## Wednesday, March 2, 2011

### Electricity, magnetism, space, and time

[One of my most popular essays is "Relativity + Electrostatics =  Magnetism", which is about how the electric force, combined with Special Relativity, explain the existence of the magnetic force.  I think this is a topic worth revisiting, because I am a better writer than I used to be, and can make better pictures.]

The magnetic force is weird.  You're all familiar with bar magnets, which have a north and south pole.  Opposite poles attract and like poles repel.  But even though bar magnets are very familiar to us, they're relatively complicated, so we'll start with a more basic example.

A simple model of two wires.  In each wire, the electrons (in blue) are moving to the left, while the atomic nuclei (in red) remain stationary.

Two wires.  Wires conduct electric current, meaning that the negatively charged electrons move to the left or the right.*  The positively charged atomic nuclei remain stationary.  If the two wires have current going in the same direction, then they attract.  If they go in opposite directions, then they repel.

*I am making a simplification, since this is not true for all materials that the wire could be made of.  For reasons that I won't get into, some materials conduct electricity by things other than electrons.

Bar magnets are sort of like circular currents.  If you have a north and south pole together, these currents are going in the same direction, and they attract each other.  If you have a north and a north pole together, these currents are going in opposite directions, and they repel.

On the left, the circular currents (dark red arrows) are going in the same direction, making the magnets attract.  On the right, the currents are going in opposite directions, making the magnets repel.  Alternatively, we can think of the magnets as having north and south poles, where opposite poles attract and like poles repel.

But let's return to the wires, because they're simpler.  The reason the wires attract is specifically because the electrons attract.  One way we know this is the electrons tend to gather towards each other at the closer edges of the wire (this is called the Hall Effect).  The electrons drag the rest of the wire with them, thus causing the wires to attract.

The reason why the electrons attract and the atomic nuclei don't is because that's just how the magnetic force works.  The magnetic force is proportional to the speed of the particle.  The electrons are moving, so they attract each other.  The atomic nuclei are not moving, so they don't attract each other.

But hold on!  Doesn't the speed of a particle depend on which way you look at it?  What if you put these wires on a moving truck, won't all the electrons and nuclei be moving faster than before?  For that matter, what if you put these wires on a moving Earth, doesn't that affect the speed?  Why is it that we can use motors and generators (which both require magnetic fields) without worrying about the earth's motion?

This question was the motivation for Special Relativity Theory.  You may have heard about Einstein's Theory of Relativity as something that radically alters our notions of space and time.  But that wasn't really the point of the theory.  The point was to explain something about magnetic fields.  The radical view of space and time was just an added bonus.

Einstein's Special Relativity is sort of like an expansion of rotations.  If you rotate your head 90 degrees, and look around you, several things change.  Ceilings become walls and walls become floors.  Right-left becomes up-down, and up-down becomes left-right.  Everything that was pointing in one direction (like gravity, which originally pointed down) is now pointing in a different direction.  But physics behaves the same way, just rotated.

The same is true if you're moving at constant velocity.  If I'm on a moving truck, things that were previously stationary are now zipping behind us.  If the air was previously still, it now becomes wind at our faces.  But physics behaves the same way, it's just that directions have changed.  In this case, we don't call it a rotation, we call it a boost.

According to Special Relativity, a boost does not just affect the motion of objects.  It also affects the electric and magnetic fields.  Just as right-left became up-down, and up-down became left-right when we rotated, electric fields can become magnetic fields and magnetic fields can become electric fields.  It's not quite like rotation (the math is harder), but the idea is the same.  A boost causes electric and magnetic fields to mix into each other.

And of course, it's not just electric and magnetic fields that mix into each other, it's space and time too.  But at everyday speeds this is difficult to observe, because they're such small boosts.  It's like rotating your head a fraction of a degree.  What was once horizontal is still mostly horizontal (but with a slight vertical component).  Similarly, at everyday speeds, what was once a distance is still mostly a distance (but with a small time component).

The mixing of electric and magnetic fields is deeply intertwined with the mixing of space and time.  But this is very difficult to demonstrate without getting into the mathematics of boosting.  There is one example where it is easy to demonstrate, and that is the example I started with.  Two wires.

In a real wire, the electrons are not really moving at the same speed.  But let's simplify and imagine that they are.  And then let's boost to a perspective where the electrons are not moving at all.

Two wires, boosted.

After we've boosted, it is no longer the case that the electrons are moving left while the atomic nuclei are stationary.  Now it is the electrons that are stationary while the atomic nuclei are moving right.

