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Bohr’s H Atom Revisited (Part 2)

In Bohr’s H Atom Revisited (Part 1) it was found that, given a proton and an electron orbiting around the origin of an inertial coordinate system, the tangential force exerted by the proton on the electron could be found to be equal and oppositely directed to the radiation reaction force, and that the atom’s electron will accordingly not  radiate. Not discussed was the proton and why the proton, also traveling in a circle, does not radiate. This article discusses the computed results of considering the case where the force, exerted by the electron on the proton, cancels the proton’s radiation reaction force, thereby eliminating the objection that the accelerating proton should radiate.

Essentially the program, written for the first article, was rewritten for the present case. (See Appendix 1). The same results were obtained. Hence it is found that the proton and electron are locked in a kind of “dance” where the tangential force, exerted on each particle by the other, cancels the radiation reaction force and the entire atom does not radiate.

It is worth mentioning that the net force, exerted on each particle by the other, also has a radial component required by Newton 2. All things considered, the model describes a “stable” state, where the atom neither flies apart nor radiates.

Now as discussed in the conclusion of Part 1, there is a companion constant (dubbed hp) to Planck’s constant, h: hp=h(me/mp). That is, Bohr’s rule, that in the electron’s case (me)(re)(ve)(2p)=h has a companion rule: (mp)(rp)(vp)(2p)=hp. The model provides insight into Newton 3. For the force felt by the proton at any given moment does not depend on where the electron is at that moment; rather it depends on the electron’s kinematical parameters at a previous, “retarded” time. The proton’s reaction in each case is to that retarded force; the proton reacts with an equal and oppositely directed force on its electromagnetic field, and this reaction is then later felt by the electron. There would appear to a certain kind of “elasticity” in the fields, where the “news” of a new force impulse on the proton is a record of the electron’s kinematics at an earlier time (and vice versa). The whole idea of such time delays is a jumping off point into the quantum electrodynamics theory of Sin-Itiro Tomonaga, Julian Schwinger and Richard P. Feynman , who shared the Nobel prize in 1965 for their seminal work.

Appendix A.

import math

c=2.99792458e8 #Speed of light

h=6.626068e-34 #PlanckConstant

eps0=8.85418782e-12 #Permittivity constant,

q=1.60217662e-19 #Elementary charge

me=9.10938356e-31 #Electron rest mass

mp=1.6726219e-27 #Proton rest mass

re=.5291772110903e-10 #Ground level electron orbit radius

ve=h/(2*math.pi*me*re) #Bohr rule for ground level electron speed

omega=ve/re #Ground level electron and proton orbital angular frequency

vp=me*ve/mp #Ground level protom speed (total atom momentum=0)

rp=vp/omega #Ground level proton orbit radius

ae=ve**2/re #Electron acceleration

maxN=1e8 #Maximun number of iterations in orbital computation

td=(re+rp)/c #Delay time

dt=-td/maxN #Time increment

loopN=0

theta=0.

alpha=0.

Lp=re

veRetarded=ve #Retarded electron speed (Electron travels at constant speed)

aeRetarded=ae #Electron retarded acceleration

FpRetarded=0. #Initial trial force of Electron on Proton

ForceOnProtonY=0.

#Now creep up until RadiationReactionForce+ForceOnProton < zero.

loopN=loopN+1

if loopN>maxN:

print('Possible infinite loop.')

input('OK?')

#If necessary, keep 'inching up’ to loop termination condition.

RetardedTime=loopN*dt

theta=-omega*RetardedTime

L=re+rp

RetardedVelocity=ve

RetardedVelocityX=-ve*math.sin(theta)

RetardedVelocityY=ve*math.cos(theta)

RetardedAcceleration=ve**2/re

RetardedAccelerationX=-RetardedAcceleration*math.cos(theta)

RetardedAccelerationY=-RetardedAcceleration*math.sin(theta)

alpha=math.atan(re*math.sin(theta)/L)

Lx=-L*math.cos(alpha)

Ly=-L*math.sin(alpha)

#Compute electric field at proton (see Griffith's text)

ux=c*Lx/L-RetardedVelocityX

uy=c*Ly/L-RetardedVelocityY

u=math.sqrt(ux**2+uy**2)

ux=u*math.cos(alpha)

uy=u*math.sin(alpha)

Ex=q/(4*math.pi*eps0)*L/(Lx*ux+Ly*uy)**3

Ex=Ex*(ux*(c**2-RetardedVelocity**2)-(Ly*(uy*RetardedAccelerationX-ux*RetardedAccelerationY)))

Ey=-q/(4*math.pi*eps0)*L/(Lx*ux+Ly*uy)**3

Ey=Ey*(uy*(c**2-RetardedVelocity**2)-(Lx*(ux*RetardedAccelerationY-uy*RetardedAccelerationX)))

ForceOnProtonX=q*Ex

ForceOnProtonY=q*Ey #Interactive tangential force on electron

gamma=1/math.sqrt(1-vp**2/c**2)

L=math.sqrt((rp+re)**2-(re*math.sin(theta))**2)

Lx=-L*math.cos(alpha)

Ly=L*math.sin(alpha)

print("ForceOnProtonY=",ForceOnProtonY)