Re: 1 FERTZ

From: Bjoern Feuerbacher (feuerbac_at_thphys.uni-heidelberg.de)
Date: 06/02/04 

Date: Wed, 02 Jun 2004 11:10:55 +0200

Y.Porat wrote:
> Bjoern Feuerbacher <feuerbac@thphys.uni-heidelberg.de> wrote in message news:<c9hgus$pr5$1@news.urz.uni-heidelberg.de>...
>
>>Y.Porat wrote:
>>
>>>Bjoern Feuerbacher <feuerbac@thphys.uni-heidelberg.de> wrote in message ne
>>>
>>>>
>>>>
>>>>>------------------
>>>>>because *you are unable to do it crook!!!
>>>>
>>>>For the 30th time: according to Maxwell's equations, a multipole moment
>>>>which is changing periodicaly with a frequency f will produce an em wave
>>>>with that same frequency f. That was proved more than 100 years ago.
>>>
>>>------------------
>
> we are not interested in your 'explanations'
> we are interested in your *calculations*- crook

The calculation I referred to above can be found in any textbook. But
since you *insist*, I will reproduce it here (and you won't understand
anything about it and will call it "obfuscation"...).

This will get a bit long, but you insisted... I won't give all of the
tiny steps in between - if you want to see those, too, you can look them
up in any book on electrodynamics. Most of them even can be found in
books on vector calculus.

Start with Maxwell's third equation:
   div B = 0 (1)
Keep in mind that B and E are vector fields, and please notice that I
am working in Gaussian units! Also, I only give the derivation for a
charge density and current density in a vacuum (after all, the earth
moves through a vacuum!); the derivation for other media would be similar.

This implies that there exists a vector field A (called the "vector
potential") so that
   B = rot A (2)
Now take the second equation:
   rot E = - 1/c del B / del t (3)
(the "del's" imply here a partial derivative)

Insert (2) into (3) and rearrange the formula:
   rot(E + 1/c del A / del t) = 0

This implies that there exists a scalar field Phi (called the "scalar
potential" or simply "potential") so that
   E + 1/c del A / del t = - grad Phi (4)

Now take the first equation:
   div E = 4 pi rho
and the fourth:
   rot B = 4 pi/c j + 1/c del E / del t
rho is here the charge density (a scalar field) and j is the current
density (a vector field)

Insert (2) and (4) here and rearrange the equations a bit; the results are:
(1/c^2 del^2/del t^2 - Del) Phi - 1/c del/del t (div A + 1/c del Phi/del
t) = 4 pi rho (5)
and
(1/c^2 del^2/del t^2 - Del) A + grad (div A + 1/c del Phi/del t)
= 4 pi/c j. (6)
I use "Del" here to denote the Laplace operator.

Phi and A are not uniquely defined; one can make a so-called "gauge
transformation" without changing E and B. I.e. one can use
   Phi ' = Phi + 1/c del/del t Lambda
and
   A' = A - grad Lambda
instead of Phi and A, with an arbitrary function Lambda. One can use
this gauge freedom to simplify the equations (5) and (6) above ("gauge
fixing"); one gauge condition which is commonly used is the "Lorentz gauge":
   div A + 1/c del Phi/del t = 0

So the equations (5) and (6) simplify to:
   (1/c^2 del^2/del t^2 - Del) Phi = 4 pi rho
and
   (1/c^2 del^2/del t^2 - Del) A = 4 pi/c j.

Now these differential equations can be solved with the help of the
so-called "retarted Green's function" (one could also use the "advanced
Green's function" or linear combinations of the two, but for emission
of radiation, the retarted Green's function is the relevant one). One
then gets the "retarted potentials":
   Phi(r,t) = int dV' rho(r, t - |r-r'|/c) / |r-r'| (7)
and
   A(r,t) = 1/c int dV' j(r, t - |r-r'|/c) / |r-r'| (8)
The integral here runs over the whole volume, r and r' are vectors, and
the combination t - |r-r'|/c is called the "retarted time".

