Miles Mathis: Bohr's Three Mistakes
| Please note that this paper is a simplification by me of a paper or papers written and copyrighted by Miles Mathis on his site. I have replaced "I" and "my" with "MM" to show that he is talking. All links within the papers, not yet simplified, are linked directly to the Miles Mathis site and will appear in another tab. (It will be clear which of these are Miles Mathis originals because they will be still contain "I" and "my".) The original papers on his site are the ultimate and correct source. All contributions to his papers and ordering of his books should be made on his site. (This paper incorporates Miles Mathis' bohr paper and magneton paper). |
Newton Planck Bohr
Abstract
Miles Mathis will show several problems with the derivation of the Bohr Magneton, but first it is necessary to reveal the first major mathematical error of Quantum Mechanics with the Bohr Model. Having have gone back to the very beginning to rerun all the equations, MM will show that the Bohr equation is compromised in several ways.
Before the Heisenberg Uncertainty Principle, before the problem of superposition, before the problem of the two-slit experiment and all the other theoretical problems, there existed the foundational math of Quantum Mechanics. This was the basis and expression of the theory, and still is. The math MM will analyze here is still taught to this day as the bedrock of QED. All physics students learn this math when they are first taught the Bohr Model.
The entire Bohr model has to be reworked, changing the mechanical and mathematical foundations of quantum mechanics and quantum electrodynamics. After this reworking, it can be shown that the Bohr radius and Coulomb’s constant are mathematically connected. Coulomb’s constant turns out to be a simple scaling transform, that takes us directly to the Bohr radius.
Finally, MM show that the current .1% gap between the Bohr magneton and the experimental value for the magnetic moment of the electron is caused by the unified field. That is to say, this paper provides proof of older papers where MM PREDICTS a .1% variance in the field at the surface of the Earth. My foundational E/M field is .009545, which is almost exactly .1% of 9.8.
Of course, the Bohr Model has long been superceded, but the math below has never been corrected. It still infects QED at the foundational level, since this very math is used to obtain all the existing maths of Quantum Theory. This math underlies and infects Schrodinger's equations and all subsequent maths and theories.
The First Mistake
In determining the allowed orbits, Bohr first equated the centripetal electrostatic force on the electron to the centripetal acceleration, by this equation:
F = ma = mv2/r (where m is mass and a is acceleration) Substituting a in Newton's second law with
Newton's Corollary1: v2/r where v is velocity and r is the radius of the orbit)
mv2/r = ke2/r2 (where k is Coulomb's constant and e is the charge
and r is the distance between the charges)
Simplifying:
(1) mv2 = ke2/r
Bohr next uses the following equations to express the angular momentum of the electron:
L = rmv (Angular momentum: where r is the radius m the mass and v the tangential velocity)
L = nh/2π (where n is the orbit number and h is Planck's constant)
rmv = nh/2π (equating these two equations)
(2) v = nh/2πmr (Expressing as v)
ke2/r = m[nh/2πmr]2 (substituting v from (2) into (1))
ke2/r = m [n2h2/4πm2r2] (Squaring through [])
r = h2n2/4π2mke2 (Simplify and expressing as r)
r1 = h2/4π2mke2
This is the Bohr radius, which we are told is in agreement with the observed size of hydrogen atoms. And we can go back and find the equation for v:
v = [h/2πm][4π2mke2 /h2] = 2πke2/h = 2.18 x 106 m/s
Next, we find the energy of the orbit:
E = K + U = mv2/2 + U
mv2 = kZe2/r
(where Z is the atomic number)
E = kZe2/2r – kZe2/r
E = -kZe2 /2r
E1 = -13.6 eV
The Problem
The problem is that for some reason Bohr has used the rotational momentum but not the rotational kinetic energy. He assumes that the momentum of the electron will be expressed by an equation that includes the radius (L = mrv), so that we have an angular momentum. But when he finds the kinetic energy of the electron, he uses a straight translational equation—one that does not include the radius. You will say that it includes the radius after we make the substitution, but that does not count. The first equation is a translational equation that contains the variable v, not the variable ω. The substitution is made on the variable v, which is not thought to be an angular velocity in either equation. In the equation mv2 = kZe2/r, v is supposed to be the instantaneous tangential velocity. In the equation E = mv2/2 + U, v is a straight-line velocity. We can make the substitution precisely because they are both assumed to be translational variables.
