Gravity vs Electrostatic

The gravity force is $20$ orders of magnitude weaker than the electric force, so it can be completely neglected for the purposes of this exercise. Simulation code is attached:

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import math

N = 4000

q = 1.0
k = 9.0*(10.0**9.0)
mult = 10.0**(-10.0)

dtheta1 = 2.0*math.pi/N
dtheta2 = dtheta1

sigma = q/(2.0*math.pi)  # angular charge density

dq1 = sigma*dtheta1
dq2 = sigma*dtheta2

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

Fx = 0.0
Fy = 0.0
Fz = 0.0

theta1 = 0.0

while theta1 <= 2.0*math.pi:

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

    theta2 = 0.0

    while theta2 <= 2.0*math.pi:

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

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

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

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

        dF = k*dq1*dq2/(D**2.0)

        dFx = dF * ux
        dFy = dF * uy
        dFz = dF * uz

        Fx = Fx + dFx
        Fy = Fy + dFy
        Fz = Fz + dFz

        theta2 = theta2 + dtheta2

    theta1 = theta1 + dtheta1

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

F = math.sqrt(Fx**2.0 + Fy**2.0 + Fz**2.0)

print N
print ""
print (mult*Fx)
print (mult*Fy)
print (mult*Fz)
print ""
print (mult*F)

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

#>>> 
#1000

#-1.78008384105e-16
#1.10614355009e-16
#-0.302574634525

#0.302574634525
#>>> ================================ RESTART ================================
#>>> 
#2000

#1.1750479177e-16
#-7.52453408026e-17
#-0.30227198973

#0.30227198973
#>>> ================================ RESTART ================================
#>>> 
#4000

#-3.04951359527e-16
#4.20477183316e-16
#-0.302120836082

#0.302120836082
#>>> 

Nice exercise comparing the strengths of two fundamental forces of nature. I have not followed the coordinate convention exactly as in the problem statement.

Let the centre of one ring be $(0.5,0,0)$ and another ring be $(-0.5,0,0)$ . A point on either ring can be parametrised as:

$\vec{r}_1 = (0.5,\cos{A}, \sin{A})$ $\vec{r}_2 = (-0.5,\cos{B}, \sin{B})$

The electrostatic force exerted by a charge element on ring 1 on that of ring 2 is:

$d\vec{F}_e = \frac{K}{4\pi^2} \left(\frac{\vec{r}_2-\vec{r}_1}{\lvert \vec{r}_2-\vec{r}_1 \rvert^3}\right)dA \ dB$

Considering only the X- component of force as the other components must cancel out due to symmetry:

$dF_{e,x} = -\frac{K}{4\pi^2} \left(\frac{dA \ dB}{\left(1 + (\cos{B}-\cos{A})^2+ (\sin{B}-\sin{A})^2\right)^{3/2}}\right)$

$F_{e,x} = -\frac{K}{4\pi^2} \int_{0}^{2 \pi} \int_{0}^{2 \pi}\frac{dA \ dB}{\left(1 + (\cos{B}-\cos{A})^2+ (\sin{B}-\sin{A})^2\right)^{3/2}}$

The same principle can be applied to gravity force. However, in this case, the gravity force is scaled by the universal gravitational constant which means that its magnitude will be several orders of magnitude weaker than electrostatic force. It is therefore neglected altogether.

The answer comes out to be:

$F_{e,x} = -13.2459\frac{K}{4\pi^2}$ $\implies 10^{-10} \lvert F_{e,x} \rvert = 13.2459 \left(\frac{10^{-10} K}{4\pi^2} \right) \approx 0.302$