I Coulomb's law taken to the extreme

[neobaud]

TL;DR Summary

Can you sum the force contribution to a charged particle for every particle in the universe.

So Coulomb's law is this simple formula for calculating the force between two charges.
Kq1*q2/(r^2)

(Disclaimer: I know this computation is not actually possible I was just wondering if it would give the correct result)

I was wondering if it is valid to sum the force vector from every charge in the universe? It occurred to me that if sum these forces you would need to figure out where every charge was going all the way back to the big bang. For example for charges near the sun I would have to compute the force using the position 8 minutes ago. Will this is summation give the right answer in terms of the final force on the charge that is consistent with the general theory of relativity? Also, if I used this summed force to compute the potential energy would this be the particle's Mass by the energy/mass relation e=mc^2?

 

[Orodruin]

No. Coulomb’s law has restrictions. First of all it describes electrostatics - electric forces in the special case where nothing is moving. This is not the case for the Universe. To be more precise you would need to use the full Maxwell’s equations.

Second, it does not take general relativistic effects into account. These are particularly relevant when you consider the entire universe.

 

[neobaud]

No. Coulomb’s law has restrictions. First of all it describes electrostatics - electric forces in the special case where nothing is moving. This is not the case for the Universe. To be more precise you would need to use the full Maxwell’s equations.

Second, it does not take general relativistic effects into account. These are particularly relevant when you consider the entire universe.

Ok right the equation is wrong. Let me adjust that then. There is some more complicated equation for the force between two charges that is a factor of their distance and velocity.

If I applied that equation to each charge for it's distance at time t-r/c, does this account for special relativity? This accounts for the time it takes for one charge to affect the other right?

 

[Ibix]

If I applied that equation to each charge for it's distance at time t-r/c, does this account for special relativity?

No. You also need to compute the magnetic field due to the motion of the charges. The general approach is the Liénard-Wiechert potentials, which would have to be summed over all the source charges, and then take the gradient to get the force.

 

[jbriggs444]

If I applied that equation to each charge for it's distance at time t-r/c, does this account for special relativity? This accounts for the time it takes for one charge to affect the other right?

The problem is worse. Electromagnetic fields can exist without any charges anywhere. Simply knowing where all charges are right now is not enough. Even knowing where all charges were in the past is not enough. For completeness, you need to know the field.

The field is a solution to Maxwell's equations. The positions, velocities and accelerations of the charges at some point in time supply some boundary conditions. But not enough for a complete, completely accurate and unambiguous solution.

 

[neobaud]

No. You also need to compute the magnetic field due to the motion of the charges. The general approach is the Liénard-Wiechert potentials, which would have to be summed over all the source charges, and then take the gradient to get the force.

Ok I just looked up Liénard-Wiechert potential and that approach is exactly what I am asking. So I guess the answer is yes

 

[neobaud]

The problem is worse. Electromagnetic fields can exist without any charges anywhere. Simply knowing where all charges are right now is not enough. Even knowing where all charges were in the past is not enough. For completeness, you need to know the field.

The field is a solution to Maxwell's equations. The positions, velocities and accelerations of the charges at some point in time supply some boundary conditions. But not enough for a complete, completely accurate and unambiguous solution.

Ok interesting. How can there be a field potential without a charge? That is where the field comes from isn't it?

 

[Ibix]

So I guess the answer is yes

No, because just by adding the electric components you haven't accounted for the magnetic part of the Lorentz force. Note also the $(1-\vec n_s\cdot\vec\beta_s)$ term in the expression for $\phi$, which is not in the Coulomb force.

 

[jbriggs444]

Ok interesting. How can there be a field potential without a charge? That is where the field comes from isn't it?

Maxwell's equations allow for wave solutions (e.g. radio waves) with no charges anywhere.

 

[neobaud]

No, because just by adding the electric components you haven't accounted for the magnetic part of the Lorentz force. Note also the $(1-\vec n_s\cdot\vec\beta_s)$ term in the expression for $\phi$, which is not in the Coulomb force.

Oh ya I had amended my original question in the thread. The equation for the force would be $$\mathbf{F} = q_1 \left[ \frac{1}{4\pi\epsilon_0} \frac{q_2}{r^2} \hat{\mathbf{r}} + \mathbf{v}_1 \times \left( \frac{\mu_0}{4\pi} \frac{q_2 (\mathbf{v}_2 \times \hat{\mathbf{r}})}{r^2} \right) \right]$$
For every particle in the universe.