Current theory always assures us that rotational motion must be expressed by rotational variables. In the chapters in physics books on rotational kinetic energy, we are given this equation for rotational kinetic energy:
K = mv2/2
(kinetic energy equation where v is velocity and m is mass)
v = rω (where v is orbital velocity)
K = m(rω)2/2 = Iω2/2
Why did not Bohr use this equation? Because he could not figure out how to make it yield the right experimental numbers.
You will say, "You can use either equation, since, as you just showed, they are equivalent: m(rω)2/2 = mv2/2."
MM answers, "Are they? Are you telling me that in the equation K = mv2/2, v = 2πr/t?" You will no doubt throw up your hands in frustration, and say, "Yes, just do the substitution, you fool!" But you willfully miss my point. Yes they are transferable, which was just shown, but they are not the same. You cannot measure a straight-line velocity with pi and a radius, since you are not given pi and a radius. And you cannot measure a rotational velocity with a distance, since you require pi and a radius.
A compensation of errors has saved Bohr in all the equations above. He has achieved the right experimental numbers only by finessing the math to fix the conceptual mistakes. The fundamental conceptual mistake is in assuming that the v variable in v = 2πr/t is an instantaneous tangential velocity. It is not. It is an orbital velocity. This velocity describes an arc of the circle; it curves; it cannot be a tangential velocity. The tangential velocity vector is a straight-line vector with its tail at the tangent. It does not follow the curve of the arc over any interval, even an infinitesimal interval. Newton never claimed that the v variable in his equation was the tangential velocity. In my paper on the equation A Correction to Newton's Equation, MM shows that this is so, by quoting Newton directly from The Principia, and by rerunning his versine derivation. Even at the ultimate interval, or at the limit, the tangential velocity and the orbital velocity are not equal.
The orbital velocity and the angular velocity are actually the same thing. The orbital velocity is just the angular velocity at a given radius. They both curve. The only difference is that the angular velocity is measured in radians and the orbital velocity is measured in meters. That is why the difference between them is just r. The variable v, as used in most places above, is no more a tangential velocity than the variable ω is. MM suggests a new letter to denote orbital velocity, but for this paper w is good enough.
v = x/t (velocity equals distance divided by time)
ω = 2π/t (angular velocity where ω is measured in radians per second and T is the period in seconds)
w = 2πr/t
a = w2/r
L = rmw
The last two equations show how Bohr was saved from his first mistake. The variable is an orbital velocity in both places, which saves his substitution. Neither velocity variable is a translational velocity nor a tangential velocity, although current textbooks still tell us they both are.
His energy equations are saved in a different way. You can see that he is trying to do a straight substitution of w for v, which cannot work. But he should be seeking a rotational kinetic energy for his electron, not a translational kinetic energy. That is to say, he should be writing his energy equations in terms of w, not v. So the equations should run like this:
mw2 = kZe2/r
E = mw2/2 + U
In which case the equations work as before.
Perhaps most physicists will see my corrections as caviling. They will not care about these subtleties, since the equations yield the right numbers, whether they are in the old form or my new form. They will dismiss my points as semantics or metaphysics, or some other easy epithet of contemporary science. But my correction here is strictly mathematical. Part of applied mathematics is variable assignment. Sloppy variable assignment must be fudged over later with sloppy equation assignment, so that you end up with proofs like those above. The misassigned variables were misassigned in the same way in Bohr’s derivation of the radius, so that he did not have to do anything else to make the equations work. But with the kinetic energy, he had to make a further misassignment to make them work out. He had to misassign an entire equation, giving his electron translational kinetic energy instead of rotational kinetic energy.
No one seems to have been embarrassed by this since then, but it is more than a cosmetic failure. As MM will show in subsequent papers, it is just this sort of error that has brought us to the theoretical impasses we now face. Bohr’s compensations were minor. In the equations above he needed only very small mathematical finesses. But very soon these finesses snowballed on QED. Before long the math was requiring huge finesses like renormalization. Most contemporary physicists seem not to be embarrassed by renormalization. But they should be. MM will show that if you are rigorous in your math at all points, you don’t need any renormalizing. The trick is to keep the equations "normal" from the start.
