This code is based on following web sites:
このページのコードは、次のページを参考にしています。
LECTURE NOTES 6, The Method of Image Charges - UIUC Physics 435 EM Fields & Sources, Prof. Steven Errede -
URL: http://web.hep.uiuc.edu/home/serrede/P435/Lecture_Notes/P435_Lect_06.pdf
Method of image charges, Reflection in a conducting sphere - Wikipedia -
URL: https://en.wikipedia.org/wiki/Method_of_image_charges
Visualizing streamlines - Number Crunch -
URL: https://www.numbercrunch.de/blog/2013/05/visualizing-streamlines/
This page shows the method to draw electric field line around a point charge adjecent to a grounded sphere using method of image charges.
このページでは、接地球近くにある点電荷から伸びる電気力線を描画します。計算は影像電荷法に基づいてします。

%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np
from scipy.integrate import ode as ode
from matplotlib import cm
from itertools import product
from matplotlib.patches import Circle

class charge:
    def __init__(self, q, pos):
        self.q=q
        self.pos=pos
 
def E_point_charge(q, a, x, y):
    return q*(x-a[0])/((x-a[0])**2+(y-a[1])**2)**(1.5), \
        q*(y-a[1])/((x-a[0])**2+(y-a[1])**2)**(1.5)
 
def E_total(x, y, charges):
    Ex, Ey=0, 0
    for C in charges:
        E=E_point_charge(C.q, C.pos, x, y)
        Ex=Ex+E[0]
        Ey=Ey+E[1]
    return [ Ex, Ey ]

def E_dir(t, y, charges):
    Ex, Ey=E_total(y[0], y[1], charges)
    n=np.sqrt(Ex**2+Ey*Ey)
    return [Ex/n, Ey/n]

def V_point_charge(q, a, x, y):
    return q/((x-a[0])**2+(y-a[1])**2)**(0.5)

def V_total(x, y, charges):
    V=0
    for C in charges:
        Vp=V_point_charge(C.q, C.pos, x, y)
        V = V+Vp
    return V

# charges and positions
a1 = 1
q1 = 1
r2 = 0.5
b2 = (r2**2)/a1
q2 = -r2/a1*q1

charges=[ charge(q1, [a1, 0]), charge(q2,[b2,0])]

# calculate field lines
x0, x1=-1, 3
y0, y1=-2, 2
R=0.01
# loop over all charges
xs,ys = [],[]

C = charges[0]
# calculate field lines only from point charge outside of the sphere
dt=0.8*R
for alpha in np.linspace(0+2*np.pi/256, 2*np.pi*127/128+2*np.pi/256, 128):
    r=ode(E_dir)
    r.set_integrator('vode')
    r.set_f_params(charges)
    x=[ C.pos[0] + np.cos(alpha)*R ]
    y=[ C.pos[1] + np.sin(alpha)*R ]
    r.set_initial_value([x[0], y[0]], 0)
    while r.successful():
        r.integrate(r.t+dt)
        x.append(r.y[0])
        y.append(r.y[1])
        hit_charge=False
        # check if field line left drwaing area or ends in some charge
        for C2 in charges:
            if np.sqrt((r.y[0]-C2.pos[0])**2+(r.y[1]-C2.pos[1])**2)<R:
                hit_charge=True
        if hit_charge or (not (5*x0<r.y[0] and r.y[0]<5*x1)) or \
                (not (5*y0<r.y[1] and r.y[1]<5*y1)):
            break
    xs.append(x)
    ys.append(y)

# calculate electric potential
vvs = []
xxs = []
yys = []
numcalcv = 300
for x,y in product(np.linspace(x0,x1,numcalcv),np.linspace(y0,y1,numcalcv)):
    xxs.append(x)
    yys.append(y)
    vvs.append(V_total(x,y,charges))
xxs = np.array(xxs)
yys = np.array(yys)
vvs = np.array(vvs)

plt.figure(figsize=(5.5, 4.5),facecolor="w")

# plot field line
for x, y in zip(xs,ys):
    plt.plot(x, y, color="k",lw=0.3)

# plot point charges
for C in charges:
    if C.q<0:
        plt.plot(C.pos[0], C.pos[1], 'bo', ms=8*np.sqrt(-C.q))
    if C.q>0:
        plt.plot(C.pos[0], C.pos[1], 'ro', ms=8*np.sqrt(C.q))

# plot electric potential
clim0,clim1 = 0,4
vvs[np.where(vvs<clim0)] = clim0*0.999999 # to avoid error
vvs[np.where(vvs>clim1)] = clim1*0.999999 # to avoid error
plt.tricontour(xxs,yys,vvs,10,colors="0.3")
plt.tricontourf(xxs,yys,vvs,100,cmap=cm.hot_r)
cbar = plt.colorbar()
cbar.set_clim(clim0,clim1)
cbar.set_ticks(np.linspace(clim0,clim1,11))
cbar.set_label("Electric Potential")

# plot grounded sphere
plt.axes().add_patch(Circle((0, 0), radius=r2, edgecolor="k",facecolor='w', linewidth=1,zorder=10))

# adjustment
plt.xlabel('$x$')
plt.ylabel('$y$')
plt.axes().set_aspect("equal","datalim")
plt.xlim(x0, x1)
plt.ylim(y0, y1)
plt.savefig("field_line_outside_a_sphere.png",bbox_inches="tight",pad_inches=0.02)
plt.show()

ほげおのクラフト

2017年12月6日水曜日

Python + Matplotlib.pyplot で影像電荷法で計算した接地球近くの点電荷から伸びる電気力線を描く