Re: fourier transform of a charge distribution

From: Igor Khavkine (igor.kh_at_gmail.com)
Date: 01/25/05 

Date: Tue, 25 Jan 2005 16:49:25 +0000 (UTC)

rancid moth wrote:
> hello,
>
> i am trying to get a handle on what the physical interpretation of a
> fourier transform of a general charge distribution would be.
>
> that is assume we have a distribution of charge in one dimension
> dependent on time p(x,t) say.
>
> then what would the function p(k,t) represent as it progressed in
> time?
>
> now i know that fourier transforms of charge distribution usually
crop
> up when you deal with scattering angles (i.e. form factors etc) -
but
> lets simply assume that p(x,t) is a general distribution of charge
> over a 'surface' for example. what is the physical interpretation
(if
> any) of p(k,t)? i have my own ideas but most of them are not very
> illuminating at the moment - any other viewpoints appreciated.

I don't think the issue of time evolution is really important for
interpretation. I like to think of the Fourier component p(k) of
the charge distribution p(x) as the amount of variation in p(x)
on the scale 1/k.

For instance, if you have a uniform charge distribution p(x) = C,
then its Fourier transform is p(k) = c delta(k), a sharp peak
at k = 0 and zero everywhere else. In other words, there is no
variation in the charge density. Suppose that p(x) varies appreciably,
has bumps or troughs, with features having characteristic size y.
Then the Fourier transform p(k) will have a peak around k = 1/y.

Another example is a square bump charge distribution. It's flat
inside the bump, so you expect a large contribution to p(k) at k=0.
But at the same time it has sharp edges (essentially a zero width
feature). So you also expect to see non-zero contributions to
p(k) for arbitrarily large k. This is in fact the case, the Fourier
transform of a square bump is the sinc function ~ sin(x)/x.

If you really want to observe something evolving in time, try this.
Suppose you start with an irregular, not very smooth, charge
distribution. Through diffusion or some other process the distribution
gets smoothed out with time. You can observe how smooth it gets by
watching the peaks move around in the Fourier transform of the charge
distribution.

Hope this helps.

Igor

Relevant Pages

  • Re: fourier transform of a charge distribution
    ... > fourier transform of a general charge distribution would be. ... what is the physical interpretation ...
    (sci.physics.research)
  • Re: The magnetic radius of the electron
    ... for force - freeness of the charge distribution because of the current - ... So you have a model with 0.62 c at the classic electron radius? ... My Modification of Dishingtons electrostatic electron ... I think Disigtons charge distribution for leptons is fundamental not only to ...
    (sci.physics.particle)
  • Re: Electrostatics question(s)
    ... "E" at a point within a charge distribution, ... I think I can find the E outside of the sphere because its some ... You will get the same answer by integrating over the charge distribution, but that's a _much_ harder way to do it. ... Would I be correct if I said that for any point on the disc, ...
    (sci.physics)
  • Re: fourier transform of charge distribution
    ... |> transform of a charge distribution would be. ... density is at what scales. ...
    (sci.physics)
  • Q on "charge generates transformation"
    ... Now charge Q is supposed to generate the transformation ... What is the physical interpretation of the last equation ...
    (sci.physics)