In my paper on Newton linked above, MM predicted a "kinetic energy meltdown" due to the form of:
F = ma = mv2/r. MM said that an improper substitution was being begged by not properly differentiating between the orbital velocity and the tangential velocity. The term mv2/r is a strictly rotational term, but there is nothing but subtle conceptual theory to keep someone from applying it to a translational kinetic energy situation. Bohr has only just avoided this catastrophe, since his electron really is in circular motion. But what if an electron in orbit ejected a photon, and that photon was assumed to be ejected from the tangent? The photon would now be in linear motion, not circular motion. What is its kinetic energy? You can see the problem.
Furthermore, there is actually not even subtle conceptual theory to keep anyone from making this improper substitution, since current theory does not theoretically disallow it. Only my theory disallows it. The subtle conceptual theory is so far mine alone.
Current theory believes that the velocity variable in a = v2/r is an instantaneous tangential velocity. That would make it the same as a translational velocity. That is why current theory uses the same variables for both. And that is why the proofs of Bohr followed the form they did and why they have never been corrected. That is why all modern physics textbooks conflate tangential velocity and orbital velocity.
This improper substitution will be found to be at the heart of some mathematical impasse in contemporary physics. In fact from this, MM has now reworked Bohr's equation completely, finding a new value for the Bohr radius, and the problem with the Bohr Magneton. Also it can be shown that Coulomb's constant is shown to be an expression of the Bohr radius. See MM's paper Coulomb's equation is a Unified Field equation in disguise.
Second and Third Mistake of Bohr
and the problem with the Bohr Magneton
Why is the Bohr Magneton not equal to the measured magnetic moment of the electron? In experiment, we find that the values are off by .1%. QED has no simple answer for this. MM has the answer.
QED proposes to explain the error by once again pouring Dirac’s virtual sea on the problem and once again waving the magic wand. The electron is said to be interacting with virtual photons, giving it a precession and thereby a g-factor. All this is just one more fudge, however. Anytime you see the word “virtual” in modern physics, it means you have left the path of reason. MM will show the simple mathematical reason for the error.
The Bohr magneton was first proposed by Procopiu in 1913. It is not a particle, but rather an expression of the magnetic field created by the individual electron. We have a simple equation for it:
μB = eh/2π2me
Unfortunately, as MM said, this gives us a number that fails by about .1%. To see why, we must study this equation more closely. We can do this by looking at angular momentum, and the easiest way to do that is by returning to Bohr’s simple math, which MM first critiqued in another paper.
L = nh/2π = rmv
We let n = 1, since we are studying the first electron in the hydrogen atom. So,
μB = eL/2me = erv/2
We find that this equation yields the wrong number. Why? Because the math is wrong as shown earlier in Bohr's First Mistake. With the equation L = rmv being wrong, current theory tries to cover this by never including momentum or velocity variables, but they were there in the beginning. Look at what tangential velocity we get, for starters:
μB = erv/2
v = 2μB/er = 2(9.274 × 10-24 J•T-1/(1.602 x 10-19C)(5.29 x 10-11m)
v = 2.19 x 106m/s
Mirroring current assumptions, MM used the Bohr radius for r. That is well under c. Why? Why doesn’t the electron maximize its orbital speed? MM will be told that it is because v is not the orbital speed, ω is. Well, v = rω. So,
ω = 4.14 x 1016/s
Is that over c? Nobody knows, because nobody understands angular speed.
You can’t just multiply or divide by a radius to make a linear velocity into an angular velocity. That doesn’t make any sense, mathematically or mechanically. Look at the equations for momentum and angular momentum closely:
p = mv
L = rmv
If the radius is greater than one, the effective angular velocity will be more than the linear velocity. If the radius is less than one, the effective angular velocity will be less than the linear velocity. That is a flagrant example of illogical scaling.
The history of physics fudges over this problem by creating a moment of inertia, but the moment of inertia is a ghost. It is the attempt to hide the fact that v = rω is wrong. You would not have a moment of inertia without v = rω.
