hecd2 said:
Well isn't that lame? It's not even vaguely plausible that he could integrate on the sphere as you describe, and get the same result as a point charge orbiting the nucleus (which is what the expressions in question describe, as plain as the nose on your face).
He could only do that if the putative extended electron had some unphysical characteristics. It would have to have uniform mass and charge density (OK, that part's easy). The magnitudes of the mass velocity and current vectors would have to be everywhere the same, and everywhere have zero divergence (in order to maintain the uniform density condition). These are not criteria which can be met on a 2-sphere (see for example the hairy ball theorem - every smooth vector field on a sphere has at least one singular point - where the field goes to zero). He is quite unjustified in using these expressions for a point particle, and I'm justified in reiterating the fundamental internal inconsistency of his "theory".
I just disagree.
Good. We'll see now how your response to this post is characteristic of your entire apologia for Mills, whether it's about the science or the engineering or the history - always attempting to justify the unjustifiable. You never say "Good point, that's a strike against Mills, I need to consider that carefully". You always put forward some attempted justification for Mills, no matter how absurd. This is a stark illustration of your bias and inability to consider his claims objectively.
So let's look at your atttempted justification in more detail.
Since the inverse square law of charge attraction is analogous to gravity, consider the following comparison. We have a ball of metal of a certain mass that is suspended above the earth at a certain distance (between the earth's centre of gravity and the ball's centre of gravity). Knowing the distance and the mass of the ball we can calculate the force of gravity acting on the metal ball. Now say that ball broken into a number of different sized pieces, and each piece is suspended above the earth at the same distance as the original ball. What is the total force of gravity acting on the pieces? It will be the same as the the force acting on the singular ball. Similarly if the metal ball was flattened into tinfoil thickness and spread across a large area. The force of gravity acting on it will be the same. So, I don't see a problem in what Mills is doing there.
You don't? Well let me help you with that. Your analogy consists of an object in the uniform field of the Earth's gravity and it is true, that under that condition, the sum of the gravitational force on an object or an assembly of objects is the same whether it consists of a ball, or a ball cut into pieces or a thin membrane, provided the total mass of the object(s) is the same.
But that condition does not come close to applying in this case, because we are not discussing an object in a uniform gravitational or electrical field. According to Mills, the electron consists of mass and charge distributed on a sphere. An atom, according to Mills, consists of the nucleus with a certain number of positively charged protons (plus in most cases some neutrons which he ignores, at least in his ionisation potential and orbital radius predictions) surrounded by a number of electrons which are spheres of charge and mass concentric with the nucleus.
Throughout Chapter 10 of his book, Mills treats the electrons as point particles. Is he justified in doing this? Let's first consider the electrons within the orbit of the electron whose orbital radius he is trying to calculate. According to Gauss's law, he is justified in treating the electons within the orbital radius of the electron in question as a point particle coincident with the nucleus
if, and only if, the charge is uniformly distributed over the sphere. You need to remember this condition.
Now, turning to the electron whose orbital radius is in question, in order to predict the radius, he sets the centrifugal force equal to the Coulomb force plus magnetic forces using expressions which are correct only for a point particle. Is he justified in doing this? Yes,
if, and only if, the charge amd the mass are uniformly distributed on the sphere
and the mass tangential speed is also uniform (basically he is claiming that every infinitesimal element of the spherical electron can be treated the same way - he is not actually making a
force balance argument but a
pressure balance argument). You need to remember this condition.
To summarise: electrons within the orbital radius of the electron in question must have uniform charge over the sphere, and the electron in question must have uniform charge, mass, and mass flow tangential speed distributions.
Although I see why you believe the change density should be uniform, you have to consider that this is charge on the move, so things are different. If I recall the product of the charge density function and the velocity function, which yields current density, has to be uniform.
Charge density has to be uniform in order to treat the electron as a point particle with regard to Coulomb's law according to Gauss's law applied to a spherical object. Mass and mass flow tangential speed have to be uniform on the sphere in order to invoke the pressure balance condition using point particles.
If you're invoking the hairy ball theorem, I'm doubtful that it applies here, because each point has multiple velocity components, that is, current overlap in different directions.
Really? The hairy ball theorem doesn't apply here? It is a mathematical theorem which applies to
all continuous vector fields on a topological sphere of even dimension. You can feel more certain that the the theorem is correct and applies to all vector fields than you can about the sun rising tomorrow. So it applies to the 2-sphere in this case, and all vector fields on the sphere. And it says that neither of the vector fields we are thinking about can be uniform (the current and the mass flow tangential speed), although they have to be for Mills to treat his model of the electron as a point. Multiple velocity components at a point can always be resolved to a single velocity. That's inherent in vector algebra - to make this argument physical, at any given point, at any given time, the wind blows in a single direction with a single speed. Current and mass flow must resolvable to a single vector at any point. Oh yes, the hairy ball theorem applies here, and it disallows the conditions which Mills needs to treat the electron as a point particle.
And we haven't even considered the additional contraint that the divergence of the mass flow and current must be zero everywhere.
The whole things requires some serious thought and rigour which I can't provide.
Indeed, but that hasn't stopped you attempting to justify Mills arguments, as you have done here, and in every other case of arguments against your position, without regard or even acknowledgement of the strength of the arguments against you. Whilst advancing truly absurd propositions as we can see above.
If you are not satisfied by Mill's reasoning and rigour (which I assume you have seen in chapter 1), I won't be able to change your mind. But at least you tried and have come to a considered opinion.
I
tried?
!
.
