原本想說用「斜率場」來畫電力線，但光靠斜率場實在太不精確，也無法對應實際的數學意義，因此決定根據公式來撰寫電場模擬。 大部分數學原理及程式碼都是參考網路上的資源。 參考資源多樣，也包含電磁學課本，恕不逐一放上。 數學(物理)原理:http://163.28.10.78/content/vocation/control/tp_nh/ee/tp_nh/7/3.htm
import numpy as np
import matplotlib.pyplot as plt
from matplotlib.patches import Circledef E(q, r0, x, y):
"""Return the electric field vector E=(Ex,Ey) due to charge q at r0."""
den = np.hypot(x - r0[0], y - r0[1]) ** 3
return q * (x - r0[0]) / den, q * (y - r0[1]) / dennp.hypot即是
# Grid of x, y points
nx, ny = 64, 64
x = np.linspace(-3, 3, nx)
y = np.linspace(-3, 3, ny)
X, Y = np.meshgrid(x, y)設定nq數量以決定模電荷子數量
nq = 2 # the number of charges
charges = []
for i in range(nq):
if i % 2 == 0:
q = 1
else:
q = -1
charges.append((q, (np.cos(2 * np.pi * i / nq), np.sin(2 * np.pi * i / nq))))charges這個list物件所append的東西是:
[正負電荷(q) , ,
]
也是決定電荷位置的關鍵!
# Electric field vector, E=(Ex, Ey), as separate components
Ex, Ey = np.zeros((ny, nx)), np.zeros((ny, nx))
for charge in charges:
ex, ey = E(*charge, x=X, y=Y)
Ex += ex
Ey += ey
fig = plt.figure()
ax = fig.add_subplot(111)
# Plot the streamlines with an appropriate colormap and arrow style
color = 2 * np.log(np.hypot(Ex, Ey))
ax.streamplot(x, y, Ex, Ey, color=color, linewidth=1, cmap=plt.cm.inferno,
density=2, arrowstyle='->', arrowsize=1.5)<matplotlib.streamplot.StreamplotSet at 0x159d03e9730>
# Add filled circles for the charges themselves
charge_colors = {True: '#aa0000', False: '#0000aa'}
for q, pos in charges:
ax.add_artist(Circle(pos, 0.05, color=charge_colors[q > 0]))
ax.set_xlabel('$x$')
ax.set_ylabel('$y$')
ax.set_xlim(-2, 2)
ax.set_ylim(-2, 2)
ax.set_aspect('equal')
plt.show()- 若想讓圖框比例自動調配可以將
ax.set_aspect('equal')中的'equal'改成'auto'
import numpy as np
import matplotlib.pyplot as plt
from matplotlib.patches import Circle
def E(q, r0, x, y):
"""Return the electric field vector E=(Ex,Ey) due to charge q at r0."""
den = np.hypot(x - r0[0], y - r0[1]) ** 3
return q * (x - r0[0]) / den, q * (y - r0[1]) / den
# Grid of x, y points
nx, ny = 64, 64
x = np.linspace(-3, 3, nx)
y = np.linspace(-3, 3, ny)
X, Y = np.meshgrid(x, y)
nq = 2 # the number of charges
charges = []
for i in range(nq):
if i % 2 == 0:
q = 1
else:
q = -1
charges.append((q, (np.cos(2 * np.pi * i / nq), np.sin(2 * np.pi * i / nq))))
# Electric field vector, E=(Ex, Ey), as separate components
Ex, Ey = np.zeros((ny, nx)), np.zeros((ny, nx))
for charge in charges:
ex, ey = E(*charge, x=X, y=Y)
Ex += ex
Ey += ey
fig = plt.figure()
ax = fig.add_subplot(111)
# Plot the streamlines with an appropriate colormap and arrow style
color = 2 * np.log(np.hypot(Ex, Ey))
ax.streamplot(x, y, Ex, Ey, color=color, linewidth=1, cmap=plt.cm.inferno,
density=2, arrowstyle='->', arrowsize=1.5)
# Add filled circles for the charges themselves
charge_colors = {True: '#aa0000', False: '#0000aa'}
for q, pos in charges:
ax.add_artist(Circle(pos, 0.05, color=charge_colors[q > 0]))
ax.set_xlabel('$x$')
ax.set_ylabel('$y$')
ax.set_xlim(-2, 2)
ax.set_ylim(-2, 2)
ax.set_aspect('equal')
plt.show()
def U(q, r0, x, y):
"""Return the electric field vector E=(Ex,Ey) due to charge q at r0."""