Where does that equation come from? It comes from 2πr/t. If v = 2πr/t, and ω = 2π/t, then v must equal rω. But, as MM has shown, v ≠ 2πr/t. In the historical derivations, v is defined as the tangential velocity. But 2πr/t is not the tangential velocity; it is the orbital velocity. The orbital velocity curves and the tangential velocity does not. The tangential velocity is a straight line vector with its tail on the curve, but it does not follow the curve.
If v = 2πr/t, then v is already an angular velocity. An orbital velocity and an angular velocity are the same thing. They both curve. Therefore, in going from 2πr/t to ω, you aren’t really going from a linear expression to an angular expression. You are going from one angular expression, expressed in meters, to another angular expression, expressed in radians.
None of the angular momentum equations in books make any sense, so MM developed his own equation to do this, going back to Newton to find the method. You can see MM's derivation in A Correction to Newton's Equation In MM's equation, v really is the tangential velocity, and therefore it is not equal to 2πr/t. It is equal to x/t.
ω = √[2r√v2 + r2) - 2r2]
This equation is logical, because using it we find that the angular velocity is always less than the tangential velocity. We don’t have any misdirection with moments of inertia, and we don’t have the illogic of having the variables change in different ways for different values of r. We have a logical progression, since as we get larger, the angular velocity approaches the tangential velocity. Obviously, this is because it loses it curvature as it increases, becoming more like the straight-line vector. By the same token, at small scales, the angular velocity gets very small compared to the tangential velocity, and this is because the curvature is so great.
Using MM's new equation for ω,
L = mω = h/2π
μB = eω/2
Now we just solve
ω = 2(9.283 x 10-24)/1.602 x 10-19
ωe = 1.16 x 10-4m/s
r = √[ω4/(4v2 - 4ω2)]
If we use c for v, we find,
re = 2.244 x 10-17m
Of course, MM have redefined the variable ω here. It is no longer measured in radians. Strictly, it stands for the orbital velocity measured in meters, not the angular velocity measured in radians. My new equations simply separate it from the tangential velocity, since, as you have seen, we need both. We do not need to be able to measure circular motion in radians, but we do need to measure it both as a tangential velocity and as an orbital velocity. Therefore, MM has jettisoned the old angular velocity in radians as a useless concept.
[To see how this affects the Stern-Gerlach experiment and the 1/2 spin of fermions, see the paper on MM's site The Stern-Gerlach Experiment.]
And now we see that the radius hidden under Bohr’s bad math is the radius of the electron, not the radius of the orbit. And the spins belong to the electron as well. But we should have known that long before. All the angular momenta have to apply to the electron, not the orbit. If the orbit was the primary cause of the various fields of the electron, then the orbit itself would show a magnetic moment and an electrical field, and so on. And if it did that, the atom wouldn’t be neutral, it would be an ion. Besides, we know that free electrons also have electrical fields and magnetic fields. So it cannot be the orbit that has all the angular momentum. The angular momentum and the magnetic moment belong to the electron, so the radius must also.
And the velocity must also belong to the electron. That is, it belongs to the spin, not to the orbit. The velocity in this equation is not a velocity of the electron in orbit, it is the velocity of the spin. It is the tangential velocity on the surface of the spin, or the linear velocity a point on the surface of the spin border would be going if it weren’t going in a circle. The magnetic moment, like the charge, belongs to the electron, not to the orbit!
MM will be told that the orbit must have a momentum of its own, angular or otherwise. Yes, there must be orbital energy, but we need not calculate an angular momentum. The only thing in the orbit is the electron, so there is no mass inside the sphere. The only thing inside the orbit is the nucleus, and it is not moving relative to the orbit, so it creates no angular momentum. This being true, we only have to look at mass and momentum at the tangent. If we do that, we don’t have to be concerned with angular anything. The electron transmits energy from the tangent, by emission, and this emission is emitted during a very small interval of the orbit. The emission leaves the electron in a straight-line vector, so we don’t have angular momentum involved. We can use the tangential velocity directly, and compute the momentum linearly, with p = mc. But this momentum does not contribute to the magnetic field. This momentum contributes to the linear energy of the emission, not its spin energy. Magnetism is not caused by linear energy, it is caused by spin energy. So we don’t have to be concerned with the orbital energy.