 

[neobaud]

Maxwell's equations allow for wave solutions (e.g. radio waves) with no charges anywhere.

Oh ya. But the wave originated from a charge at some point right? So if I wanted to figure out the total force between two charges it would be
Static force+Magnetic force+wave force?

So in my equation above I need one more term for the wave part?

 

[jbriggs444]

Oh ya. But the wave originated from a charge at some point right?

Not in an eternal space-time.

Like the Schwarzschild solution in General Relativity which has no mass anywhere (aka a vacuum solution), an eternal wave can have no charge anywhere (aka free space).

 

[neobaud]

Not in an eternal space-time.

Like the Schwarzschild solution in General Relativity which has no mass anywhere (aka a vacuum solution), an eternal wave can have no charge anywhere (aka free space).

Sorry what is an eternal space time? I was assuming spacetime originated at the big bang. Is this not the case?

 

[jbriggs444]

Sorry what is an eternal space time? I was assuming spacetime originated at the big bang. Is this not the case?

An eternal space time is one which has always existed and will always exist. Such as the Minkowski space time (flat and infinite). Or the Schwarzschild space time (curved and infinite).

The "big bang" is not the beginning of our space time, exactly. In these forums, the term is used to refer to the earliest state of our universe for which we have good evidence. A hot, dense state. One can extrapolate farther back. Possibly extending to an initial singularity. Or possibly to something else.

I lack the expertise to rule out primordial electromagnetic waves on theoretical grounds. As a practical matter, what we actually see is the cosmic microwave background which originated at the recombination epoch. Cosmological red shift has reduced the original intensity quite significantly.

 

[Orodruin]

Oh ya I had amended my original question in the thread. The equation for the force would be $$\mathbf{F} = q_1 \left[ \frac{1}{4\pi\epsilon_0} \frac{q_2}{r^2} \hat{\mathbf{r}} + \mathbf{v}_1 \times \left( \frac{\mu_0}{4\pi} \frac{q_2 (\mathbf{v}_2 \times \hat{\mathbf{r}})}{r^2} \right) \right]$$
For every particle in the universe.

No. This assumes spacetime is Minkowski, which is not a goid approximation at the scale of the Universe. On the other hand, it is a good approximation for the relevant part.

 

[neobaud]

No. This assumes spacetime is Minkowski, which is not a goid approximation at the scale of the Universe. On the other hand, it is a good approximation for the relevant part.

One of the main things I am trying to learn is if r in this equation is the r at time t-r/c, does this equation become correct?

 

[neobaud]

An eternal space time is one which has always existed and will always exist. Such as the Minkowski space time (flat and infinite). Or the Schwarzschild space time (curved and infinite).

The "big bang" is not the beginning of our space time, exactly. In these forums, the term is used to refer to the earliest state of our universe for which we have good evidence. A hot, dense state. One can extrapolate farther back. Possibly extending to an initial singularity. Or possibly to something else.

I lack the expertise to rule out primordial electromagnetic waves on theoretical grounds. As a practical matter, what we actually see is the cosmic microwave background which originated at the recombination epoch. Cosmological red shift has reduced the original intensity quite significantly.

Thanks so in the equation
$$\mathbf{F} = q_1 \left[ \frac{1}{4\pi\epsilon_0} \frac{q_2}{r^2} \hat{\mathbf{r}} + \mathbf{v}_1 \times \left( \frac{\mu_0}{4\pi} \frac{q_2 (\mathbf{v}_2 \times \hat{\mathbf{r}})}{r^2} \right) \right]$$

There is an additional factor? I get the fact that there may be some wave that existed since before the big bang. I mean the total force from a charge that is known to exist in the past.

 

[jbriggs444]

There is an additional factor? I get the fact that there may be some wave that existed since before the big bang. I mean the total force from a charge that is known to exist in the past.

If we stick with Minkowski space time, ignoring the point made by @Orodruin, then we have, in principle, the possibility of pre-existing eternal waves which trace to no finite source and are simply there. However, you wish to ignore these. Perhaps by assuming an initial condition where the electromagnetic field is asymptotically zero as one looks further and further away.

In that case, no. There is no additional factor for primordial EM waves.

But, as you point out, we do not live in a Minkowski space time.

