B How To Consistently Explain Electromagnetism With Relativity

[bob012345]

Correct.

Thanks. So if the only separation were 90 degrees to the direction of motion, simultaneous events in S could be simultaneous in S'?

 

[PAllen]

Thanks. So if the only separation were 90 degrees to the direction of motion, simultaneous events in S could be simultaneous in S'?

Correct.

 

[jartsa]

I still don't see why the rockets aren't at rest with respect to each other throughout the acceleration. Everyone keeps repeating they aren't without really explaining the cause.

Well one thing is that the motor of the trailing rocket burns fuel at slowed down rate. And why does that happen? The vicinity of the Rindler-horizon causes that kind of things. Rindler-horizon is like event horizon of a black hole, in every way, for an accelerating observer, for inertial observers it does not exist.

What do you think is the situation for the observers in the rockets after a long time, if both rockets run out fuel after a quite short time?

 

[A.T.]

I still don't see why the rockets aren't at rest with respect to each other throughout the acceleration. Everyone keeps repeating they aren't without really explaining the cause.

It was explained to you pages ago for the electrons:

If you actually attach a frame to an electron, during it's acceleration, you have time running at different rates along the wire in that frame. This effectively delays the electrons in the back, and speeds up the electrons in the front, so their distances increase.

 

[Geocentricist]

Yes, you and Ibix are right. There are no other inertial frames, where they are both at rest, even instantaneously.The OP want's to know what happens in the frame of a rocket throughout acceleration. That frame is non-inertial and has position dependent time dilation.

A and B share a frame at all times. There may be time dilation in their frame relative to some other frame but they cannot experience time dilation relative to their own frame, so there is still no cause for their separation to increase.

 

[Geocentricist]

It was explained to you pages ago for the electrons:

If you actually attach a frame to an electron, during it's acceleration, you have time running at different rates along the wire in that frame. This effectively delays the electrons in the back, and speeds up the electrons in the front, so their distances increase.

Time running at different rates along the wire? The wire frame is one frame, there cannot be multiple rates of time in this one frame.

 

[Ibix]

A and B share a frame at all times

No they don't. That's the point.

Time running at different rates along the wire? The wire frame is one frame, there cannot be multiple rates of time in this one frame.

Ditto.

 

[PAllen]

A and B share a frame at all times. There may be time dilation in their frame relative to some other frame but they cannot experience time dilation relative to their own frame, so there is still no cause for their separation to increase.

I’ve asked you to define what you mean by sharing a frame. You have refused. If you mean there is any standard type of frame (inertial or noninertial) in which they are not moving relative to each other, this is false (edit: except the original rest frame). There is no (edit: other) such frame.

 

[pervect]

Let me jump in and ask if m getting the point correctly. Since there is no size requirement on what constitutes a valid reference frame,

It might be a nitpick, but I'd add "inertial reference frame" here. There are size constraints on the size of an accelerated frame, as mentioned previously.

that's defined by velocity, so distance separated events along the direction of relative motion within one reference frame, S, can be seen as non simultaneous in another frame, S', even if they are always moving at the same velocity within frame S?

 

[Dale]

A and B share a frame at all times. There may be time dilation in their frame relative to some other frame but they cannot experience time dilation relative to their own frame, so there is still no cause for their separation to increase.

Their frame is non inertial, so the usual rules don’t apply. There can be time dilation at rest, as well as expansion and torsion.

Time running at different rates along the wire? The wire frame is one frame, there cannot be multiple rates of time in this one frame.

There can be “multiple rates of time” in one frame if the frame is non-inertial. This phenomenon is closely related to gravitational time dilation and the equivalence principle.

However, it is not likely (in my opinion) to help you learn electromagnetism. I think you have gone far off track if that is still your goal.

 

[A.T.]

Time running at different rates along the wire?

Yes:

https://en.wikipedia.org/wiki/Gravitational_time_dilation

Gravitational time dilation was first described by Albert Einstein in 1907 as a consequence of special relativity in accelerated frames of reference.

 

[bob012345]

A and B share a frame at all times. There may be time dilation in their frame relative to some other frame but they cannot experience time dilation relative to their own frame, so there is still no cause for their separation to increase.

I think what people are saying is that if the relative velocities were constant, that would, be always true but with acceleration, what one thinks of as an extended reference frame is no longer an inertial reference frame. It's all because of the acceleration not the relative velocity.