Now, the electron does interact with the field outside the orbit, but this is not a virtual field. It is the emission field of other quanta. The vacuum is awash with emission, and this emission acts as a friction on the orbit. But this doesn’t cause a precession or a g-factor. It causes the electron to have an outer spin that is opposed to its orbit, like a set of spinning cogs. The electron has a linear momentum, or tangential momentum of
p = mc = 2.7 x 10-22m2/s
The outer spin has an opposite momentum of about 10-34. This acts as a slight drag on the linear momentum, but only in the 12th decimal point. So it could not cause a .1% change in the magnetic moment. In fact, it changes nothing in the magnetic field, since the magnetic field is under the electric field. Only the orbital momentum and the electric field could be affected.
But why is the experimental number for the Bohr magneton .1% wrong? Is it just that Bohr's numbers were different than current numbers? Is it a problem of the virtual field, explained by the g-factor? No. It is caused by the unified field. In another paper MM derives a solid number for the summed charge field of the Earth. This is not the electrical field of the Earth or the magnetic field. It is what MM calls the foundational E/M field, caused by the emission of photons by all matter in the field. It causes the electric and magnetic fields, but is not equivalent to either one. It's direction is straight out from the Earth, radially; and it is always repulsive. All bodies create this field, and it is always in vector opposition to gravity proper. This field, with gravity, makes up the unified field. The average field strength at the surface of the Earth for this field is .009545 m/s2. This number was arrived at by rather simple math, by comparing the fields of the Earth and Moon. (See The Secrets of the E/M field are revealed at the Moon's surface.)
The important thing here is that the number just quoted gives us almost precisely a .1% correction to the unified field, and therefore to the Bohr magneton. We just divide that number by 9.8 to find a .1% correction, you see. We get .0974%, which is close enough for me in this problem. The reason this solves the problem of the Bohr magneton is that the experiments have all been run on the surface of the Earth, in a field not known to exist until now. This charge field has been hidden in Newton's equation, as part of the gravitational field. Newton's equation gives us the total field, but not the constituent fields. Likewise for Einstein's field equations. The charge field is ignored at the macro-level. But since MM has proved that there is a charge field in vector opposition to gravity existing at all points on the Earth, this gives us a simple explanation of the error in the Bohr magneton. This charge field must obviously affect the magnetic field of the electron directly, by straight bombardment of charge photons. This gives us the simple mechanical cause of the .1% error, with all the necessary math. As a matter of fact, all the historical and current experiments on the electron that show this .1% error are now proof of MM's theory. MM predicted a .1% variance several years ago in the linked paper, before knowing of or studying the Bohr magneton. MM has now found the pre-existing proof of it here, and have explicitly shown the necessary connection of the two numbers.
Finally, let’s check that value for the electron radius. Actually, what MM found above is the radius of the outer spin. The electron in orbit has both an axial spin and an x-spin. Therefore the radius of the electron proper is:
re = 1.122 x 10-17m
But the x-spin radius, 2.244 x 10-17m, must be the effective border of the electron, since due to the end-over-end spin, the mass will inhabit this entire radius, during motion. In the paper The Atomic World is 100 Times Larger Than We Thought, MM found the radius of the proton to be about 10-13m, and the proton is known to have a mass of about 1836 times the electron. Using those numbers, we get
r = 5.45 x 10-17m
Which is very close. We can use MM's number to re-estimate the radius for the proton, assuming it has the same density as the electron.
rP = 4.11 x 10-14m
We can fine tune that as well. Since we are finding radius here, not mass, we can use MM's spin equations from The Unification of the Proton and Electron & Finding the Electron Radius & The Fallacy of the Electron Orbit. In a nutshell, we find we need the transform 1822, not 1836. The number 1836 is a sort of mass transform, which means it must be a unified field transform. Mass is always a unified field number. But radius is not a unified field number. For that reason, we can just use the Dalton, 1822, which transforms size but not mass. In that case we get the number 4.09 x 10-14m for the proton radius. That is very interesting because it is the square root of the proton mass.