 

[SredniVashtar]

What about the creation and subsequent annihilation of charge and anti charge? No electric field before the creation of the pair, a field due to the opposite charges separating - possibly with EM wave due to the acceleration - and then no more field when they annihilate.

After the annihilation, the electric field and the EM wave are still propagating in space at the speed of light, but the charges that originated them are no longer there. (They "are" a time r/c prior the detection of the fields, tho).

 

[neobaud]

What about the creation and subsequent annihilation of charge and anti charge? No electric field before the creation of the pair, a field due to the opposite charges separating - possibly with EM wave due to the acceleration - and then no more field when they annihilate.

After the annihilation, the electric field and the EM wave are still propagating in space at the speed of light, but the charges that originated them are no longer there. (They "are" a time r/c prior the detection of the fields, tho).

If an em wave came from nowhere, that would violate conservation of energy.

 

[Orodruin]

If an em wave came from nowhere, that would violate conservation of energy.

No it would not. Maxwell’s equations allow plane waves. Such waves are eternally propagating with no source.

 

[neobaud]

He was not talking about an eternal wave. His example started at the pair annihilation. You can't have an EM wave plane or no come from nowhere.

Can you give more info about how to modify my equation at the scale of the entire universe?

 

[SredniVashtar]

Not only the wave, but the electric field itself would come from the charges - far far away "when" they existed. The fields propagate at a finite speed, so when I sense them - say one year later - I know some charge far away must have generated them. Then after say ten minutes, those (contributes to the total fields) are gone because one year minus ten minutes ago they annihilated. The field will go on, and on Alpha Centauri they will sense one year later.
But from my point of view, there is a perturbation 10 light minutes long in the EM field, with nothing before and nothing after.

 

[Orodruin]

Not only the wave, but the electric field itself would come from the charges - far far away "when" they existed. The fields propagate at a finite speed, so when I sense them - say one year later - I know some charge far away must have generated them. Then after say ten minutes, those (contributes to the total fields) are gone because one year minus ten minutes ago they annihilated. The field will go on, and on Alpha Centauri they will sense one year later.
But from my point of view, there is a perturbation 10 light minutes long in the EM field, with nothing before and nothing after.

What’s your point? The Liénard-Wiechert potential is explicitly a sum over the charges on the past lightcone.

 

[tech99]

Are we assuming that an electric field propagates with the speed of light? I am not aware of evidence for this.

 

[jbriggs444]

Are we assuming that an electric field propagates with the speed of light? I am not aware of evidence for this.

It is well known. Maxwell's equations govern how the electric and magnetic fields propagate in a vacuum.

Maxwell's equations may be combined to demonstrate how fluctuations in electromagnetic fields (waves) propagate at a constant speed in vacuum, $c$ (299792458 m/s). Known as electromagnetic radiation, these waves occur at various wavelengths to produce a spectrum of radiation from radio waves to gamma rays.

In the presence of a medium the interaction with the medium can create the effect of waves that propagate more slowly.


Of course, it is changes in the field that propagate at the speed of light. The field itself is already everywhere and has no associated speed.

 

[Nugatory]

One of the main things I am trying to learn is if r in this equation is the r at time t-r/c, does this equation become correct?

No.
The problem is that in a curved spacetime, $r$ and $t$ are only unambiguously defined in a region small enough that curvature effects can be ignored, so no equation that uses them can be correct in the sense that you're thinking.

 

[Orodruin]

It should also be mentioned that the case of three dimensions is quite special. Apart from the fact that the rank-2 field tensor can be split into a vector and a pseudo-vector representation under spatial transformations (ie, the electric/magnetic fields), the Green’s function of the wave equation in three spatial dimensions has a very peculiar property: it is a radial delta function propagating away from the source at the wave speed. In particular, unlike in one or two dimensions, there are no lingering effects once the original wave front has passed. The potential only depends on the charges on the past lightcone. In lower dimensions this would not be the case.

 

[SredniVashtar]

It is well known. Maxwell's equations govern how the electric and magnetic fields propagate in a vacuum.


In the presence of a medium the interaction with the medium can create the effect of waves that propagate more slowly.


Of course, it is changes in the field that propagate at the speed of light. The field itself is already everywhere and has no associated speed.