 

[Ibix]

I think what people are saying is that if the relative velocities were constant, that would, be always true but with acceleration, what one thinks of as an extended reference frame is no longer an inertial reference frame. It's all because of the acceleration not the relative velocity.

Not quite. The problem is that, when discussing relativity, you have to be precise in a way that sounds pedantic to laypeople. So laypeople leave out the "pointless pedantry" and end up hopelessly confused.

When you ask "is that guy moving at the same speed as me?" you actually mean "is his velocity zero in my rest frame at a time simultaneous with my watch reading 18.15.30 GMT?" Pedantic, right?

Unfortunately one of the more counterintuitive results of relativity is that simultaneity is frame dependent. So although everyone understands what you mean by the time on your watch, they don't agree on what "at the same time as" means for events at different locations. Example: if you use your James Bond watch-mounted gun to shoot someone, everyone will agree what your watch read when you pulled the trigger (same place), but not necessarily what it read when the bullet hit (different place).

Going back to the moving ships - when both are moving at constant speed none of this matters. If the answer to "how fast are they going right now?" is always 700m/s, no one cares what you mean by "now". But if they are accelerating then you will get different answers for the velocity if you mean different things by "now". Which is why there is only one frame where the ships have constant separation, which is their initial rest frame. The situation is made yet more complex by the fact that there are at least two plausible candidates for what is meant by "the rest frame of a ship" once they are accelerating. Neither resolves this problem.

 

[bob012345]

Not quite. The problem is that, when discussing relativity, you have to be precise in a way that sounds pedantic to laypeople. So laypeople leave out the "pointless pedantry" and end up hopelessly confused.

When you ask "is that guy moving at the same speed as me?" you actually mean "is his velocity zero in my rest frame at a time simultaneous with my watch reading 18.15.30 GMT?" Pedantic, right?

Unfortunately one of the more counterintuitive results of relativity is that simultaneity is frame dependent. So although everyone understands what you mean by the time on your watch, they don't agree on what "at the same time as" means for events at different locations. Example: if you use your James Bond watch-mounted gun to shoot someone, everyone will agree what your watch read when you pulled the trigger (same place), but not necessarily what it read when the bullet hit (different place).

Going back to the moving ships - when both are moving at constant speed none of this matters. If the answer to "how fast are they going right now?" is always 700m/s, no one cares what you mean by "now". But if they are accelerating then you will get different answers for the velocity if you mean different things by "now". Which is why there is only one frame where the ships have constant separation, which is their initial rest frame. The situation is made yet more complex by the fact that there are at least two plausible candidates for what is meant by "the rest frame of a ship" once they are accelerating. Neither resolves this problem.

Thanks, but can you rephrase that in one sentence with nontechnical language?

 

[Ibix]

Thanks, but can you rephrase that in one sentence with nontechnical language?

Frames agree on what "here and now" means but not what "now, over there" means, and this difference of opinion has consequences when you ask "what speed is that accelerating person over there doing now?".

That's a total cheat using a run-on sentence and an embedded question, but you'll have to live with it.

 

[bob012345]

Frames agree on what "here and now" means but not what "now, over there" means, and this difference of opinion has consequences when you ask "what speed is that accelerating person over there doing now?".

That's a total cheat using a run-on sentence and an embedded question, but you'll have to live with it.

Thanks. I can live with it till I review my Taylor and Wheeler book Spacetime Physics.

 

[Geocentricist]

I think what people are saying is that if the relative velocities were constant, that would, be always true but with acceleration, what one thinks of as an extended reference frame is no longer an inertial reference frame. It's all because of the acceleration not the relative velocity.

So if I drop two identical rocks from the same height at the same time, the rocks will disagree on who hit the ground first?

 

[Dale]

So if I drop two identical rocks from the same height at the same time, the rocks will disagree on who hit the ground first?

What does this have to do with electromagnetism? If that is still your goal then the answer to this question won’t help and asking it just encourages the thread to get further from your goal.

 

[PAllen]

So if I drop two identical rocks from the same height at the same time, the rocks will disagree on who hit the ground first?

Totally different case because the separation between the rocks is perpendicular to their acceleration rather than parallel. If you are not familiar with how fundamental this difference is in SR, then you really need to read any standard introduction to the subject, because we cannot teach you SR from the ground up.

 

[Geocentricist]

What does this have to do with electromagnetism? If that is still your goal then the answer to this question won’t help and asking it just encourages the thread to get further from your goal.