Since the magnetic force is proportional to speed, it can no longer affect the motionless electrons.  But this is easy to explain in terms of Relativity: what was once magnetic fields have now changed into electric fields.  The electric force causes the electrons in each wire to attract.  The same electric force also cause the atomic nuclei to repel, but that's okay!  The atomic nuclei have some speed, and are thus attracted by the magnetic force.  The magnetic force and electric force on the nuclei cancel out.

So we really have the same picture as before.  The electrons attract (but now due to the electric force, not the magnetic force), and they drag the entire wire along with them.

Now I will connect this to time and space.

If you look at my diagram, you'll notice I did something sneaky.  After the boost, there are more nuclei than there are electrons.  This is due to the mixing of time and space.  Previously, the protons had some spacing, some distance between them.  But as time and space mix, some of the distance changes into a time component, and the distance becomes shorter.  It's called Lorentz contraction.  When we boost from a frame where nuclei are stationary to a frame where they are moving, they become closer together.

Similarly, when we boost from a frame where electrons are moving to a frame where they are stationary, they become further apart.  It's backwards Lorentz contraction.

The end result is that each wire now has a positive charge because the nuclei are more dense than the electrons.  Electrons are attracted to positive charge by the electric force!

And so, there are two methods of solving the problem.  The first method is to mix the electric and magnetic fields through the mathematics of boosting.  The second method is to mix space and time, and then look at the resulting electric and magnetic forces.  Both of these methods are equivalent.

Hold on!  I know you have some questions.

Does that mean that the magnetic force isn't real?  No.  I'm not sure that it's meaningful to talk about whether these things are "real" or not.  What it does tell you is that the electric force, magnetic force, and Special Relativity are all connected.  Physics wouldn't make sense if you only had two out of the three.

What happened to the missing electrons?  The number of electrons does not change when we boost, so they must have gone somewhere.  The reason it seems they disappeared is because I only drew part of the wires.  For a wire to conduct electricity, you need to have the wire in a loop to complete the circuit.  The electrons will gather in the parts of the loop where they were going the opposite direction. There, they will be doubly Lorentz contracted.

What about two electrons by themselves moving alongside each other?  In the wire example, electric and magnetic fields don't change over time.  In the two electron example, they do change over time.  This significantly complicates the way things work.  But suffice it to say that the faster the electrons move, the stronger the electric force repelling them.  The magnetic force between the electrons is counteracted by this stronger electric force.

Aren't the electrons all moving at different speeds?  Yes.  But it works out the same.  It makes the math a lot more complicated.  The entire premise of this post is to simplify to an example where I don't have to show that.  If you would like to see it, you should be studying physics academically, not from blogs.

Is Lorentz contraction really that big?  No.  I exaggerated it so you can see it clearly.  In a typical wire, the electrons are moving with an average velocity of 10-5 meters per second, which corresponds to a Lorentz contraction of 1 part in 1027.  If it seems impressive that such a small Lorentz contraction can create a measurable force, that's because it is.

I think I followed what you were saying, but I'm not quite sure I got what you were trying to convey.

Are you saying that what we see/measure as magnetism is a result of time-space being altered by the flow of electrons?

miller said...

The electrons themselves don't alter space-time. Rather, we are simply looking at the electrons from different reference frames. And when we switch reference frames, both space and time are altered.

I would not say magnetism is a "result" of relativity, any more than relativity is a "result" of magnetism. I would simply say that they are connected in an intimate way, and one would not make sense without the other.

Tom Telos said...

Bless you!
For the past seven hours —shameful, but true—
I’ve been surfing the Web in search of this blog of yours,
until finally I remembered the exact phrase you used,
and there it was: the original one!

It irks me —but, pat yourself on the back— that
(of more than a hundred web pages I visited,
that deal with Einstein and/or his relativity)
NOBODY blurts this out like you do!

Something that ALL of Humanity ought to be aware of,
from Middle School onwards!

Have you ever tried to teach this to a class
in Middle School? It works! I personally guarantee it.

Anyway, I celebrate not only your original blog,
but the fact that you respect your own writing enough
to come back years later and improve it: congratulations.

Your improved version is now referenced
from a journal entry I recently wrote, here:
http://ttnn.deviantart.com/journal/Shortcut-through-space-confirmed-269770488

Thank you!

Observer said...

Considering two wires complicates things because there are both electrostatic and magnetic forces in both frames of reference.

It is much simpler to consider a wire with current and a solitary electron traveling next to it at the same speed as the electrons in the wire.

