File:VFPt metal balls neutral potential+contour.svg

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Original file(SVG file, nominally 800 × 600 pixels, file size: 126 KB)

Captions

Captions

Electric field and potential around a positively charged and a neutral sphere

Summary[edit]

Description
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 shown in the background from positive (fuchsia) to zero (yellow) together with uniformely spaced equipotential lines. Note that the field lines follow the gradient of the potential.
Date
Source Own work
Author Geek3
Other versions VFPt metal balls neutral potential.svg
SVG development
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This plot was created with VectorFieldPlot.
Source code
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Python code

# 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_neutral_potential+contour',
    commons=True, width=800, height=600, center=[400, 300], unit=u)

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

# compute series of charges https://dx.doi.org/10.2174/1874183500902010032
def make_charge_series(p0, q0, spheres):
    charges = []
    p, q = p0, q0
    i = 0
    while fabs(q) > 1e-4 * fabs(q0):
        charges.append([p, q])
        i += 1
        s = spheres[i%2]
        q = -q * s['r'] / vabs(p - s['c'])
        p = s['c'] + (p - s['c']) * (s['r'] / vabs(p - s['c']))**2
    return charges

charges1 = make_charge_series(s1['c'], 1., [s1, s2])
charges2 = make_charge_series(s2['c'], 1., [s2, s1])

# make sphere 2 neutral
charge_ratio = sum([c[1] for c in charges1[1::2]]) / sum([c[1] for c in charges2[::2]])
for c in charges2:
    c[1] = c[1] * -charge_ratio

charges = sorted(charges1 + charges2, key=lambda x: -fabs(x[1]))
field = Field([ ['monopole', {'x':c[0][0], 'y':c[0][1], 'Q':c[1]}] for c in charges])

def pot(xy):
    for s in s1, s2:
        if vabs(xy - s['c']) <= s['r']:
            return field.V(s['c'] + array((s['r'], 0)))
    return field.V(xy)

U0 = field.V(s1['c'] + array((s1['r'], 0)))
Ucorner = field.V(sc.array([4., 3.]))
doc.draw_scalar_field(func=pot, cmap=doc.cmap_AqYlFs, vmin=2*Ucorner-U0, vmax=U0)
doc.draw_contours(func=pot, linewidth=1, linecolor='#444444',
    levels=sc.linspace(0, U0, 11)[:-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])))

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), ('#ddd', 0.8),
    ('#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]['c'])),
        'style':'fill:none; stroke:#000;stroke-linecap:square', 'opacity':1})
    
    # draw rods
    if ib == 0:
        x1, x2 = -4.1 - s1['c'][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))

def startpath1(t):
    phi = 2. * pi * t
    return s1['c'] + s1['r'] * array([cos(phi), sin(phi)])

def startpath2(t):
    phi = 2. * pi * t
    return s2['c'] + s2['r'] * array([-cos(phi), sin(phi)])
    
nlines1 = 24
startpoints = Startpath(field, startpath1).npoints(nlines1)

t0 = optimize.brentq(lambda t: sc.cross(field.F(startpath2(t)),
    Startpath(field, startpath2)._dstartpath(t)), 0, 0.5)

nlines2 = 4
startpoints += Startpath(field, startpath2, t0=t0, t1=1-t0).npoints(nlines2)

#for phi in sc.linspace(-0.35, 0.35, 4):
#    startpoints.append(s1['c'] + 0.05 * sc.array([cos(phi), sin(phi)]))
#for phi in sc.linspace(-1.4, 1.4, 4):
#    startpoints.append(s2['c'] + 0.05 * sc.array([-cos(phi), sin(phi)]))

for ip, p0 in enumerate(startpoints):
    line = FieldLine(field, p0, directions='both', maxr=10.,
        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 = 0.9 * 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])
    
    doc.draw_line(line, arrows_style={'potential':pot,
        'at_potentials':[0.25 * U0, 0.55 * U0]})
doc.write()

Licensing[edit]

I, the copyright holder of this work, hereby publish it under the following license:
w:en:Creative Commons
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File history

Click on a date/time to view the file as it appeared at that time.

Date/TimeThumbnailDimensionsUserComment
current21:51, 25 May 2020Thumbnail for version as of 21:51, 25 May 2020800 × 600 (126 KB)Geek3 (talk | contribs)Uploaded own work with UploadWizard

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