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Harmonic motion and Electrical Fields

  1. Jun 13, 2015 #1
    1. The problem statement, all variables and given/known data
    Two positive charges +Q are affixed rigidly to the x-axis, one at x = +d and the other at x = -d. A third charge +q of mass m, which is constrained to move only along the x-axis, is displaced from the origin by a small distance s << d and then released from rest. (a) Show that (to a good approximation) +q will execute simple harmonic motion and determine an expression for its oscillation period T. (b) If these three charges are each singly ionized sodium atoms (q = Q = +e) at the equilibrium spacing d = 3 x 10-10 m typical of the atomic spacing in a solid, find T in picoseconds.

    2. Relevant equations
    Fnet = F1 + F2
    F = k * Q1 * Q2 / r2
    T = 2 * π * √(m/k) (Simple Harmonic Motion)
    Q = q = chargeelectron = 1.602 x 10-19 C

    3. The attempt at a solution
    I solved part (a) but am getting the wrong answer for part (b).

    Answer part a: T = π * √((m * d3) / (k * Q * q))

    (b):
    Sodium Ion (Na+)
    Molar MassNa+ = 22.9898 g/mol = 0.0229898 Kg/mol
    Mass per Na+ molecule = 0.0229898 Kg/mol / 6.02 molecules/mol = 3.8189 x 10-26 Kg
    dequilibrium = 3 x 10-10 m

    T = π * √( (3.8189 x 10-26 Kg) * (3 x 10-10 m)3 / ((8.99 x 109 N⋅m2 / C) * (1.602 x 10-19 C)2))

    = π * √( 1.031103 x 10-54 kg⋅m2 / 2.307 x 10-28 N⋅m2 )

    = π * 6.685 x 10-14 Kg/N

    = 2.1 x 10-13 seconds (?)

    = 0.2 picoseconds (the correct answer is 2 picoseconds).

    Are my units wrong? I found some mistakes while writing this post but after fixing them I am off by 1x101.
     
  2. jcsd
  3. Jun 13, 2015 #2

    mfb

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    2016 Award

    Staff: Mentor

    Your units do not match. Check the Coulombs.
    Fixing that, I get the same answer as you got. If formula (a) is right, then the 2 picoseconds could be a mistake in the given solution.
     
  4. Jun 13, 2015 #3
    I'll fix my units. The answer for question (a) is correct according to the book solutions.

    EDIT: I misread the answer, it is 0.2 ps. So there is no problem (besides my units).
     
    Last edited: Jun 13, 2015
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