Sympy求解电场能量积分无输出问题咨询

电场与能量公式 #

教材给出的电场表达式为：
$$
E(x, y, z) = \frac{q}{4\pi\epsilon_0} \left[
\frac{\hat{x}x + \hat{y}y + \hat{z}(z - z_0)}{(x^2 + y^2 + (z - z_0)²⁾{3/2}} +
\frac{\hat{x}x + \hat{y}y + \hat{z}(z + z_0)}{(x^2 + y^2 + (z + z_0)²⁾{3/2}}
\right]
$$

采用教材中的电场能量公式计算：
$$
U=\frac{\epsilon_0}{2}\int E^2d3x
$$

尝试的SymPy求解代码 #

为计算上述积分，编写了如下Python代码：

from sympy import *

# Define the symbols
x, y, z = symbols('x y z')
q = symbols('q', real=True, positive=True)
epsilon_0 = symbols('epsilon_0', real=True, positive=True)
z0 = symbols('z0', real=True, positive=True)

# Electric field components due to the image charges
E1_x = q / (4 * pi * epsilon_0) * x / ((x**2 + y**2 + (z - z0)**2)**(3/2))
E1_y = q / (4 * pi * epsilon_0) * y / ((x**2 + y**2 + (z - z0)**2)**(3/2))
E1_z = q / (4 * pi * epsilon_0) * (z - z0) / ((x**2 + y**2 + (z - z0)**2)**(3/2))

E2_x = q / (4 * pi * epsilon_0) * x / ((x**2 + y**2 + (z + z0)**2)**(3/2))
E2_y = q / (4 * pi * epsilon_0) * y / ((x**2 + y**2 + (z + z0)**2)**(3/2))
E2_z = -q / (4 * pi * epsilon_0) * (z + z0) / ((x**2 + y**2 + (z + z0)**2)**(3/2))

# Total electric field components
E_x = E1_x + E2_x
E_y = E1_y + E2_y
E_z = E1_z + E2_z

# Squaring the electric field components and summing them
E_squared = E_x**2 + E_y**2 + E_z**2

# Integrate E_squared over the half-space where z > 0
U = (epsilon_0 / 2) * integrate(integrate(integrate(E_squared, (x, -oo, oo)), (y, -oo, oo)), (z, 0, oo))

U.simplify()

遇到的问题 #

运行上述代码后，已持续1-2天无输出结果，CPU始终处于满负荷状态（设备配备8P+2E核心），无法确定该积分问题是否无法通过SymPy求解，或是计算过程过于复杂导致耗时过长。

内容的提问来源于stack exchange，提问作者Joy Jin