I've put off the electromagnetism thing for a bit until I understand why the electron spacing increases in their own frame. Also because I spent a lot of time with those graphics only to find they were wrong.

The thing about the rocks falling was supposed to be analogous to the electrons accelerating, but I think I messed that up. So I will rephrase it. If A and B are both stationary and both experience the same gravitational force (for example, a couple rocks sitting on the ground), will they disagree on what events are simultaneous?

 

[PAllen]

I've put off the electromagnetism thing for a bit until I understand why the electron spacing increases in their own frame. Also because I spent a lot of time with those graphics only to find they were wrong.

The thing about the rocks falling was supposed to be analogous to the electrons accelerating, but I think I messed that up. So I will rephrase it. If A and B are both stationary and both experience the same gravitational force (for example, a couple rocks sitting on the ground), will they disagree on what events are simultaneous?

No, because they are not accelerating relative to their starting position.

What is true is that if clocks spaced in the x direction start accelerating in the +x direction, then if they if they are to agree that their mutual distance doesn’t change, then the one further in the +x direction has to have less proper acceleration (and, in the original rest frame, they will get ever closer together). If, instead, they have the same proper acceleration, then they will both agree the distance between them is growing ( and in the original rest frame their distance will be constant).

Proper acceleration is what is measured by an accelerometer.

 

[Geocentricist]

No, because they are not accelerating relative to their starting position.

Okay, if two rocks floating in zero gravity are suddenly immersed in a gravitational field at the same time will they disagree that they were both immersed in the field at the same time?

What is true is that if clocks spaced in the x direction start accelerating in the +x direction, then if they if they are to agree that their mutual distance doesn’t change, then the one further in the +x direction has to have less proper acceleration (and, in the original rest frame, they will get ever slower together). If, instead, they have the same proper acceleration, then they will both agree the distance between them is growing ( and in the original rest frame their distance will be constant)

This does not help me understand. Sorry.

 

[Ibix]

Okay, if two rocks floating in zero gravity are suddenly immersed in a gravitational field at the same time will they disagree that they were both immersed in the field at the same time?

The answer to this depends on what magic you used to create a gravitational field out of nowhere, because that is not a situation you can describe in relativity. Also, you are struggling with special relativity. I strongly advise you to leave gravity out of your thinking because you need general relativity to describe that.

This does not help me understand. Sorry.

It's fairly straightforward. We've been discussing that the distance between two rockets grows in their own frames if they both undergo the same constant proper acceleration. So if they want the distance not to grow they have to undergo different proper accelerations.

 

[Geocentricist]

The answer to this depends on what magic you used to create a gravitational field out of nowhere, because that is not a situation you can describe in relativity.

The gravity field is uniform and just appears, poof! Like that.

Also, you are struggling with special relativity. I strongly advise you to leave gravity out of your thinking because you need general relativity to describe that.

A survey of opinion at CERN, according to Wikipedia, showed most theoretical physicists misunderstood Bell's paradox also. So I strongly advise you not to be condescending. I can use gravity since it's equivalent to acceleration.

It's fairly straightforward. We've been discussing that the distance between two rockets grows in their own frames if they both undergo the same constant proper acceleration. So if they want the distance not to grow they have to undergo different proper accelerations.

Yes, I understand the distance between A and B grows in either of their frames when they undergo identical acceleration. I get that, it's been repeated over and over in this thread. What I don't get is why, since if they were moving at an identical, constant velocity, it would not. How does acceleration make it any different?

 

[A.T.]

Yes, I understand the distance between A and B grows in either of their frames when they undergo identical acceleration. I get that, it's been repeated over and over in this thread. What I don't get is why, since if they were moving at an identical, constant velocity, it would not. How does acceleration make it any different?

This has also been explained over and over in this thread. Read up on accelerating frames of reference, and the effects occurring in those frames.

 

[Ibix]

The gravity field is uniform and just appears, poof! Like that.

Which, as I already said, is not a situation you can describe in relativity. It is inconsistent with Einstein's field equations. So this scenario is asking for the logical consequences of contradicting yourself. There is no answer.

I can use gravity since it's equivalent to acceleration.

This is a misstatement of the equivalence principle, which says that hovering in a gravitational field is indistinguishable (locally) from proper acceleration in flat spacetime. Your rocks would not be hovering, so would not be undergoing proper acceleration - so this scenario is not equivalent to accelerating rockets.

