Thinking about electricity and magnetism, and some relativity

[TheTank]

I have 2 questions. I have been thinking about these two things a while, and got no one to ask, so hope to get some info/answeres here:)

1:
When an electron is standing still, it produces an electric field (and no magnetic field), right?
So when many electrons is moving with a constant speed (a current through a wire), a magnetic field is produced around the current. Why is that? I know when you accelerate electrons you make electromagnetic waves, but electrons moving through a wire is moving with a constant speed (I think). So why is there a magnetic field produced?

I have been playing with Maxewll's equations many times, but never really tought of this..

2:
From relativity we can derive the equation for length contraction with the following thought experiment: You place a mirror at the end of a meter-stick, and take the time for the light to bounce back from the mirror back to the front of the meter-stick. You measue the time from 2 different refrence system, one standing still, and one moving with velocity v.

My question: why do you have to measure the time it takes the light to reach BACK to the start point? Why isn't it enough to just measure the time it takes the light to reach the end of the meter-stick?? It should produce the same answere, but it will not!

 

[Whovian]

1:
When an electron is standing still, it produces an electric field (and no magnetic field), right?
So when many electrons is moving with a constant speed (a current through a wire), a magnetic field is produced around the current. Why is that? I know when you accelerate electrons you make electromagnetic waves, but electrons moving through a wire is moving with a constant speed (I think). So why is there a magnetic field produced?

From what I've gathered, in a reference frame from which an electron is moving, the electric field is weaker(?), but it produces a magnetic field. From a reference frame from which it is standing still, it simply produces an electric field. This is an example of how electricity and magnetism are different manifestations of the same force. In the case where the electrons are moving through the wire, you're in a reference frame from which the electrons are moving, and so they'll produce a magnetic field. Now, from a reference frame moving with the electrons, the wire will appear to be moving backwards, and the electrons will just produce an electric field.

2:
From relativity we can derive the equation for length contraction with the following thought experiment: You place a mirror at the end of a meter-stick, and take the time for the light to bounce back from the mirror back to the front of the meter-stick. You measure the time from 2 different refrence system, one standing still, and one moving with velocity v.

My question: why do you have to measure the time it takes the light to reach BACK to the start point? Why isn't it enough to just measure the time it takes the light to reach the end of the meter-stick?? It should produce the same answere, but it will not!

Because simultaneity is relative. Two events that are simultaneous but don't occur at the same point in one reference frame are not in another, and this is pretty much the same phenomenon. We have to measure stuff from the same point or risk being backstabbed by this.

 

[Cleonis]

[...] when many electrons is moving with a constant speed (a current through a wire), a magnetic field is produced around the current. Why is that? [...]

I recommend Daniel Schroeder's discussion Magnetism, Radiation, and relativity.

Daniel Schroeder points out that one must consider both the moving electrons and the wire. There is a velocity difference between the moving electrons and the wire, which implies a difference in Lorentz contraction.

When the electrons are stationary with respect to the wire there is no magnetic effect.
With the electrons on the move there is to the difference in Lorentz contraction a corresponding charge surplus per unit of length (of the wire). This charge surplus manifests itself as magnetism.

Daniel Schroeder's discussion is far from rigorous; the intention is to show how magnetism can be understood as a relativistic side effect of electricity. This level of unification of electricity and magnetism is specific to relativistic physics.

 

[Cleonis]

From relativity we can derive the equation for length contraction with the following thought experiment: You place a mirror at the end of a meter-stick, and take the time for the light to bounce back from the mirror back to the front of the meter-stick. You measue the time from 2 different refrence system, one standing still, and one moving with velocity v.

My question: why do you have to measure the time it takes the light to reach BACK to the start point? Why isn't it enough to just measure the time it takes the light to reach the end of the meter-stick?? It should produce the same answere, but it will not!

Let me first discuss a case in Newtonian mechanics, and then point out how relativistic physics comes out differently. This may sound as an unlikely way to clear up your question, but bear with me.

Imagine two large ships (say, cruise ships) that are co-moving, at some distance from each other. Small fast boats ferry people from the leading ship to the trailing ship and the other way round. The small fast boats maintain a particular velocity relative to the water
Is the round trip time of the small boats affected by the velocity of the cruise ships?

The round trip time is shortest when the cruise ships are stationary with respect tot the water.
When the cruise ships have a velocity the two legs of the round trip are changed in opposite direction. The journey from the trailing ship to the leading ship takes longer now, the journey from the leading ship to the trailing ship takes shorter now. The two opposed effects largely cancel; but they don't quite cancel, and that is the significant part.
When the cruise ships have a velocity then the round trip time of the small boats is slightly longer.
And since the small boats have a particular velocity it follows from the longer round trip time that the small boats have traveled a longer distance in the water.Next step is to translate that scenario to sonar signals being used to tell the distance between the ships.Now, finally, arriving at the light-bouncing-between-mirrors scenario of your original question.
The way the mathematics comes out is that the relativistic effect is proportional to the change of the total distance covered. The changes in distance covered on each of the indivitual legs are much larger, but the relativistic effect is not related to that. The relativistic effect relates to the tiny remainder after canceling out the contributions of the invidual legs of the round trip.

You can try to work that out geometrically in the following way:
Starting with the cruise ship scenerio plot the worldlines of a convoy of three ships, with small boats being sent from the central ship to the outside ships and bouncing right back.Then the worldlines of the small boats will enclose an area. Examine what that enclosed area does for stationary cruise ships and for moving cruise ships.
Then examine the propagation of sonar waves scenario. (Which is identical to the small boats scenario).
Then move to the relativistic scenario of three spaceships, relaying light signals among them. Draw worldlines again, but now interpret the geometry from a relativistic point of view.