Magnetism seems absolute despite being relativistic effect of electrostatics

[Dale]

Then, in post 107, I asked you to explain how that works. Namely how "only an E-field" could be responsible for what is otherwise known as electromagnetic induction in a piece of iron by a moving electron. I'm still waiting for an explanation.

Yes, you are referring to electrostatic induction. The problem is, the question is about electromagnetic induction. You claim they are equivalent. Okay, demonstrate the equivalence. Merely claiming this or that and calling a question resolved is not satisfactory in science.

See Figure 1.1 here:
http://www.ece.drexel.edu/courses/ece-e304/e3042/CLICK_HERE_TO_VIEW.htm

It describes the scenario as seen in the electron's frame, with the additional complication that the field from the electron is spatially non-uniform, but as the conductor moves through the spatially non-uniform field the charges and currents re-distribute along the principles sketched out there. I hope it is clear now how an E-field in the electron's frame can produce the currents and charges that would be expected in the conductor's frame where there is an E- and a B-field.

 

[Dale]

What I'm asking is for you to show your work, if you feel up to it.

If you are willing to work the problem out in the conductor's frame and post it then I can transform it to the electron's frame and post how it also works there.

 

[lugita15]

I can't open PDFs on this computer.

OK, I've uploaded it to Google Docs here.

 

[Subplotsville]

If you are willing to work the problem out in the conductor's frame and post it then I can transform it to the electron's frame and post how it also works there.

Since you like to reduce things to electrostatics, in the interest of clarity and to avoid more of the same inconclusive back-and-forth, let's get rid of the conductor and rephrase the problem with only essential elements remaining.

There are two electrons in close proximity. They are stationary to each other and in parallel motion -- in a direction perpendicular to the line that joins them -- with respect to an observer. This observer sees them as having two mutual interactions: 1) electrostatic and 2) magnetic on account of their being moving charges. Yet each electron sees the other as having only an electric field. How is this reconciled?

OK, I've uploaded it to Google Docs here.

Maybe you could copy and paste the part where he addresses electromagnetic induction by an isolated charge, since this forum has a format we know everyone's computers, including mine, can actually read.

 

[chingel]

Not an expert on this subject by any means, but since the electrons are moving with respect to the lab frame, they would experience time dilation, making them accelerate less in the same amount of time as measured by a clock in the lab frame, resulting in a conclusion that they feel less force, since they accelerated less. So in the lab frame they would seem to repel each other with less force, as the magnetic fields and forces would predict.

 

[Dale]

There are two electrons in close proximity. They are stationary to each other and in parallel motion -- in a direction perpendicular to the line that joins them -- with respect to an observer. This observer sees them as having two mutual interactions: 1) electrostatic and 2) magnetic on account of their being moving charges. Yet each electron sees the other as having only an electric field. How is this reconciled?

This is a fine scenario, and I agree it captures all of the essential elements. My offer still stands, work the problem in one frame and post your work then I will transform it to the other frame and show how it works there.

 

[Subplotsville]

Not an expert on this subject by any means, but since the electrons are moving with respect to the lab frame, they would experience time dilation, making them accelerate less in the same amount of time as measured by a clock in the lab frame, resulting in a conclusion that they feel less force, since they accelerated less. So in the lab frame they would seem to repel each other with less force, as the magnetic fields and forces would predict.

Time dilation would cancel some of the electrostatic repulsion in this case, but not nearly enough to account for the magnetic attraction which increases linearly with speed and so becomes a significant factor even at slow speeds. Also, there's the situation where the two electrons are passing each other going in opposite directions, in which case their magnetism causes an increase rather than a decrease in their mutual repulsion.

 

[rbj]

Thanks, though I'm looking for an in-thread explanation, if possible. More forum readers will benefit from it that way.

The classical electromagnetic effect is perfectly consistent with the lone electrostatic effect but with special relativity taken into consideration. The simplest hypothetical experiment would be two identical parallel infinite lines of charge (with charge per unit length of \lambda and some non-zero mass per unit length of \rho separated by some distance R. If the lineal mass density is small enough that gravitational forces can be neglected in comparison to the electrostatic forces, the static non-relativistic repulsive (outward) acceleration (at the instance of time that the lines of charge are separated by distance R) for each infinite parallel line of charge would be:

a = \frac{F}{m} = \frac{ \frac{1}{4 \pi \epsilon_0} \frac{2 \lambda^2}{R} }{\rho}

If the lines of charge are moving together past the observer at some velocity, v, the non-relativistic electrostatic force would appear to be unchanged and that would be the acceleration an observer traveling along with the lines of charge would observe.

