Energy, Inertia, Electric Field and Field Density Relations: Algebra help please

1. Sep 28, 2010

Green Destiny

1. The problem statement, all variables and given/known data

I am not an expertize at algebra, and recently I was considering derivisions of equations based on postulates already known to physics. I just wanted a little guidence if any of the equations are inconsistent, which I am sure probably quite a few are, knowing my own personal track record :)

So if any mistakes are pointed out, or that I have missed something, I'd appreciate it if you recognize it aloud for me, so to say.

2. Relevant equations

The relevent equations are progressive derivations, so nothing is set in stone at first hand.

3. The attempt at a solution

Preliminaries

First deriving the property of our electric field $$\mathbb{E}$$ takes the form of:

$$\mu_0 D= c^2 \mathbb{E}$$

Rearranging gives:

$$\frac{\mu_0 D}{c^2}= \mathbb{E}$$

Squaring both sides, and then multiplying by the permitivvity gives:

$$\epsilon_0 (\frac{\mu_0 D}{c^2})^2= \epsilon \mathbb{E}^2$$

We will now employ our rules, having a conservative electric field irrespective of any external magnetic changes. Now I consider the equation:

$$\mathbb{E}^2(qt^2)^2 \int v dt= I^2$$

Again we take a similar route and do the following:

$$\epsilon_0 \mathbb{E}^2(qt^2)^2 \int v dt= \epsilon_0 I^2$$ [*]

If we take one half of the quantity $$\epsilon_0 I^2$$ where $$I$$ is the inertial moment, gives us on the right hand side a relation to the field density,

$$\frac{1}{2}\epsilon I^2=u_eq^2t^4 \int v dt$$

where the left hand side has been simplified to give one whole value of the inertial moment. This equation as I interpret it, describes the electric field strength as being inversely related to the inertial moment of the particle, and so this would be related to a certain type of electromagnetic inertia.

The equation with the star [*] is like most of the equations, describing a charge in an electric field in a state of motion.

We will soon be considering the dynamical effects of the electric field $$\mathbb{E}$$ and it's associated density $$u_e$$ in energy per unit volume of spacetime - because one can minimize the integral action, this immediately turns into a lowest frequency, or ground state solution of the vacuum. In other words, it would have exactly $$E=\frac{1}{2}\hbar \omega$$ as fluctuations in the bubbling vacuum of spacetime.

The electric field is something which describes spacetime surrounding electrically-charged particles or even a time-varying magnetic field. However, the above is seen in light of a conservative electric field with no magnetic field present. Thus the electric field in this case can be seen as exerting a force on particles.

If a field can exert a force on a particle, why may it not give rise to a force associated similar to that of an inertial force?

The force resisting acceleration could very well be related to the field densities acting as a type of electromagnetic inertia.

Inertia as Innert Energy

Einstein never ruled out completely the cause of inertia, but in a series of work, he did show it was possible that the inertial energy of a system could cause inertia, which is slightly different to the above, but shares a fascet in which that the field density can be verbally replaced but not mathematically equivalent to an energy desnity itself, since the electric field is not a field which has the dimensions of energy.

The conservative field equations I have been using, are themselves a type of equation of motion, where the integral must govern a change in position, with a velocity which by choice of the observer, can change. It's linear however, but this can still invoke a more complicated set of dervations I would have presumed which we could allow a system to experience a varying magnetic field.

Einstein, as I said, involved the genious idea that perhaps the rest energy of a material system is what causes the inertia of matter, but in a sense, energy and matter are but fascets themselves of the same manifestation under Einsteins mass-energy relationship - but, with a little thought, I question the methodologies of using energy to explain inertia alone. Afterall, saying energy is responsible for inertia in a material system is just trivial as saying matter is what causes the inertia of trapped energy - neither really are enlightening to the cause of inertia itself, other than something pointing to the inherent structure of the particle (intrinsic structures).

In response to Einsteins idea, it was then a consideration to see if we could see the above equations, in a slightly different light, this time concerning energy.