In the case of creation and annihilation, we can see the field itself as a change from zero field to nonzero field and then back to zero field. The argument of the field being 'already everywhere' should not apply here because the charges responsible for it were not always there. If we create them and do not annihilate them, we go from no field to yes field and we stay that way. The 'field' itself is the perturbation propagating at the speed of light.

What I am trying to say is that from the point of view of classical electrodynamics the existence of electromagnetic field (sorry, I had written radiation) 'on its own' is just a consequence of the finite propagation of signals. Of the three way to describe EM phenomena (charges/currents, fields, potentials) the 'more real' one is that based on where charges are and how they move. Fields are a way to describe the effect of these charges at a distance, and potentials are a mathematical commodity.

A quantum physicists might argue that from their point of view, potentials are more 'real'. But that's black magic.

 

[jbriggs444]

In the case of creation and annihilation, we can see the field itself as a change from zero field to nonzero field and then back to zero field. The argument of the field being 'already everywhere' should not apply here because the charges responsible for it were not always there.

A "field" in the sense that we use the term is a function defined over a space.

For instance, we could have a real-valued function called "height" defined over the space of a few square miles of ground. That would be a scalar field.

Or we might have a vector-valued function called "wind velocity" defined over a volume of a few cubic miles. That would be a vector field.

The electromagnetic field is a [pair of] field(s) in this sense. The field is everywhere, even though it may be zero-valued everywhere. It does not make sense to speak of the field having a speed. It is fluctuations in the field values that can have speeds.

Similar to the way that it makes sense to speak of the speed of surface waves on a lake without needing to talk about the speed of the lake. The water is already there. Its surface has a height everywhere, even before we throw a rock in.

 

[Orodruin]

The electromagnetic field is a [pair of] field(s) in this sense.

... a rank 2 antisymmetric tensor field to be precise.

 

[tech99]

I notice that when I discharge the dome of the Van de Graaff machine it discharges immediately and I do not have to await energy to travel back in from the outer reaches of space.

 

[SredniVashtar]

A "field" in the sense that we use the term is a function defined over a space.

...
Similar to the way that it makes sense to speak of the speed of surface waves on a lake without needing to talk about the speed of the lake. The water is already there. Its surface has a height everywhere, even before we throw a rock in.

That seems just semantics. But ok, the perturbation in the field between creation and annihilation is say 100 ns long and, from a distance, I have 100 feet in space where the electric field is non zero, while it is zero before and after. In the middle I only have the static electric field of the (pair of) charge(s) pointing where the source was r/c seconds before.
Is that a wave? If the charges were held fixed after the initial transient separation, all we have in the middle is an electrostatic field. No accompanying magnetic field that can justify a wave propagation.
It's as if we raised the level of the lake. Yes, there is a perturbation at the start when the tsunami hits you and you go from no field to yes field, but then - imagine you do not annihilate the charges - it no longer is a perturbation. It's like the electric field of every other existing charge.

 

[Orodruin]

The argument of the field being 'already everywhere' should not apply here because the charges responsible for it were not always there.

Sorry, but that’s just nonsense. The field is always there whether nonzero or not. Zero is a perfectly valid field value.

Charges are not responsible for the existence of the field itself, they are the cause of divergences in the field, as described by Maxwell’s equations.

 

[SredniVashtar]

Sorry, but that’s just nonsense. The field is always there whether nonzero or not. Zero is a perfectly valid field value.

Charges are not responsible for the existence of the field itself, they are the cause of divergences in the field, as described by Maxwell’s equations.

I am seeing this from the point of view of classical ED. Fields are an expression of the presence of a charge.

 

[Orodruin]

I am seeing this from the point of view of classical ED. Fields are an expression of the presence of a charge.

That is a common misconception due to Maxwell’s equations being linear. It is however not really accurate, in classical electromagnetism, the field is always present. It may be zero, but it is present.

Divergences in the field are the expressions of the presence of charges.

 

[jbriggs444]

Is that a wave?

Yes.

I am seeing this from the point of view of classical ED. Fields are an expression of the presence of a charge.

Right. You created a dipole, allowed it to exist for a short duration and then extinguished it. From a classical viewpoint, the result is a wave. A wave that persists after the dipole is gone.

 

[SredniVashtar]

That is a common misconception due to Maxwell’s equations being linear. It is however not really accurate, in classical electromagnetism, the field is always present. It may be zero, but it is present.

Divergences in the field are the expressions of the presence of charges.

