File:VFPt superconductor ball E-field potential+contour.svg

# paste this code at the end of VectorFieldPlot 2.3
doc = FieldplotDocument('VFPt_superconductor_ball_E-field_potential+contour',
    width=600, height=600, commons=True)
unit = 100.
field_direction = [0.0, -1.0]
sphere = {'p':sc.array([0., 0.]), 'r':1.2}
field = Field([['homogeneous', {'Fx':field_direction[0], 'Fy':field_direction[1]}],
    ['dipole', {'x':sphere['p'][0], 'y':sphere['p'][1],
     'px':4*pi*sphere['r']**3*field_direction[0],
     'py':4*pi*sphere['r']**3*field_direction[1]}]])

def pot(xy):
    if vabs(xy) <= sphere['r']:
        return 1e-8 * xy[1] # zero potential inside metal sphere
    return field.V(xy)

doc.draw_contours(func=pot, levels=sc.linspace(-3, 3, 11))

U0 = pot([3, 3])
doc.draw_scalar_field(func=pot, cmap=doc.cmap_AqYlFs, vmin=-U0, vmax=U0)

# draw the superconducting ball
ball = doc.draw_object('g', {'id':'metal_ball'})
grad = 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=ball)
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=grad)
doc.draw_object('circle', {'cx':sphere['p'][0], 'cy':sphere['p'][1], 'r':str(sphere['r']),
    'style':'fill:url(#metal_spot); stroke:#000; stroke-width:0.02'}, group=ball)
ball_charges = doc.draw_object('g', {'style':'stroke-width:0.02; stroke-linecap:square'}, group=ball)

n_lines = 22
for i in range(n_lines):
    a = -3.3 + 6.6 * (0.5 + i) / n_lines
    line = FieldLine(field, [a, 6], maxr=12, pass_dipoles=1,
        bounds_func=lambda xy: sphere['r'] - vabs(xy - sphere['p']))
    doc.draw_line(line, linewidth=2.4, arrows_style=
        {'at_potentials':[-2.1, 2.1]})
    
    # draw little charge signs near the surface
    path_minus = 'M {0:.5f},0 h {1:.5f}'.format(-2./unit, 4./unit)
    path_plus = 'M {0:.5f},0 h {1:.5f} M 0,{0:.5f} v {1:.5f}'.format(-2./unit, 4./unit)
    
    # check if fieldline crosses sphere surface
    tlist = sc.linspace(0., 1., 101)
    for i in range(1, len(tlist)):
        in0 = vabs(line.get_position(tlist[i-1]) - sphere['p']) <= sphere['r']
        in1 = vabs(line.get_position(tlist[i]) - sphere['p']) <= sphere['r']
        if in0 != in1:
            # find the point where the field line cuts the surface
            t = op.brentq(lambda t: vabs(line.get_position(t)
                - sphere['p']) - sphere['r'], tlist[i-1], tlist[i])
            pr = line.get_position(t) - sphere['p']
            cpos = 0.92 * sphere['r'] * pr / vabs(pr)
            if in1:
                path_d = path_minus
            else:
                path_d = path_plus
            doc.draw_object('path', {'stroke':'black', 'd':path_d,
                'transform':'translate({:.5f},{:.5f})'.format(
                    round(unit*cpos[0])/unit, round(unit*cpos[1])/unit)},
                    group=ball_charges)

doc.write()