I came to the equation:

$$\epsilon_0 \frac{(\frac{\mu_0 D}{c^2})q^2t^4}{t^2} \delta \int v dt= \epsilon_0 E_0 \ell$$

where $$E_0$$ is for inertial energy and $$\ell$$ is basically a distance travelled, with obviously dimensions of L. Here I have used a minilizing integral on the change of position, which in the sense is meant to indicate it takes the least energy required to move that distance. This means that it satisfies a ground state of energy, so I am assuming this eq. can be altered to the following form;

$$\epsilon_0 \frac{(\frac{\mu_0 D}{c^2})q^2t^4}{t^2} \delta \int v dt= \epsilon_0 (\hbar \omega) \ell$$

Where $$\hbar \omega$$ is the lowest frequency, or ground state of energy per unit volume of spacetime.

The Four Equations of Motion Relating the Electric Field, The Inertial Mass-Moment, Field density and the Energy

So far through the derivisions made and the restraints of what the theory so far permits for the respective variables we are using, there are four equations which justify each other and unify the sense of inertia being related to the density of the electric field, the rest energy of the inertia (which will be a concept challenged later), the electric field which acts like a grid not in spacetime as you would with operating with Cartesian Coordinates, but one which has itself it's own abstract field, and this will measure the density in relation to field projected by the particle in question.

The first equation is

$$\epsilon_0 \mathbb{E}^2q^2t^4 \delta \int v dt = \epsilon_0 I^2$$

This unifies the electric field with the inertial moment with a relation to the electric permitivvity.

$$\frac{1}{2}\epsilon_0 I^2=u_e q^2 t^4 \delta \int v dt$$

The second equation here unifies that half the quantity of inertia goes to the electric field strength. [A]

The Third equation now, unified the energy with the electric field density, again in quite a familiar pattern:

$$\frac{1}{2}\epsilon_0 E^{2}_{0}t^4=u_e q^2 t^4 \delta \int v dt$$

The lower script in the energy-term on the right hand side is for two useful reasons. The first is to destinguish it notationally from the electric field notation which is similar, but also to demonstrate we are talking about an inertial energy, which some define as a rest energy.

However, because it has a minililizing on the integral, the true way to express the third equation is really by this form:

$$\frac{1}{2}\epsilon_0 (\hbar \omega)^{2}_{0}t^4=u_e q^2 t^4 \delta \int v dt$$

The fourth equation comes in the form of relating the energy to the electric field itself, and it's not always completely obvious, but wrapped up in the signs $$\epsilon \mathbb{E}^2$$ does have meaning as being closely related to the displacement if it were not for the extra multitude of electric field. This being related to the energy in its cryptic-sense should not seem to bizarre, as displacement does involve the use of a kinetic energy.

$$\frac{1}{2}\epsilon_0 E^{2}_{0}t^4=u_eq^2t^4 \int v dt$$

What is interesting, is that the charge has been the one unit which has been pretty much invariant throughout each derivation of the four equations, all logical from each other, and relating their different properties in unique ways.

[A] - I have considered possible definitions of what this could mean. Because mathematics is an abstractual representation of cold rigor, sometimes it's difficult to come to a conclusive resuly in your heart what it might mean. I have come to interpret this however to mean that inertia arises as an emergent property of a charged particle moving in an electric field. This would basically confirm my original hypothesis that there is somehow a description of an electro-inertia condition. It may be even possible that the field strength has an opposing force on the charge, and I will explore that possibility in another post using coupling theory $$g(x)$$ and try and allow it to work alongside some description of a Higgs Influence.

The Coupling of the Invariant Charge and a Gravitational Mass

Since charge $$q^2$$ has been invariant throughout the equations, it might be noted that the quantum mechanical definition of a charge density is $$q(\int |\psi|^2)$$. The density of a charge will remain constant in a thermally-abundant environment. Since the electric field strength was shown to play a possibly large role in the understanding of it's connection to electric field, there maybe some indication here that perhaps this is what the thermally-abundant environment could be. The electric field istelf - this would require making a coupling constant making charge it's coefficient of reference $$g(q^2)$$, since it will be the charge in this hypothesis which couples to the mass of the particle. The equations above will lead to some description of this, but it will require more derivations.

