Re: Coulomb's Law and SR

From: Bill Hobba (bhobba_at_rubbish.net.au)
Date: 07/23/04 

Date: Fri, 23 Jul 2004 14:02:02 GMT

"Shooter" <nospam@nospam.spam> wrote in message
news:cdqqi2$pum$1@hercules.btinternet.com...
> BTW,
>
> When I teach the delta function I find this approach the best (I saw it on
> the net some time ago, but can't find the ref.):
>
> Draw a 'box-car' function on the x-y plane: that is, a rectangle centred
on
> x=0, whose left hand edge is at -e/2 and right hand edge at +e/2 and whose
> height is 1/e. Clearly its base is of width e, so its area is e times 1/e
=
> 1.
>
> The delta function is simply the limit of the box-car function as e
> approaches zero!
>
> It's easy to visualise, and most of my students say - so what's so hard
> about that!
>
> Its then easy to visualise in 3D etc. as required for point charges and so
> on.

Excellent idea. My background is math and I first came into contact with it
in my differential equations subject as a function that is zero everywhere
except at 0 where it is infinity but whose integral is one. Immediately my
hand went up and said that is not a legitimate function. The lecturer who
knew
me as a pain in the *** *** that asked awkward equations said I knew you
were going to ask that I just knew wit. Forget about it - it is just a
tool.

I left it at that then but I came across it again when I was learning QM
from Dirac's Principles of QM book - he uses it a lot. At the time I heard
a more rigorous mathematical treatment was given by Von Newman in his book
Mathematical Foundations of QM so bought that book as well - boy did he get
stuck into it there. He rigorously proves it does not exist even via
lebesque integration and poo poohs its use. Both books are great for all
sorts of reasons but I decided I should really get to the bottom of what was
going on once and for all. That's when I came across a great little book -
Theory of Distributions - a Non Technical Introduction by Richards and Youn.
It got me hooked. Explains what is going on really well. Fourier analysis
is a snap using distribution theory - should be taught that way all the
time. Just fell in love with the subject and decided I would do my masters
thesis in it. Decided to combine with another interest of mine - non
standard analysis. I did my original degree part time and was working while
doing my masters and found it a bit too demanding holding down full time job
at the same time, and having already done 5 years getting my degree, decided
to give it away. But ever since then I have had a real soft spot for it..

BTW your approach is similar to how Littlewood handled the subject (the
Littlewood of the Littlewood and Hardy fame) - he defined it as the limit of
the integral of a series of functions. But the usual rigorous approach is
to define it as the dual of test functions. It also makes sense to handle
it that way when actually trying to make rigorous sense out of how Dirac
uses it in his QM book.

Because of my background I must admit if I was teaching it I would explain
it is really part of a branch of math called distribution theory and can be
considered as a test function (ie a function that is continuously
differentiable and of compact support) that while not having the properties
attributed to it can for all practical purposes be considered to be like
that - similar to a differential being zero for many practical purposes but
is really a very small number. I found that once I looked at it that way
then I had no problems. But of course by than time I had studied the actual
theory. While I think your approach is great some *** is bound to ask -
hey how do you differentiate it if it is a discontinuous function lie a box.
If you stick to test functions you don't have that problem - it is
differentiable to all orders.

Best of luck explaining it to students.

