Re: twins paradox problem

On Mon, 20 Mar 2006 04:04:11 +0000, dseppala wrote:

On Fri, 17 Mar 2006 09:56:13 -0500, sal <pragmatist@xxxxxxxxxx> wrote:

On Fri, 17 Mar 2006 13:13:12 +0000, dseppala wrote:

On Thu, 16 Mar 2006 12:53:46 -0500, sal <pragmatist@xxxxxxxxxx> wrote:

On Thu, 16 Mar 2006 13:28:11 +0000, dseppala wrote:

On Wed, 15 Mar 2006 10:40:52 -0500, sal <pragmatist@xxxxxxxxxx>
wrote:

On Wed, 15 Mar 2006 13:26:20 +0000, dseppala wrote:

On Tue, 14 Mar 2006 15:17:07 -0500, sal <pragmatist@xxxxxxxxxx>
wrote:

On Tue, 14 Mar 2006 14:11:30 +0000, dseppala wrote:

In this problem, instead of two twins there is a battery and
a light bulb connected with a pair of wires. An ammeter at
the battery measures that one ampere (one columb per second)
is flowing there. An ammeter at the light bulb measures that
one ampere is flowing through the light bulb.

Now as in the twin's paradox we let the light bulb
accelerate to V, travel for some distance,


Here is your FIRST TASK:

Given that the current through the battery is 1 amp in the
rest frame of the BATTERY, determine the current through the
bulb as measured in the rest frame of the LIGHTBULB.
[ snip ]

If the readings don't match, what is the ratio of the readings?
Display your calculation, please.

But, here's the important question which you have glossed over:
Where are the ammeters placed, and how are they wired, in order
to obtain these readings?

And, just to emphasize it: To _which_ frame of reference does
each meter belong?

Be _precise!_ You are still just waving your hands.

[ snip ]

You need to _start_ by comparing meter readings obtained within
a SINGLE FRAME OF REFERENCE.

Okay, help me here. How do I compare meter readings obtained
within a single frame of reference? The display on one meter is
one ampere, and the display on the meter in the other frame is
also one ampere. Can't I have two observers in the same
inertial frame at different postions look at the two meter
readings?

Yes, of course. You can have any observer look at any object. But
that doesn't mean the object is in that observer's FoR.

You need to compare the readings on meters which are at rest
relative to each other. The velocity of the person looking at the
needle on the meter is obviously not relevant! :-)

The goal is to answer this question:

For _one_ observer, how much current appears to be flowing
through two different parts of the circuit?

If that observer looks at a meter which appears (to that observer)
to be in motion, it may _OR_ _MAY_ _NOT_ give the same reading as
a meter which is stationary relative to that observer.

Let's be clearer here. "Current" is electrons per second moving
past a particular point, as viewed in some particular frame of
reference.

Can you clarify your statement a little more for me. When you say
past a particular point [of space?] instead of past a particular
point of the wire,

Both make sense, either can be measured, but you need to be careful
because it can be confusing.

You can ask: "How many electrons are flowing through the wire, as
measured frame S?"

You can _also_ ask: "How many electrons are flowing through each
square inch of area at point X in space, as measured in frame S?"

The first is what you need if you want to answer the question, "How
many electrons are accumulating (or dissipating) in this spot in the
curcuit?"

The second is what you need if you want to ask "What is the magnetic
field caused by this current?" because that's determined by curl(B)
= J and it has nothing to do with whether the currents are following
a wire.


are you saying that when a battery is perpendicular to the motion
and the attached wires are parallel to the motion, observers not
in motion would say that more electrons are leaving the battery
than are going into it or vice versa?

No, of course I'm not saying that. And neither are you if you hope
to make sense.


And if the battery's orientation is parallel with the motion the
same number of electrons are going into the battery as are leaving
it. Or are you saying that when the battery is perpendicular to
the motion, observers not moving with the battery would say no
current flows into or out of the battery

What do you mean by "perpendicular" to the line of motion? If you
mean the obvious -- positive and negative terminals moving side by
side, _not_ one behind the other -- then that doesn't make sense.
Of course both frames (the battery's rest frame and the frame in
which it was moving) would agree that current was flowing through
the battery, but they wouldn't agree as to how _much_ current was
flowing through the battery.

You should be able to tell me why that's true, and since I've
already given you the answer, you should be able to tell me how
their measurements would differ.


because the battery doesn't stay at any particular point? I'm not
certain what you are saying exactly.

If I want to measure how many electrons are flowing through a point
on the wire, I'm not certain what measurement technique is used if
the wire is moving relative to my frame.

Excellent! You've recognized one of the first major issues here.

_HOW_ do you measure current through a moving object?