The coupling between the mass term $$M$$ and the charge $$q$$ will first be identified as:

$$M=q \phi$$

that should be $$M=q \phi$$

This equation is actually well known, but not illuminating. From here, my goal will be to derive the mass-term relationship to the electric field equations. But this will be no good if I have a lot of mistakes. I am very keen to learn as much as I can on calculus. I have got quite old and the memories of college just gave me a boost this year to get right stuck into all this stuff I read many years ago. It certainly does escape your memory fast.

I will appreciate any one who has the patience to study the equations I have written so far, and if they notice any mistakes, I am open-ears so that I may fix them. I am very very rusty at calculus, hence why I am keen for someone to have the patience with me and take the time out to have a wee look over them.

Green destiny

ps. This site looks good. I think I am going to enjoy it here, and maybe contribute what I can :)

Last edited: Sep 29, 2010
2. Sep 30, 2010

Green Destiny

I've not had any response, but I cannot be sure If I am making any dimensional mistakes.

But the work goes further now, so anyone with any patience is well-appreciated!

Principle of Least Charge Coupling

Dealing with charges that are invariant, will lead me to consider a coupling mechanism between the particle with a mass and the electromagnetic field, but is not a replacement of a Higgs Mechanism, but will be seen as an inertial mechanism of electromagnetic stress caused by a vorticity with a momentum description acting on the linear motion of a particle.

The wave function will be effected by the coupling by effectively displaying a deflation periodic to the inertial moment $$I$$. This would place a critical limit on how fast an electrons self-wave function interacts with the electric field, but I have not performed these calculations, as I am not sure how to.

The Zeno Effect, by definition, we will state requires the change in the probability field change is not a linear state in time $$\psi(t)=sin^2(\frac{\pi}{2}t)$$

and so $$N: \psi \frac{1}{N} \ne \psi(1)$$

Where $$\psi$$ just represents the field of its probability. So we shall have a state vector:

$$<\Psi> \approx N: \psi \frac{1}{N} \ne \psi(1)$$

To explain a known phenomenon which has not been completely decided upon by physicists, which is called the measurement problem on the self-interaction wave function on particles. If quantum mechanics theory is correct, then particles like the electron would expand to the size of the pentagon in less than a second, due to their physical distribution throughout spacetime by their wave function descriptions.

Even in absence of a large gas of particles, an electron does not expand to these physical sizes, so there is something inherently collapsing that wave function. I state it as being a non-linear description when there is a coupling in the momentum of the field on the particles charge.

This means we will have a least coupling charge $$q$$ on the potential $$A$$ and we will assume for such an interaction to occur, there needs to be a an electrodynamical vorticity to the electromagnetic field $$\sigma$$. The vorticity will be seen as a coupling in the presence of the curling field of magnetism, a concept so far which has been neglected from our conservative field equations.

Whilst vorticity is in it's own right, a force somewhat related to momentum, it is then said in quantum mechanics that even the electromagnetic field contains a momentum.

$$W=[\epsilon_0 \epsilon \frac{\mathbb{E}^2}{2}+ \frac{1}{\mu_0 \mu} \frac{B^2}{2}]$$

Where $$W$$ is the total density of the electromagnetic field per unit volume of spacetime. If our vorticity acts on a mass $$\sigma M$$ then this can be introduced into our original conservative electrodynamical equation where inertia of the electric field was proportional to the electric field density as;

$$\frac{1}{2} I^2(\sigma M)=u_eq^2t^4(\delta \int v dt \cdot M \omega)$$

Here we can see by postulating some involving of an electric vorticity of mass, we have an angular momentum $$\omega$$ present, perhaps pointing to the the possibility of a spin-related inertia-representation on the respective electric density of the field.

In order to get a better picture, we will now involve the magnetic field, since spin is invariant under some magnetic field for charges like fermions with spin 1/2 matrices, called Pauli matrices which will be well known.

