U'], 'Q =', s['Q'])
print('using', len(charges), 'mirror charges.')

field = Field([ ['monopole', {'x':c['p'][0], 'y':c['p'][1], 'Q':c['Q']}] for c in charges])

def pot(xy):
    for s in spheres:
        if vabs(xy - s['c']) <= s['r']:
            return s['U']
    return field.V(xy)

U0 = max([fabs(U_sphere(s, charges)) for s in spheres])
doc.draw_scalar_field(func=pot, cmap=doc.cmap_AqYlFs, vmin=-U0, vmax=U0)
doc.draw_contours(func=pot, linewidth=1, linecolor='#444444',
    levels=sc.linspace(-U0, U0, 25)[1:-1])

# draw symbols
#for c in charges:
#    doc.draw_charges(Field([ ['monopole', {'x':c[0][0], 'y':c[0][1], 'Q':c[1]}] ]),
#        scale=0.6*sqrt(fabs(c[1])))

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, s in enumerate(spheres):
    ball = doc.draw_object('g', {'id':'metal_ball{:}'.format(ib+1),
        'transform':'translate({:.3f},{:.3f})'.format(*(s['c'])),
        '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)
    ball_charges.append(doc.draw_object('g',
        {'style':'stroke-width:0.02'}, group=ball))

def startpath(t):
    phi = 2. * pi * t
    return s2['c'] + 1.5 * array([cos(phi), sin(phi)])
    
nlines1 = 22
startpoints = Startpath(field, startpath).npoints(nlines1)

for ip, p0 in enumerate(startpoints):
    line = FieldLine(field, p0, directions='both', maxr=1000,
        bounds_func=lambda xy: max([s['r'] - vabs(xy-s['c']) for s in [s1, s2]]))
    
    # 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['c']) < sphere['r'] and
                vabs(line.get_position(1-ci) - sphere['c']) > sphere['r']):
                # find the point where the field line cuts the surface
                t = optimize.brentq(lambda t: vabs(line.get_position(t)
                    - sphere['c']) - sphere['r'], 0., 1.)
                pr = line.get_position(t) - sphere['c']
                cpos = (-0.06 + 0.96 * sphere['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=ball_charges[si])
    
    ar_st = {'potential':pot, 'at_potentials':[-0.21*U0, 0.1*U0]}
    ar_st['scale'] = 1.2
    doc.draw_line(line, arrows_style=ar_st)
doc.write()