Thanks
Bill

>
>
> "Bill Hobba" <bhobba@rubbish.net.au> wrote in message
> news:Sl4Mc.12633$K53.9599@news-server.bigpond.net.au...
> >
> > "Shooter" <nospam@nospam.spam> wrote in message
> > news:cdqdot$qq0$1@titan.btinternet.com...
> > > Hi Bill,
> > >
> > > Even your authoritative reference
> > > http://www.cse.secs.oakland.edu/haskell/SpecialRelativity.htm refers
> > > throughout to 'source' charges and 'test' charges. Such a
distinction
> > > simply wouldn't be necessary unless the distinction needs to be made !
> >
> > So that is your basic concern - got it now. Test charges are generally
> > considered so small as to not disturb the field. This is discussed on
> page
> > 46 Landau - Classical Theory of Fields and gives details of other
sections
> > of that classic book that details how this restriction can be
> > removed.(radiation damping).
> >
> > >
> > > A P French of MIT - who is equally authoritative - states *without
> > > qualification* that Coulombs Law holds for a test charge moving at
*all*
> > > speeds, including close to the speed of light, "as long as the source
> > charge
> > > remains stationary."
> >
> > Sorry for the confusion - my error in not intepreting what you said
> > correctly - yes when consdered that way it is true. As I said above a
> test
> > charge is not consdered to peturb the field.
> >
> > >
> > > Once again, throughout his detailed discussion, he calculates the
> > > relativistic modifications to the electric force caused by motion of
the
> > > 'source' charge - he is very specific about this.
> >
> > That would seem fine.
> >
> > >
> > > Obviously, the whole thing sounds unsymmetrical and therefore
'flawed.'
> > > But it isn't - this has long puzzled me.
> > >
> >
> > It is not asymmetrical once you realize that a test charge is considered
> so
> > small as to not perturb the field. Of course such is not really
possible
> > because regardless of how close you get to any charge (regardless of how
> > small) the field can be made arbitrarily large. So the other assumption
> you
> > add is that you will not consider distances where this is problem. This
> is
> > tied up with the continuity assumption often made is classical
> > electrodynamics where charges are considered as a continuous 'dust'; the
> > continuity assumption breaks down for distances where the field becomes
> too
> > large. What you have hit upon is a well known problem with classical
> > electrodynamics - it really does not mesh well with the idea of point
> > charges - continuous charge distributions are more easily analyzed. In
> > fact to handle point charges you need to use the dirac delta function
and
> > that always must be used with a bit of care because it is not really a
> > function (mathematically it is a distribution).
> >
> > >
> > > I can only assume that the 'source' charge is being regarded as the
> > > generator of the field - and what we are really discussing is the
> > > modification of the electric field due to motion.
> > >
> > > (a test charge, of course, is traditionally regarded as not
> contributing
> > a
> > > field, simply experiencing a force due to that field.)
> > >
> > > I think therein lies the answer. But I'm still not sure, and so I'm
> > afraid
> > > my question remains unanswered...
> >
> > Yep you hit exactly on the answer. If have a look at Landaus book you
will
> > see a bit more detail on the issue (not mush more detail mind you -
after
> > all this is Laduau who has a well known terse style).
> >
> > >
> > > Thanks for the links - they're excellent.
> > >
> > > p.s. I'm an MSc Astrophysics - but there are still plenty of things
that
> > > puzzle me ;-)
> >
> > Post away. Anything relativity related is welcome here. You will find
> some
> > very knowledgeable people post (I am not one - just a guy who is
> interested
> > in relativity and related stuff). Just stay clear of the cranks - but
> they
> > are easily spotted.
> >
> > Hoping for more posts and sorry for any confusion.
> > Bill
> >
> >
> > >
> > >
> > > "Bill Hobba" <bhobba@rubbish.net.au> wrote in message
> > > news:GEYLc.12131$K53.8687@news-server.bigpond.net.au...
> > > >
> > > > "Bill Hobba" <bhobba@rubbish.net.au> wrote in message
> > > > news:QbYLc.12118$K53.5696@news-server.bigpond.net.au...
> > > > >
> > > > > "Shooter" <nospam@nospam.spam> wrote in message
> > > > > news:cdorcb$n87$1@hercules.btinternet.com...
> > > > > > Hi all,
> > > > > >
> > > > > > I'm having trouble with the treatment of 'source' and 'test'
> charges
> > > in
> > > > SR
> > > > > > (I'm not interested in GR on this point.)
> > > > > >
> > > > > > These are my problems: I'd be grateful for informed responses:
> > > > > >
> > > > > > Coulomb's Law has been verified up to speeds close to C, with
> > 'source'
> > > > > > charge (hereafter just source charge) stationary in the lab
frame
> > and
> > > > the
> > > > > > 'test' charge (hereafter just test charge) being the moving one.
> > > > >
> > > > > Coulombs law is strictly only true for stationary charges and then
> > only
> > > > > after the system has had enough time for the fields to settle
> (changes
> > > in
> > > > > them can not propagate faster than the speed of light). Thus it
is