In the frame in which the wire is stationary, there is no electrostatic force between the wire and the electron because the wire is electrically neutral. However, there is a magnetic force because of the velocity of the solitary electron and the velocity of the electrons in the current.

Next consider the frame in which the electrons are stationary. Now there is no magnetic force on the solitary electron because it is stationary. However, because of the Lorentz length contraction of the protons in the wire, the wire is now electrically positive and so there is a net electrostatic attraction between the wire and the solitary electron.

Hi,

I've done an extensive analysis for two moving point charges.
I've started with the time dilution & relativistc force law instead of the length contraction...

Hope you can review it!

Evi1M4chine said...

> The reason why the electrons attract and the atomic nuclei don't is because that's just how the magnetic force works.

Seriously??

That’s nonsensical circular reasoning. If you don’t know the actual reason, don’t write such texts. And if you know it, also don’t write such nonsense but explain it.

Because it is exactly where people stop to follow, close the thing and never come back.

miller said...

evilmachine,
The explanation was in the next few sentences.

danky said...

At one point you say:
"So we really have the same picture as before.  The electrons attract (but now due to the electric force, not the magnetic force), and they drag the entire wire along with them."

This is completely wrong. electrons repel each other electrostatically...In fact with 2 wires you cannot illustrate the role of relativity because you are stuck explaining the attraction as the magnetic force between the now moving positive nuclei.

Instead, as another comment illustrated, you can completely do away with magnetic field to explain the attraction betweeen a wire and a moving point charge via length contraction alone.

miller said...

Danky,

You're correct that the electrons don't attract directly, and in fact the direct interaction is repulsion. However, this repulsion is cancelled out by the attractive effect of the protons in the other wire. So we have to consider additional effects on top of that, such as Lorentz contraction and magnetic fields. These additional effects result in attraction (or at least they result in weaker repulsion). In this sense, the electrons attract each other.

"In fact with 2 wires you cannot illustrate the role of relativity because you are stuck explaining the attraction as the magnetic force between the now moving positive nuclei."
The positive nuclei do feel a net attractive force, even in the frame where they are in motion. There's an attractive force from the magnetic field, but it's cancelled out by the repulsive force from the net positive charge in each wire.

Perhaps you are asking, "Couldn't we view it as attraction between moving electrons in one reference frame, and attraction between moving positive nuclei in the other?" Superficially that would work as an explanation. However, it would mean that in one frame the electrons are dragging the rest of the wire along with them, while in the other frame the positive nuclei are dragging the electrons along with them. This is inconsistent, and Lorentz contraction is needed to resolve the inconsistency.

valjok said...

I still do not understand. Why moving electrons + still protons = magnetic field but still electrons + moving protons = electric field? The boost should not create double standards because it is the whole idea of relativity: you should not know who is boosted and who is not.

Why do you say in the end that "wire now has a positive charge because the nuclei are more dense than the electrons. Electrons are attracted to positive charge by the electric force!" Why don't you say that positively charged wires will repel from each other? Obviously, when you say that wires are positively charged, they must repel, not attract as you say. If we get net attraction, it is only because attractive magnetic Ampere law overweights the electrostatic repulsion, as it was from the original frame of reference. But you say nothing about magnetic attraction and attribute the attraction to the repulsive electrostatic. Everything seems upside down.

But, thanks for the idea of time-based vs. spacial separation. Might be it will help to answer my question http://physics.stackexchange.com/questions/63008/relativistic-charge-density-contraction-in-a-closed-loop

miller said...

Valjok,
The two frames are not the same. The wire must be part of some larger loop, which is electrically neutral in its own frame. So there's no electric field in the lab frame. There are magnetic fields in both frames.

In the electrons' frame, the other wire is positively charged, so they attract. It also seems like the nuclei would repel, but remember that in this frame, the nuclei are in motion. Therefore, they're affected by the magnetic fields (where the electrons were not).

miller said...

In a moving reference frame, the wire loop contracts, but only in the direction parallel to the motion. Since the electrons are moving as well, it seems like they should contract even more, making an even smaller loop. But this is not the case. On one side of the loop, the electrons will contract more than the rest of the wire; on the other side, they will contract less. The effect is that the wire loop becomes a dipole.

And no, this does not cause the electron loop to become a trapezoid. There are simply more electrons on the side with negative charge at any given moment (and this is allowable, since boosting affects what counts as a simultaneous moment).

valjok said...

Ok, but I still do not understand why do both wires, charged positively, attract? Do you see the problem? Why do you say that they attract? You say that "in each wire we have two protons and one electron, therefore, because electrons from each wire will attract the protons from the other wire, wires will attract" This pisses me off.

