I hope I am not misinterpreting your posting and driving the responses off topic - but I have never been able to to form a clear idea as to why we define potential energy (or electrostatic potential) as the negative of the integral of the product of force and displacement for what we call consrvative forces. All the conservative qualities would be there regardless of the sign - independence of path, zero integral for a closed path, etc.
It seems the only justification for the sign is that it eventually allows us to claim a conservation theorem if we add the kinetic and potential energies together whenever non-conservative forces are absent.
Does anyone know of a better reason?
Bob at PC
________________________________
From: phys-l-bounces@carnot.physics.buffalo.edu on behalf of John Denker
Sent: Fri 3/30/2007 4:24 PM
To: Forum for Physics Educators
Subject: [Phys-l] vector inconsistencies
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).