Now, if special relativity is considered, the in-motion observer's clock would be ticking at a relative rate (ticks per unit time or 1/time) of \sqrt{1 - v^2/c^2} from the point-of-view of the stationary observer because of time dilation. Since acceleration is proportional to (1/time)², the at-rest observer would observe an acceleration scaled by the square of that rate, or by {1 - v^2/c^2}, compared to what the moving observer sees. Then the observed outward acceleration of the two infinite lines as viewed by the stationary observer would be:

a = \left(1 - v^2 / c^2 \right) \frac{ \frac{1}{4 \pi \epsilon_0} \frac{2 \lambda^2}{R} }{\rho}

or

a = \frac{F}{m} = \frac{ \frac{1}{4 \pi \epsilon_0} \frac{2 \lambda^2}{R} - \frac{v^2}{c^2} \frac{1}{4 \pi \epsilon_0} \frac{2 \lambda^2}{R} }{\rho} = \frac{ F_e - F_m }{\rho}

The first term in the numerator, F_e, is the electrostatic force (per unit length) outward and is reduced by the second term, F_m, which with a little manipulation, can be shown to be the classical magnetic force between two lines of charge (or conductors).

The electric current, i_0, in each conductor is

i_0 = v \lambda

and the magnetic permeability is

\mu_0 = \frac{1}{\epsilon_0 c^2}

because c^2 = \frac{1}{ \mu_0 \epsilon_0 } so you get for the 2^nd force term:

F_m = \frac{v^2}{c^2} \frac{1}{4 \pi \epsilon_0} \frac{2 \lambda^2}{R} = \frac{\mu_0}{4 \pi} \frac{2 i_0^2}{R}

which is precisely what the classical E&M textbooks say is the magnetic force (per unit length) between two parallel conductors, separated by R, with identical current i_0.

 

[Dale]

Time dilation would cancel some of the electrostatic repulsion in this case, but not nearly enough to account for the magnetic attraction which increases linearly with speed and so becomes a significant factor even at slow speeds.

That is a very specific claim, can you prove it?

 

[Subplotsville]

That is a very specific claim, can you prove it?

The proof is implicit in what you quoted. The magnetic field is linear with speed, whereas time dilation is not and in fact increases negligibly at low speeds. Therefore, the magnetic attraction canceling the repulsion between the two electrons in parallel motion is not accounted for by time dilation. This is sufficient to refute the proposition without going into quantitative details.

 

[Dale]

The proof is implicit in what you quoted. The magnetic field is linear with speed, whereas time dilation is not and in fact increases negligibly at low speeds. Therefore, the magnetic attraction canceling the repulsion between the two electrons in parallel motion is not accounted for by time dilation. This is sufficient to refute the proposition without going into quantitative details.

Your logic would be sound if the magnetic field were linear in time dilation. Can you prove that it is?

Btw, I think that your overall point is probably correct, i.e. I think that you need all of relativity, not just time dilation. But your reasoning is unsound.

 

[Subplotsville]

Your logic would be sound if the magnetic field were linear in time dilation. Can you prove that it is?

Btw, I think that your overall point is probably correct, i.e. I think that you need all of relativity, not just time dilation. But your reasoning is unsound.

Proof at relativistic speeds is unnecessary. It is only necessary to look at the linearity of the magnetic field with changing speed at the low end of speed: where relativistic effects are close to flat and can thus be ignored over small changes in speed. Meaning, the two (mutually at rest) electrons exchange magnetic forces that vary linearly (to the observer) even where relativistic effects vary negligibly.

 

[Dale]

Proof at relativistic speeds is unnecessary.

Agreed. I was talking about small speeds.

It is only necessary to look at the linearity of the magnetic field with changing speed at the low end of speed: where relativistic effects are close to flat and can thus be ignored over small changes in speed. Meaning, the two (mutually at rest) electrons exchange magnetic forces that vary linearly (to the observer) even where relativistic effects vary negligibly.