I see that both wires have positive net charge, so they will repel each other (despite each hides a negative charge that could attract the other wire). The positive charge is stronger. So, this logically implies repulsion, not attraction.

Covering the key electric force factor by magnetic aspect also makes me to think that magnetic force completely eliminates the repulsion of moving charges so that only attraction between opposite charges of two wires remains. Meantime, it is not obvious to which extent this happens.

These are two very stupid things that my mind revolts against.

valjok said...

regarding the loop, why to consider it moving? Only charges are moving, in the loop frame of reference.

miller said...

"Covering the key electric force factor by magnetic aspect also makes me to think that magnetic force completely eliminates the repulsion of moving charges so that only attraction between opposite charges of two wires remains."
Yes, that is what's happening here. The magnetic force eliminates the repulsion of the moving ions, so that only the attractive force on the electrons remains.

valjok said...

Why electrons? What do they attract? They can attract either electrons or nuclei. Both is nonsense. Electrons cannot attract other electrons because they all have the same polarity and always repel. Saying that electrons attract nuclei is like saying that Earth attracts the Sun.

Here we have two logic errors. At first, attraction/repulsion is a symmetric process. Sun and Earth both attract each other. The only asymmetry is because one is much larger and we say that lighter is attracted to each other. Electrons cannot attract other electrons or nuclei. Ok?

Saying that magnetic force eliminates the positive repelling would be justified in two cases: when it exactly cancels the relativistic extra charge or when all nuclei loose their electric charge completely. However, in our case, magnetic contribution overweithts the "nuclei repulsion" (cancellation is not exact) but does not eliminate it completely. We have two boys figth against one but you comment it like "3rd boy counteracts the first but does not help the second." It is underestimation of situation. Ok? Now, the 3rd boy does not eliminate the first one (completely) because if that would happen, then nothing would prevent the atoms from collapsing into black holes. Yes, there are also electrons. But you say that they only attract in this frame of reference.

miller said...

The electrons are attracted to the positive charges on the other wire. Those positive charges also feel an attractive force towards the electrons, it's just that this force is cancelled by other forces.

To find the net force, you just need to add all the total forces. Electric force from ions in other wire + electric force from electrons in the other wire + magnetic forces + forces preventing electrons from jumping out of the wire. In the electron's frame, the electrons feel an attractive force due to ions in the other wire, and a repulsive force due to electrons in the other wire. But the attractive force is stronger, because of the net positive charge. In the electron's frame, the ions feel an attractive force due to the electrons in the other wire, and a repulsive force due to the ions in the other wire. This is a net repulsive force, but once we add the magnetic force, the total is zero.

The electrons feel an attractive force in all inertial frames, and the ions always feel zero net force (excluding the forces preventing electrons from escaping the wire), it's just that in different frames this is caused by different combinations of electric and magnetic forces.

I don't really understand what you are saying with the boys, but it sounds like you are complaining that even in the shifted frame there is a need to refer to magnetic forces. I can think of examples where this is not the case (eg a wire made purely of electrons), but this would not be nearly as elegant or realistic. Instead of the magnetic field cancelling forces completely, the magnetic field would merely cancel the boost-induced change in the electric force, and it would be hella confusing.

valjok said...

> But the attractive force is stronger, because of the net positive charge

This is nonsense. Ask anybody. Everybody knows that net positive charge, distributed uniformly, causes repulsion, not attraction.

Larry, The Barefoot Bum said...

Anybody???? I'm an econ/poli sci student; I have no fucking clue about net positive charges.

Word of advice: you're starting to come across as a jackass.

miller said...

A net positive charge causes an attractive force on negative charges. But I'm repeating myself now.

valjok said...

> Anybody???? I'm an econ/poli sci student; I have no fucking clue about net positive charges. Word of advice: you're starting to come across as a jackass.

It is good to see the ignorants in opposition. The sad fact is that they are going to rule us (or, already do)

> A net positive charge causes an attractive force on negative charges. But I'm repeating myself now.

You silence the fact that negative charges repel, once again. Saying only half of the truth is a typical method of manipulation. It changes the sign, from good to evil. You cannot say that net positive charges attract electrically. It does not matter whether negative charges exist in them or not. Net charge is only important.

miller said...

I can copy/paste the relevant sentence for you again.

"In the electron's frame, the ions feel an attractive force due to the electrons in the other wire, and a repulsive force due to the ions in the other wire. This is a net repulsive force, but once we add the magnetic force, the total is zero."

