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Magnetic dipole moment derivation

  1. Feb 1, 2016 #1
    mdp3.png (2)




    Hi. I am having a problem with understanding how to approximate 1/R in the forms of equations written above.

    I took this equations from a blog, and it tells that I can use talyor polynomial. but I don't get there somehow.
     
  2. jcsd
  3. Feb 1, 2016 #2

    blue_leaf77

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    $$
    \sqrt{(x-x')^2+(y-y')^2+z^2} \approx \sqrt{x^2+y^2+z^2-2xx'-2yy'}
    $$
    where the terms square in ##x'## and ##y'## have been omitted. Then make substitution ##r^2 = x^2+y^2+z^2##,
    $$
    r\sqrt{1-2\frac{x}{r^2}x'-2\frac{y}{r^2}y'}
    $$
    and apply Taylor expansion truncating the third term.
     
  4. Feb 1, 2016 #3
    Hello, I am really sorry but could you provide me the taylor expansion of it?
     
  5. Feb 2, 2016 #4

    blue_leaf77

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    To simplify the appearance, you can make the substitution ##-2\frac{x}{r^2}x'-2\frac{y}{r^2}y' = u## so that
    $$
    r\sqrt{1-2\frac{x}{r^2}x'-2\frac{y}{r^2}y'} = r(1+u)^p
    $$
    where ##p=1/2##. Now look up online or in your textbook examples of Taylor series, especially the series which corresponds to a form ##(1+u)^p## with ##|u|<1## as is the case here.
     
  6. Feb 3, 2016 #5
    Thanks. I got it!
     
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