Electrostatic generator

[EOW]

I calculated the sum of energy with a charged particle in a linear translation (axis X) with a circular dipole (2 charges) in rotation. I think with electrostatic charges for example and the law is like 1/d². The single particle has a mass 'm' and has a velocity 'v0' at start. A disk turns only (no translation) has an inertia 'j' and 2 charged particles are fixed on it (one positive, one negative). I calculated the energy for one turn only, it's enough but I calculated for several rounds and the energy is not the same.

The energy must be the same for one turn, why ? because the single particle don't need an energy for move far away or move closer to the disk: the disk has one charge positive and one charge negative !

But with a disk in rotation and a charge with a velocity the energy is not the same. And it's logical because the single particle accelerates/decelerates with a difference of forces but the disk accelerates/decelerates with the sum of forces.



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I give the program in python for test my results. I used the lib gmpy2 for better precision, so install it before.

import gmpy2
import matplotlib.pyplot as plt
import matplotlib.cm as cm
from gmpy2 import mpfr


dt=mpfr("1e-4")      # step time converge from 1e-4 s

pi=mpfr("3.14159265358979323846264338327950288419716939937510")   # pi 50 digits

m   = mpfr("1")    # mass of single particle
j   = mpfr("3")      # inertia of disk
v0  = mpfr("1")      # initial velocity of single particle
w0  = mpfr("1.2")      # initial angular velocity of disk
d   = mpfr("100")   # distance between the single particle and the disk
r   = mpfr("0.05")   # radius of the disk
o0  = mpfr("0")      # angular at start
k   = mpfr("1")      # coeficient of the force

t   = mpfr("0")      # the time
f1  = mpfr("0")      # force from the single particle and the attractive particle on disk
f2  = mpfr("0")      # force from the single particle and the repulsive particle on disk
f1x = mpfr("0")      # force f1 on the axis X
f2x = mpfr("0")      # force f2 on the axis X

v   = mpfr("0")      # the linear velocity of the single particle
a   = mpfr("0")      # the acceleration of the single particle
d0  = mpfr("0")      # the distance at start
x   = mpfr("0")      # the position of the single particle

w   = mpfr("0")      # the angular velocity
s  = mpfr("0")      # the angular acceleration
o   = mpfr("0")      # the angle
f1r = mpfr("0")      # torque
f2r = mpfr("0")      # torque

l1  = mpfr("0")      # the lenght between in axis x from the single particle and the attractive particle on the disk
l2  = mpfr("0")      # the lenght between in axis x from the single particle and the repulsive particle on the disk
yc1 = mpfr("0")      # the root of length between the single particle and the attractive particle on the disk
yc2 = mpfr("0")      # the root of length between the single particle and the repulsive particle on the disk
h   = mpfr("0")      # the vertical distance of a particle on the disk

el  = []      # list of each linear velocity of the single particle
wl  = []      # list of each angular velocity of the disk
tl  = []      # list of each time


with gmpy2.local_context(gmpy2.context(), precision=200) as ctx:

   v=v0      # set v at V0
   w=w0      # set w at w0
   x=d0+d      # set position of the single particle at d0+d
   o=o0      # set the start angular position (not necessary)

   while o < 2*pi :
   
       h=r*( abs( gmpy2.cos(o)))      
      l1=x+r*(1-gmpy2.cos(o-pi/2.0))
      l2=x+r*(1+gmpy2.cos(o-pi/2.0))      
      yc1=pow( l1 ,2 ) + pow( h ,2 )
      yc2=pow( l2 ,2 ) + pow( h ,2 )
      f1=k/yc1         # calculate the force from single particle to attractive particle on the disk
      f2=k/yc2         # calculate the force from single particle to repulsive particle on the disk
      f1x=f1*l1/gmpy2.sqrt(yc1)
      f2x=f2*l2/gmpy2.sqrt(yc2)
      a=(f1x-f2x)/m         # calculate the linear acceleration of the single particle
      v=v+a*dt         # calculate the linear velocity of the single particle
      x=x-( 1/2.0*a*dt*dt+v*dt )

      f1r=f1*(x+r)*abs(gmpy2.cos(pi/2.0-gmpy2.acos(l1/gmpy2.sqrt(yc1)))) # the torque from single particle to attractive on disk
      f2r=f2*(x+r)*abs(gmpy2.cos(pi/2.0-gmpy2.acos(l2/gmpy2.sqrt(yc2)))) # the torque from single particle to repulsive on disk
      if o < pi/2.0:
         s=(f1r+f2r)/j         # calculate the angular acceleration
      elif o >= pi/2.0 and o < pi:
         s=(-f1r-f2r)/j         # calculate the angular acceleration
      elif o >= pi and o < 3.0*pi/2.0:
         s=(-f1r-f2r)/j         # calculate the angular acceleration
      elif o >= 3.0*pi/2.0 and o < 2*pi:
         s=(f1r+f2r)/j         # calculate the angular acceleration
      w=w+s*dt         # calculate the angular velocity of the disk
      o=o+w*dt+1/2.0*s*dt*dt      # calculate the angle   

      el.append(1/2.0*m*v*v-1/2.0*m*v0*v0)
      wl.append(1/2.0*j*w*w-1/2.0*j*w0*w0)         
      tl.append(t)


      t=t+dt            # add step time
      

   # print various datas
   print "v=",v
   print "x=",x
   print "w=",w
   print "o=",o
   print "t=",t
   print "f1x=",f1x
   print "f1y=",f2x
   
   print "E at start = ",1/2.0*m*v0*v0+1/2.0*j*w0*w0    # print the energy at start
   print "E at final = ",1/2.0*m*v*v+1/2.0*j*w*w       # print the energy at end

   
   # plot energies
   fig, ax1 = plt.subplots()
   ax2 = ax1.twinx()
   ax1.set_xlabel('Time (s)')
   ax1.set_ylabel('E lin (J)', color='b')
   ax1.plot(tl,el,'b')
   ax2.set_ylabel('E w (J)', color='r')
   ax2.plot(tl,wl,'r')
   plt.show()
   

[Low-Q]

Nice. Just not hoping you believe anything of this is over unity ;-)
Any charge, wether it is electrons or mechanical potential energy, everything is accounted for.

Vidar