We won't deal with such matrices, however, we will turn to Canonical description of the momentum vacuum solution as a least principle of charge coupling on the field $$A$$ as a linear equation of motion $$\delta \int v dt$$ involving the magnetic field density per unit volume of spacetime, and our results give us a momentum description in a vacuum vorticity related to the inertia which is itself already identified as a product of the electric field and it's respective density:

$$\frac{1}{2}\epsilon_0 I^2+u_b(\sigma M +qA)\Psi=u_e+ \frac{1}{\mu_0 \mu}\frac{B^2}{2}q^2t^2(\delta \int v dt \cdot M \omega + qA)\Psi$$

Where $$\Psi$$ is our distributive wave probability, which is non-linear and $$u_b$$ is our magnetic field density. It would seem that the linear motion $$\delta \int v dt$$ through the density of the field $$u_e$$ acts on the field state of the probability $$\Psi$$ as a non-linear system. The wave function therefore deflates in the coupling of the charge in respect to the electromagnetic field.

The latter equation can be further simplified knowing;

$$W=\frac{1}{2}\epsilon_0 \mathbb{E}+\frac{1}{\mu} \frac{B^2}{2}$$

From now on however, we will view the permitivvity and permeability in relative notation. And knowing the canonical momentum as;

$$\mathbf{P}=mv+qA$$

So our equation takes a new simplified light of:

$$\frac{1}{2}\epsilon_0 I^2+u_b(\sigma M +qA)\Psi=Wq^2t^2 \int \mathbf{P}\Psi$$

So the momentum and the field densities are the inverse properties of the inertia experience added with the vorticity coupling momentum expression in the appearance of a charge.

Last edited: Oct 1, 2010
3. Sep 30, 2010

RoyalCat

Could you please post where you got all of those equations from?

4. Sep 30, 2010

Green Destiny

They are my own.

5. Sep 30, 2010

Green Destiny

Well, some of them, sorry.

6. Sep 30, 2010

RoyalCat

Yes, I understand that you derived them, but by what means? For instance, you said,
What rules would those be? Where did that equation come from?

7. Sep 30, 2010

Green Destiny

I'm sorry, I picked you up wrong.

You want the history, correct?

I first imagined the presence of an electric field, and an electric field $$\mathbb{E}$$ creates a charge $$q$$ on a particle with mass $$M$$ ~ I could have used an ionization notation $$\bar{e}M$$ but I didn't. I stuck with the expression simply as $$\mathbb{E}q^2t^2$$ and the extra $$qt^2$$ came from another derivation involving the inertial mass-moment as $$Et^2=I$$ where here $$E$$ is the energy. So energy was in my opinion, an inherent property of the system, but I wanted to measure the energy as the expression $$E \ell t^2$$ and this expression would invoke the idea of some possible linear motion from one point to another.

Instead of representing this in a superspace or even usual Cartesian Coordinates, I represented the linear motion $$\int v dt$$ on the particles charge $$q$$ as being part of the final equation,

$$\mathbb{E}^2q^2t^4 \int v dt = I^2$$

It just so happened that the dimensions (or the page in which I am calculating these dimensions from, since I can never remember them by memory) told me this was the same quantity of the inertial mass-moment of the particle squared. This was my motivation to move onto the following equations.

8. Sep 30, 2010

Green Destiny

Another set of rules, is that the electric field is conservative, meaning that it cannot variate in the integral, so remains a constant, alongside the charge.

I forgot to mention that one.

9. Sep 30, 2010

Green Destiny

I have one last part to go. I take a while each part because the latex takes a long time as I am sure you know.

I'll say as a side-note, that the vorticity of a field would be given as $$\vec{\sigma}= \vec{\nabla} X (v)$$ which is the curl on a velocity vector. So if the equation:

$$\frac{1}{2}\epsilon_0 I^2+u_b(\sigma M +qA)\Psi=Wq^2t^2 \int \mathbf{P}\Psi$$

where $$\sigma M$$ exist, it's best given as:

$$\frac{1}{2}\epsilon_0 I^2+u_b(\vec{\sigma M} +qA)\Psi=Wq^2t^2 \int \mathbf{P}\Psi$$

where the $$\vec{\sigma M}$$ represents a deeper meaning of the vorticity ebing connected to the mass through a momentum term $$\vec{\sigma M}= \vec{\nabla}X(Mv)$$.

