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Re: Potential of a point charge(gravitaional variety)

From: John Denker <jsd@AV8N.COM>
Date: Fri, 27 Feb 2004 12:33:19 -0800

In response to SSHS KPHOX <kphox@CHERRYCREEKSCHOOLS.ORG>:

Rather than answer the question(s) as stated, I'm going to
make some remarks that I hope are helpful ... perhaps not
infinitely helpful, but the best I can do.

There are some thorny problems here. At the root of it all,
mathematics offers us some lousy notation. Some of it is
self-inconsistent, so that almost anything we write therewith
is open to misinterpretation. Fortunately, some better notation
is also on offer.

Suggestion: When I am trying to think clearly or write clearly,
I find it helpful to avoid writing "dx" or "d(anything)" except
in one of the following three very specific contexts:
1) The notation for integrals (i.e. integral y dx) is
pronounced "integral of y with respect to x" and we
say dx denotes the measure.
2) The notation for derivatives "dy/dx" is pronounced
"derivative of y with respect to x"
3) By itself, dx can represent a vector, namely the exterior
derivative of x.
In particular, I have learned to avoid interpreting "dx" as
some sort of "infinitesimal" and I do not willingly consider
dy/dx to be the ratio of infinitesimals.

Note that in *none* of my three cases does it make sense to
talk about whether dx is positive or negative. For example,
if we have a derivative dy/dx that happens to be negative,
I don't know whether that's because dy is negative while
dx is positive, or vice versa. The derivative is a well-
defined well-behaved creature, while dx in this context is
not. We would have to pronounce "dx" something like "with
respect to x" which is clearly an uninterpretable fragment
of a sentence.

Similarly in an integral, I have no idea what dx means until
you tell me whether we are integrating from A to B or from B
to A ... and if A and B are points in a multidimensional
space you need to tell me the path along which we are integrating.

Possibly constructive suggestion: whenever you see something
like "dx" or "dr" by itself, try to turn it into an honest
derivative (rather than a differential) perhaps by "dividing"
(ugh!) by dt or something. The other possibility is to try
interpreting it as an exterior derivative, which makes dx a
vector (specifically, a one-form, if you care to distinguish
one-forms from pointy vectors).

Applying these suggestions to the gravitational potential Phi
(which is a scalar field) we obtain del Phi, which is a well-
defined vector field, directed everywhere radially outward.
The force is also a well-behaved vector field
F = - del Phi
and F is directed everywhere radially inward.

Infinitesimals bad. Vectors good.

(BTW: plain text good for list.
Markup bad for list, especially when it's invalid markup.)

===================
Original message:

<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
<html><head><meta http-equiv=3D"Content-Type" content=3D"text/html; c=
harset=3DISO-8859-1"></head><body bgcolor=3D"#FFFFFF">
<div style=3D"margin-left:10px;margin-right:10px;text-indent:0px;padd=
ing-left:0px;padding-right:0px;margin-top:0px;">Forum for Physics Educators <<a href=3D"mailto:PHYS-L@lists.n=
au.edu">PHYS-L@lists.nau.edu</a>> writes: 
Does anyone e=
lse get as confused as I do with negative signs when trying to 
derive the potential for a point charge? 
 
 
I have always had the same difficulty when doing this with gravity an=
d my view of what I read in texts (HRW etc) seems like hand waving. L=
et me express my logic and what I think I have learned this week: 
 
1. delta U =3D -work done by internal force (gravity) 
 
2. when an object is allowed to move from infinity to some point A th=
e gravity force and dr are in the same direction. (This is where I th=
ink the problem is in my head) 
 
3. the work done by gravity then is GMm*integral from infinity to R o=
f dr/r^2 
 
4. I get the result for this of -GMm/R 
 
5. but deltaU =3D -Work so should be GMm/R 
 
As I have read some of the posts on the Electrical case I think I am =
seeing that my confusion is in what I am calling dr and what is dr. r=
is a vector directed from earth to mass m. So when it moves dr is ne=
gative. In what I wrote above, instead of dr I should have written ds=
which is parallel to F making my dot product positive. But when I re=
place ds with dr, I must say that dr =3D -ds? 
 
Do I now have this right? It has bugged me for a long time.

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