What are Debye potentials?
PuZHA...@gmail.com
Happy new year!
I'd like to get a clear concept of Debye potentials.
For the sake of this, I searched around the internet and
checked several classic textbooks, like Jackson's and
Stratton's, but no satisfactory results. Instead I get
several papers describing Debye potentials published
decades before ("Debye potential representation of
vector fields").
From those papers I find out that:
Debye potentials have something to do with the special
case of Helmholtz Theorem with divergenceless vector
fields. It's proved then this field can be represented by
two scalar potentials:
F = Lø + curl(L=÷),
where F is the vector field and L is the standard orbital
angular momentum operator. It's said these two scalar
potentials are Debye potentials. (Is this obsolete? Why
isn't there any like content in today's textbooks)
Except this I also get various descriptions, but I can't
figure out a unified idea. Could anyone suggest some
detailed reading?
BTW, it seems that Debye potentials have close
relation with multipole expansion. Is this true and what's
that?
Thanks for any reply!
Thomas Smid
wrote:
> Dear all,
>
> Happy new year!
>
> I'd like to get a clear concept of Debye potentials.
> For the sake of this, I searched around the internet and
> checked several classic textbooks, like Jackson's and
> Stratton's, but no satisfactory results. Instead I get
> several papers describing Debye potentials published
> decades before ("Debye potential representation of
> vector fields").
>
> From those papers I find out that:
> Debye potentials have something to do with the special
> case of Helmholtz Theorem with divergenceless vector
> fields. It's proved then this field can be represented by
> two scalar potentials:
> where F is the vector field and L is the standard orbital
> angular momentum operator. It's said these two scalar
> potentials are Debye potentials. (Is this obsolete? Why
> isn't there any like content in today's textbooks)
>
> Except this I also get various descriptions, but I can't
> figure out a unified idea. Could anyone suggest some
> detailed reading?
>
> BTW, it seems that Debye potentials have close
> relation with multipole expansion. Is this true and what's
> that?
>
> Thanks for any reply!
Hi,
In plasma physics, the Debye potential is the potential arising from
the screening of a test charge by the free charges in the plasma (see
http://farside.ph.utexas.edu/teaching/plasma/lectures/node7.html ).
Note however that a fundamental assumption in this derivation is the
existence of a thermodynamic equlibrium i.e. a Boltzmann energy
distribution. This implies a collisionally dominated isothermal
situation where the pressure gradient exactly cancels the force due to
the electric field. The Debye potential is therefore the consequence
of the implicit assumption of collisions in thermodynamic equilibrium
preventing the purely electrostatic screening which would hold in a
collisionless plasma. However, collisions (and the related pressure
forces) should only be relevant in a plasma if the collision frequency
is higher than the plasma frequency (which determines the timescale
for the electrostatic re-arrangement of charges). Unless one is
dealing with a very low degree of ionization, this condition is only
satisfied for extremely high plasma densities as encountered in
solids, fluids or the interior of the sun.
Thomas
Timo A. Nieminen
Given the vector Helmholtz equation, how can we find a general solution?
For the scalar Helmholtz equation, we "simply" go ahead and use separation
of variables. In spherical coordinates, this gives us the scalar spherical
wave functions (ie, scalar wave multipoles). What we need is a recipe to
convert these to the vector solutions.
(I'm going from memory here, so beware error!)
If A is a solution of the scalar equation, then
L = rA (r = position vector)
is a solution of the vector equation. curl L = 0, so not of much use for
electromagnetics. However,
M = curl L
is a divergence free solution. Also,
N = (1/k) curl M
is also a divergence free solution. Note that M = (1/k) curl N.
So, at this point, we might expect that we can write an arbitrary solution
to the vector Helmholtz equation as
E = curl(rA) + (1/k) curl(curl(rB)) [1]
which is, apart from the (1/k), the Debye representation, for a
monochromatic field. We haven't proved this yet, since we haven't shown
that we can do this for all E.