Here are two basic approaches that I can see.

a) Arrange to have a stationary meter read current at the endpoints of
a region in which objects are moving. For instance, stick the
equipment in a centrifuge, so it's moving, but provide current from
the outside via sliprings on a central shaft. You can measure the
current going into and out of the sliprings using stationary meters.

This is not perfect, obviously, for a number of reasons.


b) Measure the current in the _moving_ frame, and convert that to a
count of electrons/second past a point in the _moving_ frame. The
count of electrons is, of course, absolute; it doesn't depend on
the frame. However, the time it took for them to pass _does_
depend on the frame. So, you can scale the electrons/second rate by
gamma to obtain electrons/second in the "stationary" frame. You
can then convert electrons/second in the stationary frame to a more
conventional current measurement.

This is not entirely satisfactory either, because it's a computed
reading rather than a direct measurement.


There may be some other, better way, but I can't think of it off hand.


If I put an ammeter at the point on the wire that is moving with the
wire (its part of the circuit) that ammeter measures coulumbs per
second in the moving frame. I'm interested in how many columbs pass
a given point in a moving wire. Can't all observers simply take the
moving ammeter times the time measured by the moving clock?

No. As I said above, the total electrons flowing past a point between
two events will be the same in all frames, but the _time_ they took is
not the same in every frame.

Look: If Joe sees 3 coulombs/sec flowing _into_ a pool for 10 seconds,
but Harry sees 2 coulombs/sec flowing _out_ for 15 seconds, who saw
more electrons go past? Did the number of electrons in the pool
change?

In order to determine if (and where) electrons might be accumulating
in some part of the circuit, you need to know the _rate_ at which
they're flowing into and out of each part of the circuit in a single
frame of reference. For that, you need electrons _per_ _second_ in
_one_ frame. And it's the per-second part that's frame dependent.

As above, if Joe sees them flowing in at 3 coulombs/sec and Harry (who
is in motion relative to Joe) sees them flowing out at 2 coulombs/sec
... well, so what? You can't conclude anything from that. You need to
know how fast Joe sees them flowing _out_ as well as in; you need to
know how fast Harry sees them flowing _in_ as well as out.


If not, what measurement technique is used to count electrons
passing a given point in the wire?

See above for ways one could plausibly do it.

As to what technique "_is_ used", I'm not sure this is commonly done,
so I don't know that there is any generally applied technique.

By the way, you also have not considered the power dissipated in each
part of the circuit, nor have you considered the voltage measured in a
moving frame versus the voltage measured in a stationary frame, nor
have you considered what happens to a moving resistor -- does its
resistance change due to the motion, or not? What about a lightbulb
-- is it equally bright when viewed by two observers who are in
relative motion?

This has been discussed at some length at least once before in this
newsgroup. It's an interesting problem (or set of problems).
However, you have, as yet, not gotten to first base in grasping either
its intricacies or its solution.


For the purposes here, you can view electrons as marbles rolling
around inside the wires. You can, then, ask how long it takes for
some number of them to roll past a particular point in the _wire_,
and you can ask how long it takes for some number of them to roll
past a particular point in _space_.

Either of those can be measured in any frame of reference.

You should be able to say what the relationship between the
measurements of these values in any two frames would be. Please try
it, and see what you get.


David

[ snip ]

[ds]
Can you please clarify what you mean by comparing readings
obtained in a single frame of reference?

[sal]
Again, the _meters_ _themselves_ must be relatively at rest.

I'll make it as easy as I can, by actually including a tiny
calculation:

Current is _not_ invariant. It's particles/second, and elapsed
time isn't invariant: elapsed time goes as 1/gamma, where
gamma=1/sqrt(1-v^2). So current transforms as gamma (inverse of
time).

Suppose there's a wire which is stationary relative to me. If I
measure the current in that wire as "a", then someone else moving
at velocity v relative to me will measure the same current in the
same wire at the same time as being "a*gamma".

While 10 electrons traverse the wire, the moving observer will
measure 1/gamma times as much _time_ going by; same number of
electrons, in 1/gamma times as much time, means the current was
observed to be gamma times larger.

But this raises another question: _HOW_ do you measure current in
a _moving_ wire? It's hard to set up the problem in a way that
makes sense. Any setup that moves the ammeter with the wire
results in getting the reading from the rest frame of the wire.
One possibility might be to put some of the equipment into a
centrifuge and use wires running to sliprings on the shaft to make
contact with the outside; you then take the voltage and current
measurements at the sliprings. That's not entirely satisfactory
but it's a start.

And how do you tell when electrons have traversed a wire segment?
The easiest way is to assume the wire is oriented perpendicular to
the line of motion. Otherwise you've got problems with
simultaneity and clock skew between the ends of the segment.


--
Nospam becomes physicsinsights to fix the email
I can be also contacted through http://www.physicsinsights.org

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