I Coulomb's law taken to the extreme

[neobaud]

TL;DR

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.