So far, the derivation has been completely general. Now we come to
charge and current densities which change periodically:
    rho(r,t) = rho_0(r) e^(-i omega t) (9)
and
    j(r,t) = j_0(r) e^(-i omega t) (10)
(please notice that rho_0 and j_0 are not entirely arbitrary - they obey
the continuity equation). omega here is the "circular frequency"; it is
simply an abbreviation for 2 pi f, where f is the frequency.

Insert (9) and (10) into (7) and (8) and rearrange the equations a
bit; then one gets:
Phi(r,t) = e^(-i omega t) int dV' rho_0(r) e^(i omega |r-r'|/c)/|r-r'|
and
A(r,t) = e^(-i omega t) 1/c int dV' j_0(r) e^(i omega |r-r'|/c)/|r-r'|.

So we see that the potentials have the *same* time dependence as the
charge and current density, i.e. both are periodic functions of the time
with the *same* frequency as the densities. Since we are only interested
in the time dependence, I will now introduce abbreviations for the
integral and write simply
   Phi(r,t) = e^(-i omega t) Phi_0(r)
and
   A(r,t) = e^(-i omega t) A_0(r).

Now insert this into the equations (2) and (4); this gives:
   E = e^(-i omega t) (-grad Phi_0(r) + i omega/c A_0(r))
and
   B = e^(-i omega t) rot A_0(r).

So we see that the fields have the *same* *periodic* time dependence as
the densities. In other words: if one has a charge and/or current
density which changes periodically with a frequency f, there will be
electric and magnetic fields which *also* change periodically with the
*same* frequency f. This result is *completely* general, it does not
depend on the frequency f in any way.

The motion of the earth around the sun produces a charge and current
density which changes periodically with a frequency of 1/1 year. Thus
there will be electric and magnetic fields with that same frequency. In
other words, there is an electromagnetic wave with a frequency of 1/1 year.

If you want to know more details about that wave, you have to do
the integrals (i.e. calculate A_0 and Phi_0), and then do the
derivatives to get E and B. From that, you can get the Poynting vector
   S = c/4 pi (E cross B),
which tells you how much energy is radiated in which directions.

But all these details are rather irrelevant, since the basic thing was
already proven above...

> 1 fertz is a very acurate number

That's a rather meaningless statement. 1 Hz, for example, is *also* a very
accurate number.

> (a mistake of your calculation of the order of 20 timwes
> will be also accepted as a good result!!!....)

Well, my proof shows that the motion of the earth around the sun
produces electric and magnetic fields which have the *same* periodicity
as the motion of the earth around the sun, i.e. a frequency of *exactly*
1/1 year. There is no error at all here - the frequency comes out *exactly*.

> -------------
> crackpot
> Y.Porat
> ---------------

Thanks that you keep pointing that out.

Bye,
Bjoern

Relevant Pages

  • Re: 1 FERTZ
    ... charge density and current density in a vacuum (after all, ... the derivation for other media would be similar. ... Phi and A are not uniquely defined; one can make a so-called "gauge ... So we see that the potentials have the *same* time dependence as the ...
    (sci.physics)
  • Re: phase and conservation of charge in QM
    ... find the motion in terms of the potentials, themselves, you get: ... This expresses the motion of the charge in terms of its CANONICAL ... The geometry is best described NOT by a principal bundle, ... The difference between the two maps is called a gauge transformation. ...
    (sci.physics.research)
  • Re: Measuring the vector potential
    ... E of a charge moving in a charge distribution acts upon that ... The L-W potentials are derived using a 'small' volume filled with ... increasing the charge density by at r. ... Still awaiting your derivation. ...
    (sci.physics.electromag)
  • Re: Measuring the vector potential
    ... E of a charge moving in a charge distribution acts upon that ... The L-W potentials are derived using a 'small' volume filled with ... Still awaiting your derivation. ...
    (sci.physics.electromag)
  • Re: why lorentz transformation?
    ... determined by the condition that the charge is at position r-R at time ... >> electromagnetic field associated with it. ... guarantee the existence of the required potentials A and \phi. ... When we take the gauge transformation, ...
    (sci.physics.relativity)