You are assuming that the relativistic effects can be ignored. You cannot assume the very point in question, that is a logical fallacy called begging the question.

In your argument above you started with three correct premises:
1) that the time dilation would cancel some of the electrostatic force
2) that the magnetic force is linear in v
3) that time dilation is not linear in v

But the fact that time dilation is not linear in v is not relevant. Time dilation is not force. What we are interested in is not whether or not time dilation is linear in v but whether or not the force canceled out by time dilation is linear in v.

Again, I think your conclusion is likely to be sound, but your argument is invalid.

 

[lugita15]

Maybe you could copy and paste the part where he addresses electromagnetic induction by an isolated charge, since this forum has a format we know everyone's computers, including mine, can actually read.

The excerpt I gave before was about how the magnetic force arises from the elecrostatic force and relativity, which is what I thought you were asking about. If you want to know about electromagnetic induction, here is another excerpt from Purcell, both in PDF format (attached) and in Google Docs, containing a relativistic analysis of Faraday's law. It may not be possible for me to copy and paste from it. Why exactly are you not able to view the Google Docs version?

 

[Tantalos]

Agreed. I was talking about small speeds.

You are assuming that the relativistic effects can be ignored. You cannot assume the very point in question, that is a logical fallacy called begging the question.

In your argument above you started with three correct premises:
1) that the time dilation would cancel some of the electrostatic force
2) that the magnetic force is linear in v
3) that time dilation is not linear in v

But the fact that time dilation is not linear in v is not relevant. Time dilation is not force. What we are interested in is not whether or not time dilation is linear in v but whether or not the force canceled out by time dilation is linear in v.

Again, I think your conclusion is likely to be sound, but your argument is invalid.

What about length contraction? Why do we not take into account the fact that the charge is length contracted on the lines when they move?

 

[Dale]

What about length contraction? Why do we not take into account the fact that the charge is length contracted on the lines when they move?

Yes, that is why I suspect that the conclusion is probably correct. I don't think time dilation alone can explain it in all cases, I suspect that all relativistic effects are required.

 

[rbj]

What about length contraction? Why do we not take into account the fact that the charge is length contracted on the lines when they move?

Yes, that is why I suspect that the conclusion is probably correct. I don't think time dilation alone can explain it in all cases, I suspect that all relativistic effects are required.

the simple analysis i did that also got to the correct conclusion, but used nothing other than time-dilation. the mass per unit length \rho gets bumped up by a factor of \gamma = \left(1-\frac{v^2}{c^2}\right)^{-1/2} because of length contraction. and it gets bumped up by another factor of \gamma because of relativistic mass. that's in the denominator.

in the numerator there is the charge per unit length gets bumped up by \gamma because of length contraction. the question is should the quantity of charge itself be affected by \gamma? if not, there is \gamma^2 in the numerator and \gamma^2 in the denominator and it comes out in the wash. so the question is, is charge invariant under special relativity? i think i was told it was, but it is curious that mass is affected but charge is not, both intrinsic properties of a particle like an electron.

 

[Subplotsville]

But the fact that time dilation is not linear in v is not relevant. Time dilation is not force. What we are interested in is not whether or not time dilation is linear in v but whether or not the force canceled out by time dilation is linear in v.

The logical step connecting time dilation and a change in force was omitted because it was already described in someone's post. The idea being that the electrons have a slower clock and therefore measure less acceleration by the electrostatic repulsion between them. At low speeds this effect is virtually flat with a change in speed, while the magnetic force goes up in proportion to speed. On this basis alone, the two effects are not equivalent. This leaves the lab observer unable to account for the two electrons not seeing the decreased net repulsion between themselves which he attributes to the magnetic attraction caused by their parallel motion.

BTW, I'm not sure a slower clock would measure less acceleration. Since the time variable is in the denominator, less time means more acceleration. But this doesn't matter since the non-linearity problem disqualifies it anyway.

Why exactly are you not able to view the Google Docs version?

This computer is restricted as to what online file types it can access, for security reasons. Pretty much only html, images and a few others. Sorry about that.