Yes, I understand the distance between A and B grows in either of their frames when they undergo identical acceleration. I get that, it's been repeated over and over in this thread. What I don't get is why, since if they were moving at an identical, constant velocity, it would not. How does acceleration make it any different?

I explained in #89, #96, #107, #113 and #115. Others have also answered. What did you not understand about those explanations? Please be specific. Simply repeating that you don't follow the explanations given has got you nowhere in the last 35 posts.

 

[jbriggs444]

IIf A and B are both stationary and both experience the same gravitational force (for example, a couple rocks sitting on the ground), will they disagree on what events are simultaneous?

Do they have a non-zero separation in the direction of the gravitational force? Is, for instance, one at the first floor and the other on the second floor? If so, the reference from @A.T. to gravitational time dilation is appropriate.

Note that gravitational time dilation is not a function of local gravitational acceleration. It is a function of potential.

 

[Dale]

I've put off the electromagnetism thing for a bit until I understand why the electron spacing increases in their own frame.

That has very little to do with rocks falling in gravity.

Also because I spent a lot of time with those graphics only to find they were wrong

That is because this approach is fundamentally wrong.

You are trying to draw the graphics in order to avoid the math. The correct approach is to draw graphics based on the math in order to understand the math. You have to start with the math before the graphics because the graphics are right or wrong based on if they accurately represent the math.

The graphics can help you understand the math, they cannot help you avoid the math.

 

[Geocentricist]

Which, as I already said, is not a situation you can describe in relativity. It is inconsistent with Einstein's field equations. So this scenario is asking for the logical consequences of contradicting yourself. There is no answer.

If I kick a soccer ball, does not a gravitational field suddenly (poof!) appear in the soccer ball frame?

This is a misstatement of the equivalence principle, which says that hovering in a gravitational field is indistinguishable (locally) from proper acceleration in flat spacetime. Your rocks would not be hovering, so would not be undergoing proper acceleration - so this scenario is not equivalent to accelerating rockets.

The rocks are hovering in the sense that the ground is preventing them from falling. It seems to me the situations are equivalent. They are experiencing 1G which could be described validly as either a gravitational field or an acceleration, depending on whether we use Earth's frame or the rocks' frame, no?

I explained in #89, #96, #107, #113 and #115. Others have also answered. What did you not understand about those explanations? Please be specific. Simply repeating that you don't follow the explanations given has got you nowhere in the last 35 posts.

I've been told in A's frame, B does not accelerate at the same time as A because their definition of simultaneity does not agree. But why does it not agree?

That is because this approach is fundamentally wrong.

There's nothing fundamentally wrong about my approach, since it can arrive at the correct answer as well as any mathematical formula. These are just two ways of describing the same idea. You can argue my idea is inefficient but to say it's wrong is incorrect. I enjoy doing it this way and I will be more confident that I understand it this way. As soon as I understand why electron spacing in their own frame increases, I will get back to my graphics. Or maybe before, if I give up on understanding the electron spacing thing.

 

[jbriggs444]

If I kick a soccer ball, does not a gravitational field suddenly (poof!) appear in the soccer ball frame?

First of all, the soccer ball is not a rigid object. So there is some ambiguity about when it starts moving and how fast it is moving.

Second, even if one picks a point on the ball and uses that point to define a reference frame, that reference frame is, per your scenario, accelerating. The motion of a single point can define an inertial reference frame. For an accelerating frame, picking a single point is not enough to define the frame.

[Which is to say that speaking of "the soccer ball frame" is premature -- no such frame is uniquely defined]

Third, if you have complete your definition of a particular accelerating frame, you will have had to come up with a simultaneity convention. How were you planning to make the simultaneity convention in the accelerating frame match the simultaneity convention in the inertial frame so that it would still correct to say that "poof", everything starts accelerating at once? Having adopted such a convention, what makes you sure that every point on a [Born-rigid] soccer ball will be accelerating at the same rate?

Fourth, all you have done is changed coordinates. A gravitational field in general relativity is more than just a change in coordinates. It is an invariant that is captured by a metric. You have not changed the metric -- you've just re-labelled the points on a [sub-]manifold.

 

[Ibix]

If I kick a soccer ball, does not a gravitational field suddenly (poof!) appear in the soccer ball frame?

