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Re: [Phys-l] non-conservative --> non-grady ???



Your capacitor example would work the same even if you left the field on for
the entire time since the electric field outside the capacitor is zero.

I'm not quite sure what the issue is that you are trying to address. Has
someone claimed that conservative forces cannot do work? Of course they
can. Gravity pulling a falling ball downward does work on the ball. If you
define the ball alone to be your system, then its energy increases as it
falls. The ball by itself isn't a conservative system.

If you enlarge the system to include the ball plus the earth, then you can
use the fact that gravity is conservative to define a potential energy and
add it to the kinetic energy to give a total energy for the entire system
that will be conserved as the ball falls.


Your charge flying through the capacitor can be treated similarly. Suppose
you add "reflectors" that will reverse the direction of motion of the flying
charge outside the capacitor on both right and left (they could be anchored
point charges of opposite sign to the flying charge, for example).

The flying charge will enter the capacitor through the hole in the left
plate, speed up while inside, then exit the right plate at higher speed just
as before. But the anchored charge on the right will turn it around and it
will go back into the capacitor and be slowed down while inside. Then it
will come back out the left side at low speed, get turned around by the
other anchored charge, and the process will repeat over and over.

Taking just the flying charge as the system you can say that its energy is
changing as work is done on it by the electric field. But if you include
everything in the system you can define a total energy that will stay
constant all the time, even when the charge is outside the capacitor.


So far I have imagined leaving the electric field turned on all the time.
But suppose you did turn the field off each time the capacitor was on the
rebound from the anchored charge on the right. Then the flying charge would
not slow down at all on the way back. When you turned on the field again
for the trip to the right it would gain additional speed. In this way you
could pump the thing up to any speed you'd like as long as your timing was
good.

Clearly this system is not a conservative system. Extra energy is getting
in. Where's that energy coming from?

Steve Highland
Duluth MN




On 02/19/2008 08:28 PM, LaMontagne, Bob wrote:
Isn't the changing electric flux introducing a non-conservative
magnetic interaction that the charged particle interacts with once it
starts to move? Perhaps I'm not visualizing this in the sense that
you meant, but it doesn't seem to be purely conservative.

Consider the following geometry
http://www.av8n.com/physics/img48/accelerator.png
and the following method of operation:
As before, the charged particle is deemed "the system".
At time t_A the particle is at location A moving slowly to
the right. The KE is small. The field is off.
At time t_B the particle is at location B. We turn on
the field. This does not affect the particle, because
it is outside the capacitor, where the field strength
is small, and can be made arbitrarily small by suitable
engineering. Any /magnetic/ effects are doubly negligible,
firstly because the field is small, and secondly because
the particle is on the axis of symmetry ... which /direction/
would the magnetic force have????
At time t_C the particle is at location C. It is being
vigorously accelerated by the field.
At time t_D we turn off the field. This has no effect on
the particle. The particle retains its large KE.
Magnetic effects are doubly negligible.
At time t_E we observe the potential energy to be the same
as at t_A, and the KE to be much larger.

At no time was the particle exposed to a non-grady field of
any significance. Yet energy was transferred across the
boundary of the system.


==========

On 02/19/2008 08:36 PM, Alfredo Louro wrote:
But surely the forces can't be conservative, if they're time dependent?

The field in my particle accelerator is /piecewise/ time-independent.
-- During the entire time that the particle sees a non-negligible field,
the field is time-independent.
-- During the times that the field is changing, its effect on the particle
is negligible.


========================================

This business of being able to do work using only grady force-fields
is important in many real-world situations. Consider for example
one of the first physics demos many people encounter: contact
electrification, commonly called "static electricity". The typical
contact electrification scenario involves only forces that are
the gradient of some potential, yet you can do quite a bit of work
with such forces.

For that matter, water is held in a ladle essentially by grady
forces, yet you can do work on the water, ladling it from a
lower bucket to a higher bucket.

I chose the less-common example of the capacitive particle accelerator
because it is amenable to precise analysis. In contrast, rigorous
analysis of contact electrification is a more demanding task. But
even that is doable; if you're interested, check out
http://www.av8n.com/physics/contact-electrification.htm

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