What about length contraction? Why do we not take into account the fact that the charge is length contracted on the lines when they move?

Length contraction shouldn't be a factor in this case. It is only in the direction of motion. The electric and magnetic forces between the two electrons are perpendicular to the direction of their motion. The lab sees the electrons with a velocity perpendicular to the line joining the electrons, which see each other as at rest.

 

[Dale]

The idea being that the electrons have a slower clock and therefore measure less acceleration by the electrostatic repulsion between them. At low speeds this effect is virtually flat with a change in speed, while the magnetic force goes up in proportion to speed. On this basis alone, the two effects are not equivalent.

No, at low speeds the gamma factor is virtually flat, that does not imply that the less acceleration effect is virtually flat. That is the part which you have not proved and which is not implied by the comments you have made. You are simply assuming your conclusion, aka begging the question.

Btw, there is another way to resolve this issue, simply take me up on my offer from above.

 

[Dale]

so the question is, is charge invariant under special relativity? i think i was told it was, but it is curious that mass is affected but charge is not, both intrinsic properties of a particle like an electron.

The integral of charge density over all of space is invariant. Charge density itself is part of the four-current.

The idea of relativistic mass is not in much use today, so most people wouldn't say that mass is affected. They would just use the invariant mass which is obviously invariant. Using relativistic mass can be particularly problematic in this kind of problem when you are doing accelerations in directions perpendicular to the motion.

 

[Subplotsville]

No, at low speeds the gamma factor is virtually flat, that does not imply that the less acceleration effect is virtually flat.

Acceleration (the second derivative of position versus time) varies inversely with the square of time. The slowed clock effect is so flat at low speeds (see Lorentz equation) that the corresponding change in measurements of acceleration is also virtually flat. The equations are what imply the virtual flatness. But this is beside the point anyway, since it turns out that a slower clock measures greater acceleration, while the lab observer is trying to account for (the electrons' observation of) the decrease in acceleration he ascribes to their magnetic fields.

 

[Dale]

The slowed clock effect is so flat at low speeds (see Lorentz equation) that the corresponding change in measurements of acceleration is also virtually flat.

Begging the question again.

 

[Subplotsville]

Begging the question again.

See my previous post. See also the relevant Lorenz equation, as mentioned. It is not begging the question. All the information you need is there. I don't know how else to explain it. Do you not understand how changing measurement of time effects measurement of acceleration, or how squaring next-to-nothing yields next-to-nothing in the present case, or how this is an irrelevant digression anyway because time dilation increases measured acceleration while the lab observer is looking for a decrease?

 

[Dale]

But this is beside the point anyway, since it turns out that a slower clock measures greater acceleration, while the lab observer is trying to account for (the electrons' observation of) the decrease in acceleration he ascribes to their magnetic fields.

This is incorrect. The time dilation is in the correct "direction". Consider that the charges are initially at rest wrt each other and some apparatus with distances marked along the path they will travel. Suppose that in the apparatus' frame the electrons leave the initial mark at t=0 and reach the first mark at t=1. In the moving frame (moving perpendicular to the apparatus) they were released at t'=0 and reached the first mark at t'=γ>1. So since it took more time to get to the same location that corresponds to a reduced acceleration in the frame where the charges are moving. This is the correct "direction" for the magnetic force.

 

[Dale]

It is not begging the question. All the information you need is there.

Maybe, but you haven't proved it.

Similarly, I can simply point you to Maxwell's equations and the Lorentz transform and the writings of Einstein and Purcell and others. All the information you need is there.

 

[Dale]

Oh, one other thing. The magnetic force does not increase linearly with speed at low speeds. For a constant external magnetic field the magnetic force on a charge is proportional to the speed of the charge. However, in this scenario the external magnetic field is not constant but depends on the speed of the other charge.

 

[Subplotsville]

This is incorrect. The time dilation is in the correct "direction". Consider that the charges are initially at rest wrt each other and some apparatus with distances marked along the path they will travel. Suppose that in the apparatus' frame the electrons leave the initial mark at t=0 and reach the first mark at t=1. In the moving frame they were released at t'=0 and reached the first mark at t'=γ>1. So since it took more time to get to the same location that corresponds to a reduced acceleration in the frame where the charges are moving. This is the correct "direction" for the magnetic force.

