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Approximating the force on a dipole Taylor series

  1. Jan 13, 2019 #1
    1. The problem statement, all variables and given/known data
    Show that the magnitude of the net force exerted on one dipole by the other dipole is given approximately by:$$F_{net}≈\frac {6q^2s^2k} {r^4}$$
    for ##r\gg s##, where r is the distance from one dipole to the other dipole, s is the distance across one dipole. (Both dipoles are of equal length and both have charges of magnitude q).

    2. Relevant equations
    ##F=\frac {kq_1q_2} {r^2}##
    ##f(x)=\sum_{n=0}^\infty \frac {f^{(n)}(0)} {n!} x^n##

    3. The attempt at a solution
    I worked out the net force that a dipole would be acting on another as: $$F_{net}=kq^2(\frac {1} {(r-s)^2}+\frac {1} {(r+s)^2}-\frac {2} {r^2})$$ This equation is right because I plugged in values for r and s given r is much greater than s, and got the same value for if I used the approximated equation at the top.

    I just lack the knowledge of using a taylor series to approximate my equation into the desired one.
     
  2. jcsd
  3. Jan 13, 2019 #2

    haruspex

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    First step is to get everything inside parentheses into the form 1±α, by suitable division, where α<<1.
     
  4. Jan 14, 2019 #3
    Ok, so then I get $$F_{net}=kq^2(\frac {1} {(1-α)^2}+\frac {1} {(1+α)^2}-\frac {2} {r^2})$$ Do I then expand both the parentheses into polynomials?
     
    Last edited: Jan 14, 2019
  5. Jan 14, 2019 #4

    haruspex

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    I wrote "get it into the form, by suitable division", not "arbitrarily change it". It still has to follow from the equation you wrote in post #1.
    (Or did you make a mistake in typing out the post?)
     
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