den = np.hypot(x - r0[0], y - r0[1])
return q / den
nq = 2
charges = []
for i in range(nq):
if i % 2 == 0:
q = 1
else:
q = -1
charges.append((q, (np.cos(2 * np.pi * i / nq), np.sin(2 * np.pi * i / nq))))
U_array = np.zeros((nx, ny)) #set empty to '0'(or something)
for charge in charges:
delta_U = U(*charge, x=X, y=Y)
U_array += delta_U
Ey, Ex = np.gradient(-U_array)
fig = plt.figure()
ax = fig.add_subplot(111)
# Plot the streamlines with an appropriate colormap and arrow style
color = 2 * np.log(np.hypot(Ex, Ey))
ax.streamplot(x, y, Ex, Ey, color=color, linewidth=1, cmap=plt.cm.inferno,
density=2, arrowstyle='->', arrowsize=1.5)
# Add filled circles for the charges themselves
charge_colors = {True: '#aa0000', False: '#0000aa'}
for q, pos in charges:
ax.add_artist(Circle(pos, 0.05, color=charge_colors[q > 0]))
ax.set_xlabel('$x$')
ax.set_ylabel('$y$')
ax.set_xlim(-2, 2)
ax.set_ylim(-2, 2)
ax.set_aspect('equal')
ax.set_title('$ \\vec{E} =- \\nabla V $')
plt.show()
如下方程式
import numpy as np
import matplotlib.pyplot as plt
from matplotlib.patches import Circle
# Grid of x, y points
nx, ny = 64, 64
x = np.linspace(-3, 3, nx)
y = np.linspace(-3, 3, ny)
X, Y = np.meshgrid(x, y)
def U(q, r0, x, y):
"""Return the electric field vector E=(Ex,Ey) due to charge q at r0."""
den = np.hypot(x - r0[0], y - r0[1])
return q / den
nq = 2
charges = []
for i in range(nq):
if i % 2 == 0:
q = 1
else:
q = -1
charges.append((q, (np.sin(2 * np.pi * i / nq), np.cos(2 * np.pi * i / nq))))
''' sin and cos decide where the charges were placed '''
U_array = np.zeros((nx, ny)) #set empty to '0'(or something)
for charge in charges:
delta_U = U(*charge, x=X, y=Y)
U_array += delta_U
Ey, Ex = np.gradient(-U_array)
fig = plt.figure()
ax = fig.add_subplot(111)
# Plot the streamlines with an appropriate colormap and arrow style
color = 2 * np.log(np.hypot(Ex, Ey))
ax.streamplot(x, y, Ex, Ey, color=color, linewidth=1, cmap=plt.cm.inferno,
density=2, arrowstyle='->', arrowsize=1.5)
# Add filled circles for the charges themselves
charge_colors = {True: '#aa0000', False: '#0000aa'}
for q, pos in charges:
ax.add_artist(Circle(pos, 0.05, color=charge_colors[q > 0]))
ax.set_xlabel('$x$')
ax.set_ylabel('$y$')
ax.set_xlim(-2, 2)
ax.set_ylim(-2, 2)
ax.set_aspect('auto')
ax.set_title('$ \\vec{E} =- \\nabla V $')
plt.show()