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.
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
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))
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.