Re: twins paradox problem
- From: sal <pragmatist@xxxxxxxxxx>
- Date: Fri, 17 Mar 2006 09:56:13 -0500
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:[ snip ]
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.
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.
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]
[sal]Can you please clarify what you mean by comparing
readings obtained in a single frame of reference?
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.
--
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