Disc and Flux in a lane

Consider a point on the charged surface:

$\vec{r}_1 = \left(0,r_1 \cos{\alpha},r_1 \sin{\alpha}\right)$

Consider an elementary charge on that surface around this point which is:

$dQ = \frac{r_1 \ dr_1 \ d\alpha}{\pi}$

Consider a point on the second surface:

$\vec{r}_2 = \left(1,r_2 \cos{\beta},r_2 \sin{\beta}\right)$

A surface area element on this surface is:

$d\vec{S} = r_2 \ dr_2 \ d\beta \hat{i}$

Electric field due to the elementary charge is:

$d\vec{E} = \frac{dQ}{4 \pi}\left(\frac{\vec{r}_2 -\vec{r}_1}{\lvert \vec{r}_2 -\vec{r}_1 \rvert^3}\right)$

Elementary flux is:

$d\phi = d\vec{E} \cdot d\vec{S}$

This simplifies to:

$d\phi = \frac{1}{4 \pi^2}\left(\frac{r_1 \ r_2 \ dr_1 \ d\alpha \ dr_2 \ d\beta}{\left(1 + \left(r_2 \cos{\beta}-r_1 \cos{\alpha}\right)^2+\left(r_2 \sin{\beta}-r_1 \sin{\alpha}\right)^2\right)^{1.5}}\right)$

$\phi = \frac{1}{4 \pi^2}\int_{0}^{1} \int_{0}^{1} \int_{0}^{2 \pi} \int_{0}^{2 \pi} \frac{r_1 \ r_2 \ dr_1 \ dr_2 \ d\alpha \ d\beta}{\left(1 + \left(r_2 \cos{\beta}-r_1 \cos{\alpha}\right)^2+\left(r_2 \sin{\beta}-r_1 \sin{\alpha}\right)^2\right)^{1.5}}$

Code attached below. Here, $w = r_1$ , $x = r_2$ , $y = \alpha$ , $z = \beta$ .

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clear all
clc

% Numerical resoloution
dw = 1/200;
dx = 1/200;
dy = (2*pi)/200;
dz = (2*pi)/200;
I = 0;

for w = 0:dw:1
    for x = 0:dx:1
        for y = 0:dy:2*pi
            for z = 0:dz:2*pi

                % Numerator:
                N = (w*x*dw*dx*dy*dz);

                % Denominator:
                D = (1 + (w*cos(y) - x*cos(z))^2 + (w*sin(y) - x*sin(z))^2)^1.5;

                % Integrand:
                dI = N/D;

                % Integration:
                I  = I + dI;

            end
        end
    end
end


Answer = I/(4*pi^2)

Nice problem. This one ends up being a quadruple integral. Given that these are disks, I recommend describing the surfaces as $y^2 + z^2 \leq 1$

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import math

# Constants

N = 200

e0 = 1.0
Q = 1.0
R = 1.0

k = 1.0/(4.0*math.pi*e0)

A = math.pi*(R**2.0)
sigma = Q/A

##################################

dr1 = R/N
dtheta1 = 2.0*math.pi/N

dr2 = R/N
dtheta2 = 2.0*math.pi/N

##################################

# Object 1 is the charge disk
# Object 2 is the test disk

phi = 0.0

nx = 1.0
ny = 0.0
nz = 0.0

r1 = 0.0

while r1 <= R:

    theta1 = 0.0

    while theta1 <= 2.0*math.pi:

        x1 = 0.0
        y1 = r1*math.cos(theta1)
        z1 = r1*math.sin(theta1)

        dA1 = r1*dr1*dtheta1
        dq1 = dA1 * sigma

        r2 = 0.0

        while r2 <= R:

            theta2 = 0.0

            while theta2 <= 2.0*math.pi:

                x2 = 1.0
                y2 = r2*math.cos(theta2)
                z2 = r2*math.sin(theta2)

                dA2 = r2*dr2*dtheta2

                Dx = x2 - x1
                Dy = y2 - y1
                Dz = z2 - z1

                D = math.sqrt(Dx**2.0 + Dy**2.0 + Dz**2.0)

                ux = Dx/D
                uy = Dy/D
                uz = Dz/D

                E = k*dq1/(D**2.0)

                Ex = E * ux
                Ey = E * uy
                Ez = E * uz

                dot = Ex*nx + Ey*ny + Ez*nz

                dphi = dot * dA2

                phi = phi + dphi


                theta2 = theta2 + dtheta2

            r2 = r2 + dr2

        theta1 = theta1 + dtheta1

    r1 = r1 + dr1

##################################

print N
print phi

#>>> 
#100
#0.116629970353
#>>> ================================ RESTART ================================
#>>> 
#200
#0.116370400456
#>>>