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Re: [Phys-l] vector inconsistencies



Thanks, you answered my question - wish I had seen your post before I sent mine.

Bob at PC

________________________________

From: phys-l-bounces@carnot.physics.buffalo.edu on behalf of Dan Crowe
Sent: Fri 3/30/2007 4:40 PM
To: phys-l@carnot.physics.buffalo.edu
Subject: Re: [Phys-l] vector inconsistencies



A gradient indicates the direction of the maximum rate of increase. A
conservative force is the negative of the gradient of potential energy.
I don't see that as being inconsistent, although it might cause
conceptual difficulties for some students. The real issue is that
potential energy is defined to be the negative of the work done by a
conservative force. The reason that the definition includes a negative
sign is because it is desirable to define mechanical energy as the sum
of kinetic energy and potential energy, rather than as a difference
between the two.

Daniel Crowe
Loudoun Academy of Science
dan.crowe@loudoun.k12.va.us

jsd@av8n.com 3/30/2007 4:24 PM >>>
Hi Folks --

Case 1: Consider an electrostatics problem in one dimension.
If the electric potential is increasing left-to-right,
the electric field vector will be pointing right-to-left.


5 6 7 8 9
----+----+----+----+----+---- electric potential
A B
<---------- electric field


The field vector must point "downhill" because it is related
to a force, and there is a minus sign in the equation for
PVW (principle of virtual work).

So far so good, right?


==========

Case 2: suppose we are talking about plain old position rather
than electric potential. The number line can be considered
the "east potential" i.e. a measure of how far east a given
point is.

Consider two points on the number line, A and B.

5 6 7 8 9
----+----+----+----+----+---- number line
A B
----------> displacement vector
from A to B


We can represent A by x(OA) which is a vector from the origin
to A, and represent B by x(OB) which is a vector from the
origin to B.

Then the difference can be represented by
x(AB) = x(OB) - x(OA)
which is a displacement vector from A to B.

The tip and tail of x(AB) must be as shown in the drawing
above. This is a consequence of the law of vector addition,
assuming the vectors x(OA) and x(OB) have their tails at
the origin, as is universally conventional.

So you see that for a positive displacement (B>A) the
displacement vector points "uphill" along the number
line.

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

The foregoing two cases are covered by clear-cut conventions,
... but still, such inconsistencies drive students up the
wall. The fact that it is no problem for us professionals
makes it all the more of a problem for the students (because
it is easy for us to lose sight of their problem).

But wait, there's more (as Ron Popeil would say). What about
other cases? Is there a rule that says in which cases we draw
uphill arrows as opposed to downhill arrows?

Marginal cases can be found in thermodynamics: A pressure
gradient is like a force, so it should probably (?) point
downhill ... but what about a temperature gradient, or a
concentration gradient?????

This is not a rhetorical question; I am genuinely uncertain
as to how I should think about this myself (let alone explain
it to students).

Any suggestions?
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