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X>>R assumptions in magnetism

  1. May 13, 2015 #1
    1. The problem statement, all variables and given/known data

    I am asked to show that

    [tex]B = \frac{\mu_0Q\omega}{2\pi R^2}[\frac{R^2+2x^2}{(R^2+x^2)^{1/2}}-2x][/tex]

    simplifies to this

    [tex]B \approx \frac{\mu_0}{2\pi}\frac{\mu}{x^3}[/tex]

    if x>>R

    where [itex]\mu[/itex] is the magnetic dipole moment for a disk spinning with angular velocity [itex]\omega[/itex], which is

    [tex]\mu = \frac{Q\omega R^2}{4}[/tex]


    2. Relevant equations
    GYza5r.png



    3. The attempt at a solution


    I ignored the R^2 in the denominator since it has become a small quantity. Then I have (R^2+2x^2)/(sqrt(x^2)) -2x
    From this I get R^2/x but this equation lacks the third degree of the x.
    The book has a solution for this problem as I have posted but I didn't understand it either.
     
  2. jcsd
  3. May 13, 2015 #2

    mfb

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    2016 Award

    Staff: Mentor

    Small, but not irrelevant - you subtract two "large" numbers with a small difference from each other, so the first order of the difference is relevant. See the second line in (2), where the (inverse) denominator gets expanded up to second order (the part with "...").
     
  4. May 13, 2015 #3

    NascentOxygen

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    Staff: Mentor

  5. May 13, 2015 #4
    I learned it in high school and it was only for integers. This is something new but I'll try to handle it, thank you. What exactly has to be done after writing the series expansion?
     
    Last edited: May 13, 2015
  6. May 13, 2015 #5

    NascentOxygen

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    Staff: Mentor

    You substitute the first few terms of the infinite series (writing it in the numerator) in place of the square-root expression in the denominator, then multiply and simplify.
     
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