So, let us go to the multiple expansion. Let us start with the scalar
multipoles. OK, we have S_nm, n = 0 to infinity, as our scalar multipoles.
We can write an arbitrary solution of the scalar Helmholtz equation as
U = sum_{n=0}^{n=infinity} sum_{m=-n}^{m=2} a_nm S_nm. [2]
We can use the above recipe on S_nm and obtain L_nm, M_nm, N_nm, which
together must be a complete basis set for solutions of the vector
Helmholtz equation. If we have div E = 0, we don't need L_nm, and we can
write
E = sum_{n=0}^{n=infinity} sum_{m=-n}^{m=2} a_nm M_nm + b_nm N_nm. [3]
I haven't gone through the details, and don't want to type them anyway,
but if you go through the details, you should be able to use the
definitions of M_nm and N_nm to convert [3] into [1], with A and B written
in the form of [2]. This shows that E can be written as [1] in all cases.
It also shows that, in general, A doesn't equal B (since usually
a_nm != b_nm).
Thus, we see that in the Debye representation, the potential A is the TE
part of the solution, and B the TM.
There might be an easier way to show that [1] is general, but I'm only
familiar with Debye potentials in the context of multipole expansions.
Related potentials are the Hertz potentials and the Bromwich potentials.
Stratton covers Hertz potentials. Why don't textbooks cover this in
detail? It's specialised. When do you use it? Debye and Bromwich
potentials are useful in Mie theory, and electromagnetic scattering in
general when using spherical coordinates. Usually only the advanced
textbooks cover Mie theory.
Potentials of these types are in current use. See for example, N. A.
Gumerov and R. Duraiswami, A scalar potential formulation and translation
theory for the time-harmonic Maxwell equations, Journal of Computational
Physics 225 (2007) 206-236.
For a nice compact review of the recipe and associated maths, see
Brock B. Using vector spherical harmonics to compute antenna mutual
impedance from measured or computed fields. Sandia report,
SAND2000-2217-Revised. Sandia National Laboratories, Albuquerque, NM,
2001.
--
Timo Nieminen - Home page: http://www.physics.uq.edu.au/people/nieminen/
E-prints: http://eprint.uq.edu.au/view/person/Nieminen,_Timo_A..html
Shrine to Spirits: http://www.users.bigpond.com/timo_nieminen/spirits.html
J. J. Lodder
> Dear all,
>
> Happy new year!
>
> I'd like to get a clear concept of Debye potentials.
> For the sake of this, I searched around the internet and
> checked several classic textbooks, like Jackson's and
> Stratton's, but no satisfactory results. Instead I get
> several papers describing Debye potentials published
> decades before ("Debye potential representation of
> vector fields").
>
> From those papers I find out that:
> Debye potentials have something to do with the special
> case of Helmholtz Theorem with divergenceless vector
> fields. It's proved then this field can be represented by
> two scalar potentials:
> F = Lø + curl(L=÷),
> where F is the vector field and L is the standard orbital
> angular momentum operator. It's said these two scalar
> potentials are Debye potentials. (Is this obsolete? Why
> isn't there any like content in today's textbooks)
The name of Debye is usually not associated with this decomposition.
Instead you can find the fields derived from the potentials
described as toroidal and poloidal fields.
They play a role in the theory of for example
the earth's dynamo, the sun's magnetic field,
or magnetically confined fusion devices.
> Except this I also get various descriptions, but I can't
> figure out a unified idea. Could anyone suggest some
> detailed reading?
>
> BTW, it seems that Debye potentials have close
> relation with multipole expansion. Is this true and what's
> that?
The basic toroidal fields are sigular on the Z-axis,
and the multipole expansion gets replaced by the same thing
with the Legendre polynomials Plm
replaced by the Legendre functions of the second kind,
Qlm.