Are we still talking of classical electrodynamics? Because, to my knowledge, charges are the sources of the electric field, and currents are the sources of the magnetic field. They still are the sources (in the sense of the reason for the field presence - or if you prefer it "the nonzero value of the ever-present field") even when the field lines curl around each other. The 'curling' is the result of the relativistic transformation of the electric field of charges in a moving frame reference and of the finite speed of propagation of light.

 

[SredniVashtar]

Yes.

Right. You created a dipole, allowed it to exist for a short duration and then extinguished it. From a classical viewpoint, the result is a wave. A wave that persists after the dipole is gone.

Ok, let's change the timescale: instead of 20 ns, let's make it 4 billion years. We sit in the middle of this event. Would you call that perturbation that has been static for 2 billion years minus the few microseconds or milliseconds during the initial transient a "wave"? I wouldn't . I call that a static field. If I create a dipole in the lab, by bringing two spheres with opposite charge close to each other in the arc of say ten seconds, in the time I turn around to the instrument to measure the field, there is an electrostatic field, not a wave. We should rewrite ALL books of physics if that were not the case.

Granted, the distinction between statics, quasi-statics, and dynamics can be seen as artificial, since it is essentially centered on human timescales, but it is useful nonetheless. As John Percy Hammond put it:

[missing quote from one of Hammond's books or papers where he explain why it is useful to differentiate between static, quasi-statics, and dynamics - I have been looking for that for days but I can't find it]

If we want to put all these phenomena under the same category, instead of "wave" (which is indeed the solution to the full set of Maxwell equations, and not selectively maimed subsets) I'd use the term "solution of Maxwell's equations". Because after the first million years, I am pretty sure we can consider the transient that changed the value of the electric field from zero to a nonzero value spent and no longer relevant to all practical purposes.

And this is what I mean when I say that the electric field travels at the speed of light from the source: the information about the updated value travels with that speed. It is the convective part of the Lienard-Wichert formula. It is preceded by a wave-like transient corresponding to the radiation part of the Lienard-Wiechert formula. The existence of a mathematical structure that represent a 'global' electric field extending to all space does not seem that important in classical ED.

Anyway, even if we use this concept of all-present electric field with the special value of zero where some would say there is no electric field, like a the surface of a lake at zero height, the point I was trying to make is that the level of the lake cannot rise all 'together'. There needs to be faucets and sinks from where the water either enters or exits the lake, and the rest of the lake will change its level after a time r/c from these sources/sinks.

 

[renormalize]

Because after the first million years, I am pretty sure we can consider the transient that changed the value of the electric field from zero to a nonzero value spent and no longer relevant to all practical purposes.

This is nonsense. In an otherwise empty universe that initially contains an electron and positron at rest, they will attract and annihilate, thereby emitting electromagnetic radiation. Afterwards that universe is charge-free, but the outward-propagating radiation persists for all time, as it must since energy is conserved.

 

[jbriggs444]

Ok, let's change the timescale: instead of 20 ns, let's make it 4 billion years. We sit in the middle of this event. Would you call that perturbation that has been static for 2 billion years minus the few microseconds or milliseconds during the initial transient a "wave"?

2 billion light years from the position of the temporary dipole there will be a wave, yes. What is your point?

 

[SredniVashtar]

This is nonsense. In an otherwise empty universe that initially contains an electron and positron at rest, they will attract and annihilate, thereby emitting electromagnetic radiation. Afterwards that universe is charge-free, but the outward-propagating radiation persists for all time, as it must since energy is conserved.

Who ever said they are in an empty universe and they are free to recombine?

 

[SredniVashtar]

2 billion light years from the position of the temporary dipole there will be a wave, yes. What is your point?

My point is that the wave is the transient, and the static field we perceive after the transient has died out is... a static field. Not a wave.

 

[jbriggs444]

Who ever said they are in an empty universe and they are free to recombine?

If the universe is non-empty then an argument that the local region has a static field becomes invalid. If they are not free to recombine then how do we explain the hypothetical situation in which they did recombine.

If your only point in posting here was to point out that we can have a large local region over which the electromagnetic field is static and at least approximately zero-valued then I think we are in agreement.

 

[renormalize]

Who ever said they are in an empty universe and they are free to recombine?

I did. It's a simplified model that is both consistent with physics and represents a counterexample that clearly invalidates your reasoning and assertions.