Explaining things to people online is a service. I don't need to take antagonism. You can read a textbook instead.

valjok said...

Which textbook? All textbooks are saying that your main text, where you write "The reason the wires attract is specifically because the electrons attract. One way we know this is the electrons tend to gather towards each other at the closer edges of the wire" is nonsense.

miller said...

Jackson is well-known.

BlueDrink9 said...

Ok, I see this was last commented in May, but I came across this idea elsewhere and it blew my mind, but then I had some questions and went searching.
My main question was that if the opposite wire had a net positive which attracted your electron, why wasn't it cancelled out by electron's attraction to the net positive in your wire? Same idea as why the net positives didn't repel?
You have explained this by saying the magnetic force on the moving positives causes them to attract, cancelling this repulsion. However this says there is such thing as a magnetic force.
I was under the impression that this concept was explaining how magnetic force is simply electric force which only acts on moving charges because of contraction and charge condensation, and the magnetic force was a false force just to simplifying this.

Also the explanation doesn't help when considering the magnetic attraction between two parallel moving electrons. (These do attract magnetically, right?)

miller said...

Bluedrink9,
"You have explained this by saying the magnetic force on the moving positives causes them to attract, cancelling this repulsion. However this says there is such thing as a magnetic force."

I do not agree that the magnetic force is simply a false force. Rather, I would say that in any particular object's own reference frame, the magnetic force on that object is zero (since magnetic force is proportional to velocity). However, the object's own reference frame is not the only valid reference frame, as all reference frames are equally valid.

If you insist on considering the forces on each object in their own reference frame, you'd have to consider the forces on the positive charges in their own reference frame, which is a different reference frame from that of the electrons.

"Also the explanation doesn't help when considering the magnetic attraction between two parallel moving electrons. (These do attract magnetically, right?)"

As I mentioned in the post, yes. If you consider it in the rest frame, the electrons generate magnetic fields which cause them to attract. However, changing magnetic fields create electric fields, and this results in extra electric field which cancels out the magnetic attraction. This is dissimilar to the two wire problem, because in the wire problem there are no changing electromagnetic fields.

BlueDrink9 said...

Ookaaay, I think I kind of understand.

I think I get the thing about the positives. Moving positives create an attractive magnetic field overcoming repulsion. Still have one question you didn't really answer, if the opposite wire had a net positive which attracted an electron, why wasn't it cancelled out by that electron's attraction to the net positive in its own wire?

And the other thing, with regards to the lone particles, do we observe a net attraction between them? To me, what you are saying it that, just as there is electric repulsion in the rest frame, there is a field created opposing magnetic attraction when they are moving. There is never any net attraction, but the reasons seem to change depending on reference frame.

I think I may have to get a better understanding about what causes a magnetic force first, so if you could recommend a link that'd be great.

miller said...

"if the opposite wire had a net positive which attracted an electron, why wasn't it cancelled out by that electron's attraction to the net positive in its own wire?"

Yeah, actually the electron is attracted to the positive charge in its own wire, and that's why the electrons don't leave the wire. Instead they drag the whole wire along with them.

"And the other thing, with regards to the lone particles, do we observe a net attraction between them? To me, what you are saying it that, just as there is electric repulsion in the rest frame, there is a field created opposing magnetic attraction when they are moving. There is never any net attraction, but the reasons seem to change depending on reference frame."

That's correct. There is always a net repulsion between the charges, but the balance of electric and magnetic forces is different depending on the reference frame.

"I think I may have to get a better understanding about what causes a magnetic force first, so if you could recommend a link that'd be great."

I don't have good links on the hand, sorry. My feeling is that here I'm already pushing the limit of what a person can understand without going into the math.

miller said...

Thanks buddy. This is what I was looking for. I had gotten an intuition of this from trying to decipher electrostatic forces and magnetism and the relationship of time and space. I *suspected* they were at 90 degrees to each other but all the convoluted descriptions I found elsewhere had it completely buried. I'm not a mathematician and I don't speak "math" - I'm a programmer, a writer - I think in pictures not symbols (well, english language symbols and musical symbols but not mathematical symbols - math always seemed like excessively compacted computer programs to me) -

Thanks for breathing some clarity into this. I'm now educating my 5000+ followers on Vine, another 6000 followers on Google+ and a few thousand on Facebook. Will any of them care? Well, a few will. And that's enough.

miller said...

Great article. How have I only discovered this recently? Why don't textbooks explain the relationship between magnetism and electricity like this? It makes much more intuitive sense than any other way.