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Multipole expansion. Problems with understanding derivatives

  1. Jun 26, 2013 #1
    Hi everyone

    1. The problem statement, all variables and given/known data

    I want to find the multipole expansion of

    [tex] \Phi(\vec r)= \frac {1}{4\pi \epsilon_0} \int d^3 r' \frac {\rho(\vec r')}{|\vec r -\vec r'|}[/tex]



    2. Relevant equations

    Taylor series

    3. The attempt at a solution

    My attempt at a solution was to use the Taylor series. I tried to approach 1/|r-r'| around r'/r (that's what the task told me to, because r>>r').

    I found the taylor series for [tex] \frac {1}{\sqrt{(1-x)}}[/tex]

    Which I can use I guess for this problem, where my x is:
    [tex]2 \frac {\vec r\vec r'}{r^2} -\frac {r'^2}{r^2}[/tex]

    so I get

    [tex] \frac {1}{| \vec r - \vec r'|}= \frac {1}{r} \frac {1}{\sqrt{ 1- 2 \frac {\vec r \vec r'}{r^2} + \frac {r'^2}{r^2}}} [/tex]


    But now I'm stuck. I don't know how to handle the derivates. Do I only have to derive the vector r' with nabla or r'^2 aswell?

    Thanks for your help in advance.
     
  2. jcsd
  3. Jun 26, 2013 #2

    TSny

    User Avatar
    Homework Helper
    Gold Member

    If I understand your question, you will need to take partial derivatives of the r'2 term as well as the r##\cdot##r' term with respect to x', y', and z'. Just think of the argument of the square root as some function of x', y', and z'.
     
  4. Jun 26, 2013 #3
    Use the vector form of Taylor Expansion i.e.
    f(x+h) = f(x) + (h.grad)f(x) + [(h.grad)^2]f(x) + ...

    where x and h are vectors, grad is the usual gradient operator and "." indicates the dot product.
     
  5. Jun 27, 2013 #4
    Thanks for your help. I know that this has something to do with Nabla, but I don't understand why I have to use Nabla here actually.
     
  6. Jun 27, 2013 #5
    you just have to use the binomial expansion for the term which you got in op.Different higher order terms in r'/r represents monopoles,dipoles,quadrupoles etc.
     
    Last edited: Jun 27, 2013
  7. Jun 27, 2013 #6
    I think what you're referring to as nabla is what I call grad.

    Just do the vector taylor expansion as I mentioned, this was the box standard thing to do back in electrodynamics exams. Oh, and use summation convention to make life easier.
     
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