Last edited: Oct 1, 2010
10. Sep 30, 2010

RoyalCat

Where did you learn all of this? It is not true.
Specific reservations are in bold within the quote.

11. Sep 30, 2010

Green Destiny

''This is not true. In the presence of an external electric field, an initially neutral conductor would have a certain charge distribution on it, depending on the space and time variation of the external field. However, the total charge would be 0.
An electric field does NOT create a charge. Charges are the source of the electric field, not the other way around.''

I'm not axactly dictating that. This applied for a mass $$M$$. An ionization on the mass $$\bar{e}{M}$$ interacts with the field $$\mathbb{E}$$. The result later of this is theoretically due to a vorticity on the mass.

''..? Where did time come into this? Just because the units match, doesn't mean that the expression makes physical sense. If I took your word for it, that the moment of inertia of an object is related to its energy according to , that would mean that its moment of inertia increases without bound as time goes on! This is nonsensical.''

How do you deduct that from the equation?

''You cannot post expressions without equating them to anything. What is equal to? What is this expression? It cannot be the energy, since you'd be saying which is nonsense. So if you meant that E is the electric field, then it makes no sense dimensionally. In general, the motion of objects is not linear, there is no reason to expect it to be so.''

It's equal to $$i \hbar \frac{\partial}{\partial t} \cdot t^2 \int v dt$$. And E is energy this time around.

' Just because the dimensions match does not mean that the equation is true! If I told you that for any mass in a gravitational field, and that therefore the mass is defined by , you would rightly say that it is nonsense. The only context in which '

It's a hypothesis. If it is any consideration, I wouldn't believe what you wrote either.

When we say that a field is conservative, we don't mean that it never changes. When a field doesn't change over time, we call it static, and when it doesn't change over space as well as time, we would call it a field of constant magnitude and direction.
A conservative field is one that conserves energy. That is, if I take a charged particle and move it about, eventually returning it to its initial position, the net work performed on it would be 0.
Two mathematical formulations of this would be, since you show some familiarity with the concept of closed-loop integrals and the curl derivative:

My field is a conservative field. Look at the integrals, the main ones use a minimilizing constant on the integrals.

12. Sep 30, 2010

RoyalCat

For most materials, unless you have an extremely strong electric field, ionization will be minimal. Even if I accept that we're working with an ionized mass, the moment of inertia has NOTHING to do with the charge, electric field, time elapsed, or distance traveled.

You just equated a real quantity with an imaginary one.

I'm afraid you've made an absolute salad of some equations you've read. I suggest you pick up a well-structured physics book in order to continue your studies. As it stands, you are simply playing around with equations and dimensionally correct (And often incorrect) expressions.

A conservative field still can, and in general does, vary in space, and therefore cannot be taken out of the integral!

I suggest some of these books for all around intros to physics:
Fundamentals of Physics By David Halliday, Robert Resnick, Jearl Walker - John Wiley & Sons (2010) - Hardback - 1328 pages - ISBN 0470469080 Link
Sears and Zemansky's University Physics (10th Edition) https://www.amazon.com/Sears-Zemanskys-University-Physics-10th/dp/0201603225

Last edited by a moderator: Apr 25, 2017
13. Oct 1, 2010

Green Destiny

You are absolutely correct.

The only way to keep the electric field constant is to have a fixed constant for the radius term, if one has one, for an enclosed charge because it varies radially for a point charge.

Thank you!

Last edited by a moderator: Apr 25, 2017
14. Oct 1, 2010

Green Destiny

' Even if I accept that we're working with an ionized mass, the moment of inertia has NOTHING to do with the charge, electric field, time elapsed, or distance traveled.'