> > > > > meaningless to say that coulombs law has been tested to speeds
close
> > to
> > > C.
> > > > > They can be applied as a reasonable approximation in certain
> > > circumstances
> > > > > to moving charges but it is certainly false for speeds anywhere
near
> > the
> > > > > speed of light. To analyze that one must use the full machinery
of
> > > > > classical electrodynamics eg Maxwell's equations.
> > > > >
> > > > > >
> > > > > > In AP French's otherwise excellent book, he explains that
> Coulomb's
> > > Law
> > > > > > holds experimentally at all speeds.
> > > > >
> > > > > It does not. The correct equations can be found in the Feynman
> > Lectures
> > > > on
> > > > > Physics Volume 1 Chapter 28 page 2. It is very complicated and
> > depends
> > > on
> > > > > retarded time and only takes the form of Coulombs law for small
> > > velocities
> > > > > not accelerating much - certainly nothing close to the speed of
> light.
> > > > > Think about it for a minute. Information can not travel faster
than
> c
> > > > thus
> > > > > the force on a stationary charge can not depend on the
instantaneous
> > > > > distance between it at the other charge as demanded by Coulombs
law.
> > > > Hence
> > > > > if charges are moving the force must depend in where the charge
was
> > > > > previously otherwise you have violated the fundamentals of
> relativity.
> > > > >
> > > > > > But then takes a relativistic approach
> > > > > > where, as one example:
> > > > > >
> > > > > > The source charge (please remember my inverted commas !) is
moving
> > > > > relative
> > > > > > to the rest frame, towards a test charge stationary within that
> rest
> > > > > frame.
> > > > > >
> > > > > > The relativistic principle is applied, and the frame shifted to
> one
> > > > where
> > > > > > the source charge is stationary, and the test charge now moving
> > > towards
> > > > > the
> > > > > > source.
> > > > > >
> > > > > > Relativistic transformations are now made, and then transformed
> back
> > > > into
> > > > > > the original rest frame etc. - but now the force is modified by
> > > > factor(s)
> > > > > of
> > > > > > gamma etc.
> > > > > >
> > > > > > I'm genuinely puzzled - because there now appears to be a lack
of
> > > > symmetry
> > > > > > in the force. And I can't see how the test charge can't be
> > considered
> > > > the
> > > > > > source charge (in the lab frame) with the stationary 'source'
> charge
> > a
> > > > > test
> > > > > > charge moving wrt it.
> > > > >
> > > > > I have seen this type of analysis in many sources - eg Griffith.
> And
> > > > > certainly there is a lack of symmetry in the force (as compared to
a
> > > > > stationary charge) because in relativity ordinary force transforms
> in
> > a
> > > > > funny way and the analysis only works for velocities small
compared
> to
> > > > light
> > > > > eg the drift velocities of electrons in a wire.
> > > > >
> > > > > >
> > > > > > I'd be very grateful for some expert advice here. Does the
> concept
> > of
> > > > > > source and test charge continue to have meaning in SR - that is,
> one
> > > is
> > > > > > considered to provide the field - and the other to experience
the
> > > field?
> > > > >
> > > > > I do not have that book but I stand by what I said above. I think
> the
> > > > best
> > > > > way to understand what is happening is to see a correct treatment
of
> > > > > relativity - http://arxiv.org/abs/physics/0110076 and then how
that
> > can
> > > be
> > > > > applied to derive electrodynamics and Maxwell's equations form
> > coulombs
> > > > law
> > > > > and SR - see
> > > > http://www.cse.secs.oakland.edu/haskell/SpecialRelativity.htm.
> > > >
> > > > I wish to add I have not looked at the above paper for a while so I
> took
> > a
> > > > quick peek. There is at least one 'hidden' assumption (very mild)
but
> > for
> > > > completes I thought I would mention it. If you get caught in trying
> to
> > > spot
> > > > it drop a post and I will explain it.
> > > >
> > > > PS this time Greywolfs response was OK buy generally speaking
greywolf
> > > does
> > > > not see eye to eye with some highly knowledgeable posters (people of
> the
> > > > caliber of Steve Carlip -
> > > http://www.physics.ucdavis.edu/Text/Carlip.html).
> > > > It goes without saying you can trust what Feynman says.
> > > >
> > > > Thanks
> > > > Bill
> > > >
> > > >
> > > > > That way you get the full development right from the start. Also
do
> > > read
> > > > > chapter 28 of the Feynman Lectures. In fact for physics you
should
> > > really
> > > > > get the complete set. If your studying at a university simply ask
> you
> > > > > lecturer what he/she thinks - I am sure they will recommend it
too.
> > In
> > > > fact
> > > > > if your teacher is telling you it holds for velocities approaching
c
> > > then
> > > > > take my explanation along to then ask them to clarify what is
> > happening.
> > > > >
> > > > > Thanks
> > > > > Bill
> > > > >
> > > > >
> > > > > >
> > > > > > Many thanks
> > > > > >
> > > > > > (No lunatic answers please - the problem is with my own
> > understanding,
> > > > not
> > > > > > with SR)
> > > > > >
> > > > > >
> > > > >
> > > > >
> > > >
> > > >
> > >
> > >
> >
> >
>
>

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