No. If the presence or absence of a gravitational field were frame dependent I could fly by using a frame where there was no gravity. In the non-inertial frame of the ball inertial forces do appear, just as they do in the non-inertial frame of someone standing on the surface of the Earth. The use of a non-inertial frame is the similarity, not gravity.

The rocks are hovering in the sense that the ground is preventing them from falling. It seems to me the situations are equivalent. They are experiencing 1G which could be described validly as either a gravitational field or an acceleration, depending on whether we use Earth's frame or the rocks' frame, no?

You are modifying your scenario - there was no mention of a ground in your #122. In the case that there is a ground under the rocks then they are undergoing proper acceleration, but all the gravitational field is doing is making the situation more complex and removing the possibility of a global inertial frame to work in (@jbriggs444 has made this all the points so far in my post in significantly more detail).

Even in your revised form, the two rocks is a different scenario to the two rockets. They are accelerating in a Born rigid manner, which the rockets are not doing.

I've been told in A's frame, B does not accelerate at the same time as A because their definition of simultaneity does not agree. But why does it not agree?

It's a direct consequence of the two postulates of relativity. The usual demonstration is Einstein's train thought experiment, as pervect mentioned in #90.

 

[Geocentricist]

[4 reasons why kicking a soccer ball is too ambiguous]

A point-particle soccer ball (electron) is kicked (accelerated) on a soccer field (particle accelerator). Does not a gravitational field appear in the electron frame?

No. If the presence or absence of a gravitational field were frame dependent I could fly by using a frame where there was no gravity.

You could not. You'd be accelerating downwards.

You are modifying your scenario - there was no mention of a ground in your #122. In the case that there is a ground under the rocks then they are undergoing proper acceleration, but all the gravitational field is doing is making the situation more complex and removing the possibility of a global inertial frame to work in

So it's impossible to answer my question, do the two rocks agree on what events are simultaneous?

Even in your revised form, the two rocks is a different scenario to the two rockets. They are accelerating in a Born rigid manner, which the rockets are not doing.
It's a direct consequence of the two postulates of relativity. The usual demonstration is Einstein's train thought experiment, as pervect mentioned in #90.

Consider the two rocks as two electrons if it helps you answer my question.

I do not see how the increased electron-frame electron spacing follows from the two postulates of Special Relativity. Which postulate would be contradicted if the electron-frame electron spacing did not increase during the electrons' acceleration?

 

[Ibix]

I do not see how the increased electron-frame electron spacing follows from the two postulates of Special Relativity. Which postulate would be contradicted if the electron-frame electron spacing did not increase during the electrons' acceleration?

Do you understand Einstein's train thought experiment?

 

[jbriggs444]

A point-particle soccer ball (electron) is kicked (accelerated) on a soccer field (particle accelerator). Does not a gravitational field appear in the electron frame?

Again, what electron frame? A single accelerating point does not a frame define. Further, slapping the label of "gravity" on a coordinate chart does not change anything. It's still the same underlying physical reality, no matter what coordinates you use to refer to the events. All you are doing is changing coordinates.

 

[Geocentricist]

Do you understand Einstein's train thought experiment?

It seems to demonstrate the finite speed of light. What does it have to do with relativity?

A single accelerating point does not a frame define.

What information is lacking? The rate of acceleration? Direction?

 

[jbriggs444]

What information is lacking? The rate of acceleration? Direction?

Foliation (aka simultaneity convention).

For an inertial frame, there is one natural simultaneity convention. For an accelerating frame, things are not that simple.

If you do not understand Einstein's train experiment (as you do not) then you have not understood special relativity.

 

[Geocentricist]

If you do not understand Einstein's train experiment (as you do not) then you have not understood special relativity.

Enlighten me.

 

[Ibix]

It seems to demonstrate the finite speed of light.

No. It uses the finite speed of light to demonstrate that "two things happen at the same time" is not a complete sentence. You need to add "...as measured in frame S", and other frames will not agree. Please go and study the train thought experiment until you understand - or at least, accept - that. It's critical to understanding what's going on with the accelerating rockets.

What does it have to do with relativity?

Pretty much everything.

 

[jbriggs444]

Enlighten me.

Show us that you are willing to make the effort to understand it.

Edit: Look at https://en.wikipedia.org/wiki/Relativity_of_simultaneity (for instance), attempt to understand it and then ask more focused questions. "Enlighten me" does not measure up.