There so no theory which describes it as such, that is why. My attempt is to try and suggest it through my theory.

15. Oct 1, 2010

RoyalCat

As admirable as trying to come up with a theory yourself is, you still need to have an understanding of the basics, and a grounding in reality, for it to have any value.
As it stands, your theory contradicts A LOT of well established physics, and leads to nonsensical results (Infinitely increasing moment of inertia for any mass in an electric field, to name one)

16. Oct 1, 2010

Green Destiny

The increase of inertia must be renormalizable somehow, if that is truely what these equations are stating, unless they are subject to relativity, in which case inertia would increase with an increase of mass.

That part admittedly requires some extra thought.

17. Oct 1, 2010

Green Destiny

What seems fascinating to me however, is that we can find the field to contain a vorticity on the particle's momentum with a least coupling of charge in the potential.

What am I saying... I
I'm an idiot. My wave function negates the linear increasing of inertia because it's not a linear function of time.

Solved.

18. Oct 1, 2010

RoyalCat

Please reconsider everything you've posted. It is wrong from the very first line (c2=1/(μ0ε0)). It shows a deep confusion with respect to the relevant concepts.
I urge you to take the subject up seriously and methodically, you seem to me like you're very passionate about the subject and about exploring it for yourself, and again, I applaud you for it. It's admirable. But as it stands I think you would enjoy physics much much more once you start dealing with REAL physics, and not the made-up variety.

You can't post a formula you've invented, and then say that you are fascinated by what you found. You haven't found anything. Your hypotheses are not supported by experimental evidence, and contradict well-founded theory. Meaning, that to a great degree of certainty, they are dead-wrong.

I suggest you consult a textbook on the subject to help sort things out in your mind, since as it stands, you are making a salad of everything.

And if you're enjoying the math more than the application to the real world, then there are plenty of ways to educate yourself on the math involved as well, since as it stands, you are equating scalars with vectors, and real quantities with imaginary ones, as seen in these posts:

And that shows a shallow understanding of the math involved as well. Again, as admirable as self-teaching these subjects is, I strongly suggest that you do it methodically and with well-established theory, rather than making it up as you go along.

Last edited: Oct 1, 2010
19. Oct 1, 2010

Green Destiny

I really do appreciate the time you've taken out to address these issues. I do accept this theory is probably flawed to the understanding of physics, but I have taken quite a bit of time to see where it does contradict the theory. The increasing inertia was an interesting one, because it wasn't apparent to me, but I did decide the wave function state vector conditioning on the self-interaction would negate this. I still am not sure, as you haven't answered that question.

I do ask you that in despite of it being contradictory to what we think we know, I do ask a little humor on the theory, because it truely was to see if I could mathematically-perform these things, rather than the theory on a whole. For instance, there was some conversation about the definition of a conservative electric field, and you were right in what you said, and it's been noted, but because my equations finally come to an interpretational methodology of the inertia related to the momentum of the electromagnetic field could be seen in terms of an electromotive decription.

The rate of rotation of a field element is proportional to shear stress in fluid dynamics. So shear stress gives rise to rate of rotation of a field and so the electromagnetic field contains an angular momentum which acts on an ionized particle.

In much the same sense, the electromagnetic field acts like a rotating fluid as a particle has a motion through it so the momentum of the vorticity of the field is akin to the fluid dynamics of moving particles in a fluid-medium. The electromotive force truely is by definition "that which tends to cause current (actual electrons and ions) to flow." [1] and so must be indestinguishable I thought to an inertial body in acceleration. The flow must be subject to the inertial moment of the mass, and inertial moments I have postulated need not take the usual classical definition since quantum particles are not classical by definition, only in an approximated value for slow speeds.

$$\frac{1}{2}\epsilon_0 I^2+u_eq \mathbf{\epsilon} \Psi=Wq^2t^2(\int P - \vec{\sigma M})\Psi$$

Where the electromotive force is $$\mathbf{\epsilon}$$. You might know that conservative electric fields are very important in electromotive descriptions.

[1] http://en.wikipedia.org/wiki/Electromotive_force