Of course this means the Bohr radius is wrong as well. Bohr’s math is completely compromised by now, so everything has to be redone. The problem with angular velocity has infected all the math, and nothing will stand. Let’s correct the Bohr equation:
mv2/r = ke2/r2
First of all, a ≠ v2/r . If we want to use the angular velocity, we must use this equation
a = ω2/2r
Which gives us,
mω2/2 = ke2/r
Using Bohr’s method, we find
L = mω = h/2π
ω = h/2πm
hω/4π = ke2/r
ω = 4πke2/hrh/2πm = 4πke2/hr
r = 8π2mke2/h2
This gives us the inverse of Bohr’s radius. You can see that because of the correction to the equation L = rmv, we get the radius on the wrong side of the equation, skewing the math. This means that Bohr’s math depends on using that false equation. If you use the right math for angular momentum, the rest of Bohr’s math fails. It fails because L = h/2π applies to the electron, but Bohr is trying to apply r and mω to the orbit. So these substitutions we are making can’t work.
Think of the orbit like a big spinning particle, of radius r. That big particle has an angular momentum. The electron also has an angular momentum. Bohr has conflated the two. His equations are a mixing of both values.
So he has made two big errors. One, he has used the wrong equation for angular momentum, based on a mistaking of tangential and orbital velocity. Two, he has a false equality. If our first equation (mω2/2 = ke2/r) is correct, then the angular velocity ω must apply to the orbit, not to the electron. If it applies to the orbit, then mω ≠ h/2π. This is because h/2π applies to the electron in orbit, not to the orbit.
We will also see in Rewriting Schrodinger’s equations that they do not solve this problem. Schrodinger has Bohr’s principal quantum number and also a separate angular momentum quantum number, but he does not assign these physically. Because we don’t get the mechanics, the math is unclear. And Schrodinger’s equations retain the errors of Bohr in going from linear to angular velocity. That is, Schrodinger still uses a false angular momentum equation. L = rmv was not corrected by Schrodinger, and it has never been corrected since.
Can we still find a Bohr radius? Let us assume that the first equation is right, after correcting the momentum equation.
mω2/2 = ke2/r
But we have two unknowns, r and ω, and only one equation. We can’t solve without another equation, and Bohr’s momentum equations are false. Let us first try using c for v. We will assume the tangential velocity of the electron is maximized.
So we simply return to the equation ω = √[2r√v2 + r2) - 2r2], using c for v. Since the electron is bigger than the photon, it must have a limit just under c, but since that limit is in the fifth decimal point of c, we will ignore it here.
ω = √[2r√c2 + r2) - 2r2]
mω2/2 = ke2/r
r√c2 + r2) - r2 = ke2/mr
We can simplify that by noticing that the left side will be dominated by c, allowing us to omit values of r.
cr = ke2/mr
r = √ke2/cm] = 9.19 x 10-4m
That’s way too large, so we know something is still wrong. Let’s try using Bohr’s radius to find the angular velocity.
ω = √2ke2/mr] = 3.09 x 106m/sInteresting that that is almost what we got for the tangential velocity v using Bohr’s math above: remember that using the Bohr radius, we found v = 2.19 x 106m/s. But if we use the right velocity equations, we find
r2 = ω4/(4v2 - 4ω2)
v = ω2√[4r2 + ω2] = 3 x 1019m/s
So that can’t be right either. From these calculations, it would appear that the Bohr radius is either greater than or equal to about a millimeter, or we are using the wrong values for the electron, or the equation is still wrong.
It turns out that the equation is still wrong. The problem is simple: the constant k doesn’t apply at the quantum level. Coulomb’s equation is for use at the macro-level, and the constant is a scaling constant.
In G is the Key to the Secret of Gravity, MM showed with G in another paper, k takes us from one level of size to another, so that we can compare fields that have different mediating particles or accelerations. Coulomb was working with little pith balls, not electrons, and his balls were nine orders of magnitude larger than the orbital radius of the electron, as you will see.
