Re: What are Debye potentials?

On Wed, 2 Jan 2008, PuZHANG0702@xxxxxxxxx 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:
F = Lų + 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?

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

Relevant Pages

  • Re: What are Debye potentials?
    ... Debye potentials have something to do with the special ... It's said these two scalar ... Given the vector Helmholtz equation, how can we find a general solution? ...
    (sci.physics.research)
  • Fun with vector coupling
    ... I will play with the current coupling term from EM. ... I thought there was nothing to do with a scalar term so ... so let's split the transverse part ... different kind of symmetry than the transverse potentials. ...
    (sci.physics.research)
  • Re: Aharonov-Bohm & scalar wave applications
    ... >waves interaction with atomic nuclei described by QCD?? ... >Also the scalar potential can propagate instantaneously everywhere ... othe particle or do they appear neutral like the potentials of two ...
    (sci.physics)
  • Re: Tapping the Vacuum
    ... > Scalar EM potentials are comprised of temporally bidirectional EM wave ... where the pairs are equal harmonics and become ...
    (sci.physics)