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Usefulness of multipole expansion of skalar potential

  1. Sep 10, 2009 #1
    Upper undergraduate here. Lot of time spent in studying, but can't find acceptable answers in what follows.

    To be more specific, my questions are related on "Classical Electrodynamics", Jackson 2nd edition, Sect. 4.2. (and 4.1 of course), titled "Multipole expansion of the Energy of a Charge distribution in an External field".

    1. There is relation (4.24) which is expansion of energy:
    [tex]W=q\Phi(0)- \textbf{p}\cdot\textbf{E}(0)-\frac{1}{6}\sum\sumQ_{ij}\frac{\partial E_{j}}{\partialx_{i}}(0)+...[/tex]

    And after that follows "This expansion shows characteristic way in which the various multipoles intreact with an external field (the charge with potential, the dipole with the electric field, the quadrupole with the field gradient, and so on".

    I am afraid that I cant understand what means, for example, that "dipole interact with el. field" and how can that be usefull? Can anyone explain this please.

    2. In my (incomplete) notes from lectures there is statement about aplication of above - that "choosing different external el. field we can measure terms in expansion... For example, if E=const, third and higher terms vanish, because there is no changing in E, so W depends only of monopole and dipole terms".

    These are very rough notes, so please if I can get more elaborate explanation.

    3. (And third question is probably included in above two) How multipole expansion of scalar potential can give us some information about shapes of charge distribution?

    4. If one ask me generally what multipole expansion is for? What concise answer should I give?

    Sorry for too much questions, but this drives me crazy. I cant find answres in books, and cant get reasonable explanations in my mind, inspite of lot of time already invested.

    Thanks in advance.
  2. jcsd
  3. Sep 10, 2009 #2


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    This is really useful as an approximation - if you are far away from a bunch of charges, what will it look like to you? How fast will it fall off with range? If you are really far away, the first non-zero term in the multipole expansion will give you a good approximation to the answers of these questions.

    Such expansions turn out to be useful for radiation calculations as well - the algebra is a little more complicated but it is the same idea. Jackson covers that in later chapters, but if you get the basic idea from statics then you understand it.

    I am an engineer and these kind of approximations are key to doing quick and dirty feasibility estimates.

    So it is worth learning, for sure. If Jackson is too opaque, other simpler books such as Griffiths' well known book also covers multipole expansions in a friendlier way.

    good luck,

  4. Sep 11, 2009 #3


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    I think jasonRF got it. Multipole expansions can be a good way of accurately approximating the field due to various charge or current distributions (currents can have multipole expansions too). Increasing the number of terms of the poles will give you terms that increase their dependence on r-i[/sup. So from an appreciable distance, the higher order terms will not contribute and you can estimate the source as a summation over a few terms of the multipole expansion. This is a very useful technique in simulations of electromagnetics and gravitational bodies we can condense the far field effect of a group of sources into a multipole expansion which can save immensely on the amount of CPU time and memory needed.
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