This does not fit the present case. The electrons have the slower clock and therefore measure less time for whatever increase in distance due to electrostatic repulsion takes place between them. (Since this distance is perpendicular to their velocity according to the lab observer, it is unaffected by their motion.) Time dilation is doing the opposite of what the lab observer seeks to explain how the two electrons account for the decreased repulsive force between them.

Oh, one other thing. The magnetic force does not increase linearly with speed at low speeds. For a constant external magnetic field the magnetic force on a charge is proportional to the speed of the charge. However, in this scenario the external magnetic field is not constant but depends on the speed of the other charge.

What? The electrons are at rest wrt each other. The only speed we're looking at is the speed of the two electrons wrt the lab observer. That is the cause of their magnetic fields.

 

[Dale]

This does not fit the present case. The electrons have the slower clock and therefore measure less time for whatever increase in distance due to electrostatic repulsion takes place between them. (Since this distance is perpendicular to their velocity according to the lab observer, it is unaffected by their motion.) Time dilation is doing the opposite of what the lab observer seeks to explain how the two electrons account for the decreased repulsive force between them.

Yes, that is what I said above. The electrons measure less time (greater acceleration, electrostatic force only). The lab measures more time (less acceleration, electrostatic force minus magnetic force). That is correct, not opposite.

What? The electrons are at rest wrt each other. The only speed we're looking at is the speed of the two electrons wrt the lab observer. That is the cause of their magnetic fields.

Understood, and so the magnetic force in the lab is not proportional to the velocity in the lab because the magnetic force depends both on the velocity of the charge in the lab and the strength of the external field, which in turn depends on the velocity of the other charge in the lab.

 

[Subplotsville]

Yes, that is what I said above. The electrons measure less time (greater acceleration, electrostatic force only). The lab measures more time (less acceleration, electrostatic force minus magnetic force). That is correct, not opposite.

This is correct. The previous poster who brought this up originally was right to do so. Nevertheless, the fact that this effect is virtually flat at low speeds per the Lorentz equation, compared to how the magnetic field varies directly with speed, remains relevant. Do you not accept this discrepancy? I'm still not sure exactly what you disagree with. If you look at the Lorentz equation (which this browser has trouble with, so I won't try to post it), you will see how time dilation varies with speed. Though acceleration does vary with the square of time, you're still nowhere near how the magnetic field varies with the speed of the moving charge generating it. If you could clarify precisely where you differ with this reasoning rather that just saying "begging the question," that would be great.

Understood, and so the magnetic force in the lab is not proportional to the velocity in the lab because the magnetic force depends both on the velocity of the charge in the lab and the strength of the external field, which in turn depends on the velocity of the other charge in the lab.

This is not about an electric charge moving through a magnetic field. The lab sees the two electrons as 1) mutually stationary like charges causing a repulsion and 2) mutually stationary magnets with fields aligned to cause an attraction. On the other hand, the electrons only see each other as like charges.

 

[Dale]

Nevertheless, the fact that this effect is virtually flat at low speeds per the Lorentz equation, compared to how the magnetic field varies directly with speed, remains relevant. Do you not accept this discrepancy?

Clearly not. That is why continuing to assert it is begging the question. You are asking me to accept as a premise the very point under discussion without any justification other than your continued assertion.

You need to do more than just state that time dilation is "virtually flat", you need to show that it does not account for the magnetic force.

Though acceleration does vary with the square of time, you're still nowhere near how the magnetic field varies with the speed of the moving charge generating it.

This is the part that you haven't shown. You are simply asserting that we are "nowhere near" the required force without showing it.

This is not about an electric charge moving through a magnetic field. The lab sees the two electrons as 1) mutually stationary like charges causing a repulsion and 2) mutually stationary magnets with fields aligned to cause an attraction.

They are mutually stationary, but that is not relevant in Maxwell's equations nor the Lorentz force law. There is no "mutual velocity" term in either of those equations. All of the velocity terms in Maxwell's equations and the Lorentz force term are measured wrt an inertial reference frame.

So, in the lab frame you have two moving electrons, each generating an electric and a magnetic field. Each electron then experiences an electric and a magnetic force due to their motion through the field from the other electron.