Is this solvable using sympy? - no output after 1 day

Question

The original question:

My approach: The electric field is already given in the textbook as

\begin{equation}
  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]
\end{equation}

So I used this equation in the textbook:

\begin{equation}
  U=\frac{\epsilon_0}{2}\int E^2d^3\!x=
\end{equation}

To calculate this, I asked GPT-4 to write a python program to compute this using sympy. Below is the code:

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()

I ran the code and expected to get a result in some day. I've left it running for around 1-2 days and there is no output. It is constantly using 100% CPU (I have 8P+2E cores). I'm not sure if this is not solvable in sympy or is it just really complicated.

Answer 1

Your problem has axial symmetry, around the axis from the point charge to the ground plane (and perpendicular to the ground plane).

You need to convert it to cylindrical or 2d polar coordinates that explot the symmetry, rather than cartesian. The triple integral in cartesian coordinates becomes a double integral in cylindrical coordinates which would probably work in sympy.

Alternately, you can use the method of images https://en.wikipedia.org/wiki/Method_of_images and consider two point charges rather than a point charge and ground plane. That system still has cylindrical symmetry and the double integral is probably calculable by hand - look at a few textbooks to see if there is a worked example.