MM showed that G is a scaling constant that takes us from the size of emitted photons to the size of the atom. Yes, the B-photon is G times smaller than the proton. We find it in Newton’s equation, of all places, because Newton's law is a Unified Field of Gravity and E/M and thus a unified field equation in disguise. It contains both the gravitational acceleration and the foundational E/M field (or charge field). Well, the same applies to Coulomb’s equation. It looks just like Newton’s equation because it is the same unified field equation in a different disguise. Newton’s equation is hiding the E/M field, and Coulomb’s equation is hiding the gravitational field. Because neither Newton nor Coulomb understood the fields under their equations, they only provided us a math that works. Their equations work because they compress the unified field into one field, and the transform between the two fields is the constant.
Since in Bohr’s equation the field is the actual field the electron is moving in, we don’t need a transform or a scaling constant. The electron is already moving at the proper scale. In the little illustration in the book, we see the electron circling the nucleus, and the electron and the orbital radius are in the same field. We have to do very little scaling (between the charge and the field it is in) in order to draw the picture, and this is not beside the point. The proton is actually repulsing the electron down that very radius.
And so we get this very simple equation:
r = √(e2/mc) = 9.69 x 10-9m
That is the corrected Bohr radius. The value of Coulomb’s constant is 9 x 109. That is a scaling transform that takes us directly from the Bohr radius to our own world. But Coulomb's balls were not one meter in radius. His pith balls were about 6mm, which is about 170 times smaller than 1 meter. That number 170 is not a coincidence either, since MM has just found that the Bohr Radius is 177 times larger than we thought. In fact, if we divide 1 meter by 177, we get 5.65 mm. Coulomb himself tells us that his balls were "2 to 3 lines in diameter", which is 4.5 mm to 6.8 mm.
What all this means is that Coulomb's constant is not a constant. It is taking us from one size to another, so it cannot be applied across a range of sizes. This was to be expected, since MM has shown that Coulomb's equation, like Newton's equation, is a unified field equation that includes both the E/M field and the gravitational field. Furthermore, MM has shown that the two fields are not the same relative size to each other, as we scale the equations up and down.
So the connection between Coulomb’s constant and the Bohr radius is not a coincidence. Although the current numbers are wrong, it is no coincidence that the current Bohr diameter is thought to be about 1/k meters. It is the mistake in the Coulomb equation that led directly to the mistake in the Bohr radius, and they are connected both mathematically and historically. It is also not a coincidence in MM's new math, since if you multiply the old Bohr radius and the real diameter of Coulomb’s ball by 177, you get MM's new Bohr radius and 1 meter. For more on this, see Coulomb's equation is a Unified Field equation in disguise.
The paper The Atomic World is 100 Times Larger Than We Thought shows that a misreading of the scattering equations meant we have the atomic size about 100 times too small, so MM's new equation also fits that prediction and correction very well. The Bohr radius is 177 times larger than we thought, and you can now see all the math and logic behind the correction.
Addendum [February, 2010]: MM finally noticed that the corrected Bohr radius equation looks a lot like the classical electron radius.
MM's corrected Bohr radius = √(e2/mc)
classical electron radius = e2/mc2
This is because the classical electron radius was derived from these same faulty angular momentum equations that MM has had to correct. Notice that the classical electron radius was never logical. The number 2.82 x 10-15m has always been too large, since if we scale radius to mass, we should be able to multiply the electron radius and get the proton radius. That would make the proton radius 2.82 x 10-15m x 1836 = 5.2 x 10-12m. That is only a factor of ten below the current Bohr radius, so it is way too large. The electron should never have been calculated to be 2.82 x 10-15m. The numbers we have had up to now never matched. The current Bohr radius is 100 times too small, and the current estimate for the electron is 100 times too large. The Compton radius of the electron is now written in terms of the fine structure constant, but it is still the same value as the classical electron radius. This means that both the classical radius and the Compton radius of the electron are way off, due to the faulty equations of Bohr and others. (See Planck's Constant and the Fine Structure Constant to see it dissolve.)
MM will show in a new paper on the Compton effect on MM's site that MM's smaller radius for the electron is much better, but it should have been seen long before that the electron could not be as large as 2.82 x 10-15m.
To read more about the problems of Bohr, you may now read MM's third paper on the subject, called More Problems with Bohr. on MM's site. This paper shows another half-dozen fatal errors in his equations, and leads us into a correction to the Rydberg Formula on MM's site.