 

[Geocentricist]

I actually just solved my problem to my own satisfaction. Here is my reasoning. The two spaceships accelerate at the same time in S, so their separation (between midpoint of each spaceship) remains constant, while the length of each spaceship (and the string) contracts. This breaks the string.

In order to break the string in the spaceship frame where the spaceships are motionless and thus not contracted themselves, the only way is to increase the separation of the spaceships.

So now I understand the separation in the rest frame must increase in order to agree with phenomena in the S frame.

Finally I can get back to electromagnetism. How much will the separation between electrons increase if they accelerate to 0.87 c? Will it double? I want to correct my graphics.

 

[Ibix]

That explanation works as long as you are happy to ignore the acceleration phase and just consider "before and after". There are a number of complications during the acceleration.

Finally I can get back to electromagnetism. How much will the separation between electrons increase if they accelerate to 0.87 c? Will it double?

Yes. Note that this is a highly unrealistic electron drift velocity. Typical speeds are around 10^-5m/s, yielding a separation change of around one part in 10²⁶.

 

[Dale]

There's nothing fundamentally wrong about my approach, since it can arrive at the correct answer as well as any mathematical formula.

I disagree. As was pointed out earlier, you needed other people (using the mathematical formulas) to tell you the strength of each interaction. Your approach can help you understand the math, but it simply cannot substitute for the math.

You can argue my idea is inefficient but to say it's wrong is incorrect

It is not only inefficient, it is also insufficient.

But I am not the one who said your graphs were wrong, that was you. I am just telling you why: they are wrong because they are the outcome of a fundamentally flawed approach.

Another fundamental flaw of this approach is that you are attempting to “explain electromagnetism with relativity” when you don’t know relativity. How does that make sense as an approach? In order to avoid learning electromagnetism directly you are now trying to learn general relativity, but general relativity is a substantially more difficult subject.

If an advanced second grader asked you to explain multiplication tables using logarithms when they don’t know logarithms, would you not recommend a different approach to multiplication?

How much will the separation between electrons increase if they accelerate to 0.87 c? Will it double? I want to correct my graphics.

Case in point...

 

[Geocentricist]

That explanation works as long as you are happy to ignore the acceleration phase and just consider "before and after".

Awesome! That's fine since I'm not illustrating the acceleration.

As was pointed out earlier, you needed other people (using the mathematical formulas) to tell you the strength of each interaction. Your approach can help you understand the math, but it simply cannot substitute for the math.

You're right.

 

[Geocentricist]

How's this? Is this correct? Obviously didn't put as much effort as I did in the other ones.

 

[A.T.]

Is this correct?

DrGreg's diagram is better because it shows at least two electrons in each case, so you see their spacing:

For two identical currents duplicate the lower part of the loop.

For two opposite currents look at the lower and the upper part of the loop.

 

[Geocentricist]

DrGreg's diagram is not suited for my purposes since there are too many particles and also, not a simple ratio of electrons to protons in each frame. Is mine correct?

 

[A.T.]

DrGreg's diagram is not suited for my purposes since there are too many particles ...

Just enough to have at least two electrons in each case, so you see their spacing. The perfect amount.

...and also, not a simple ratio of electrons to protons in each frame.

What is a simple ratio? What non simple ratios do you see in DrGregs diagram?

Is mine correct?

It doesn't show enough particles to tell.

 

[Geocentricist]

Just enough to have at least two electrons in each case, so you see their spacing. The perfect amount.What is a simple ratio? What non simple ratios do you see in DrGregs diagram?

In one frame the ratio is 0.25 which is harder to work with than mine which is 0.5. Not a big difference but I want to keep things simple as possible!

It doesn't show enough particles to tell.

Sure it does, the ratio is all that matters and you must assume the segment I've shown is representative of the ratio of the entire wire

 

[A.T.]

In one frame the ratio is 0.25 which is harder to work with than mine which is 0.5.

The ratios are 1:1, 1:2 and 1:4. Doesn't get simpler than that, if you want to show all cases (3 ratios).

 

[Geocentricist]

The ratios are 1:1, 1:2 and 1:4. Doesn't get simpler than that, if you want to show all cases (3 ratios).

Ah you're right.

I have a question about the magnetic force. If the strength of the electric repulsion between two electrons co-moving at 87% the speed of light is 1 then is the strength of the magnetic attraction between them 0.5? And if the electrons are moving in opposite directions is the magnetic force repulsive or is it simply halved in strength and still attractive?