Since the Legendre functions of the second kind
lack the nice orthoganality properties
of the corresponding Legendre polynomials
this multipole expansion of the second kind is less useful.
Best,
Jan
PuZHA...@gmail.com
> On 2 Jan, 10:54, "PuZHANG0...@gmail.com" <PuZHANG0...@gmail.com>
> wrote:
>
>
>
>
> > Happy new year!
>
> > I'd like to get a clear concept of Debye potentials.
> > For the sake of this, I searched around the internet and
> > checked several classic textbooks, like Jackson's and
> > Stratton's, but no satisfactory results. Instead I get
> > several papers describing Debye potentials published
> > decades before ("Debye potential representation of
> > vector fields").
>
> > From those papers I find out that:
> > Debye potentials have something to do with the special
> > case of Helmholtz Theorem with divergenceless vector
> > fields. It's proved then this field can be represented by
> > two scalar potentials:
> > where F is the vector field and L is the standard orbital
> > angular momentum operator. It's said these two scalar
> > potentials are Debye potentials. (Is this obsolete? Why
> > isn't there any like content in today's textbooks)
>
> > Except this I also get various descriptions, but I can't
> > figure out a unified idea. Could anyone suggest some
> > detailed reading?
>
> > BTW, it seems that Debye potentials have close
> > relation with multipole expansion. Is this true and what's
> > that?
>
> > Thanks for any reply!
>
>
> In plasma physics, the Debye potential is the potential arising from
> the screening of a test charge by the free charges in the plasma (seehttp://farside.ph.utexas.edu/teaching/plasma/lectures/node7.html).
>
> Note however that a fundamental assumption in this derivation is the
> existence of a thermodynamic equlibrium i.e. a Boltzmann energy
> distribution. This implies a collisionally dominated isothermal
> situation where the pressure gradient exactly cancels the force due to
> the electric field. The Debye potential is therefore the consequence
> of the implicit assumption of collisions in thermodynamic equilibrium
> preventing the purely electrostatic screening which would hold in a
> collisionless plasma. However, collisions (and the related pressure
> forces) should only be relevant in a plasma if the collision frequency
> is higher than the plasma frequency (which determines the timescale
> for the electrostatic re-arrangement of charges). Unless one is
> dealing with a very low degree of ionization, this condition is only
> satisfied for extremely high plasma densities as encountered in
> solids, fluids or the interior of the sun.
>
> Thomas
Thanks!
Actually the Debye potential I care is that related to Helmholtz
Theorem.
Now I'm clear what Debye potential is in my sense. Here's a list of
helpful
papers:
1, Debye potential representation of vector fields
2, Multipole expansions of electromagnetic fields using Debye
potentials
3, Debye Potentials by Wilcox
PuZHA...@gmail.com
> On Wed, 2 Jan 2008, PuZHANG0...@gmail.com wrote:
> > I'd like to get a clear concept of Debye potentials.
> > For the sake of this, I searched around the internet and
> > checked several classic textbooks, like Jackson's and
> > Stratton's, but no satisfactory results. Instead I get
> > several papers describing Debye potentials published
> > decades before ("Debye potential representation of
> > vector fields").
>
> > From those papers I find out that:
> > Debye potentials have something to do with the special
> > case of Helmholtz Theorem with divergenceless vector
> > fields. It's proved then this field can be represented by
> > two scalar potentials:
> > where F is the vector field and L is the standard orbital
> > angular momentum operator. It's said these two scalar
> > potentials are Debye potentials. (Is this obsolete? Why
> > isn't there any like content in today's textbooks)
>
> > Except this I also get various descriptions, but I can't
> > figure out a unified idea. Could anyone suggest some
> > detailed reading?
>
> > BTW, it seems that Debye potentials have close
> > relation with multipole expansion. Is this true and what's
> > that?
>
Thanks! Your reply is very illuminating.
Is your mentioned Hertz potential equivalent to polarization
potential? I think
Polarization potential is another thing bah.