# Simple Harmonic Motion of Charges

1. Jul 18, 2010

### phatmomi

1. The problem statement, all variables and given/known data
A negatively charged particle -q is placed at the center of a uniformly charged ring of radius a having positive charge Q. The particle, confined to move along the x-axis, is moved a small distance x along the axis (x << a) and released. Show that th eparticle oscillates in simple harmonic motion with a frequency given by
f = (1/2pi)(kqQ/ma^3)^(1/2).

2. Relevant equations
F = kq1q2/r^2
torque = r x F
torque = (I)(alpha)
I (moment of inertia) = mL^2 (L = distance from point about which rotation occurs, in this case, approx. L = a)
alpha = d^2(theta)/dt^2
*Where theta is angle between a and the hypotenuse in the triangle with base and height a and x.

3. The attempt at a solution
I tried finding net force on -q as exerted by the ring (relevant force is only in x direction)
F = -kQq/a(a-x) + kQq/a(a+x)
F = -2kQqx/a^3

then I plugged this into torque = r x F
where r is approx. a
and equated with torque = I(alpha)
which gave me the motion for simple harmonic motion d^2(theta)/dt^2 + 2kQqx(theta)/ma^4
(I made use of the limit lim(∆theta-->0) sin(theta)/theta = 1)

and from this, the angular frequency should be (2kQqx/ma^4)^(1/2)
which gives f = (1/2(pi))(kQqx/ma^4)^(1/2)

What am I doing wrong?
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Jul 18, 2010

### vela

Staff Emeritus
It appears what you're doing wrong is blindly using formulas without understanding what they mean.

You didn't explain how the ring is oriented relative to the x-axis. Does it lie in the yz-plane with its center at the origin?