# Simple harmonic motion of charged particles

## Homework Statement

Two identical particles, each having charge +q, are fixed in space and separated by a distance d. A third particle with charge -Q is free to move and lies initially at rest on the perpendicular bisector of the two fixed charges a distance x from the midpoint between the two fixed charges. See image.

a) Show that if x is small compared with d, the motion of -Q is simple harmonic along the perpendicular bisector. Determine the period of that motion.

b) determine the period of that motion

c) How fast will the charge -Q be moving when it is at the midpoint between the two fixed charges if initially it is released at a distance a<<d from the midpoint?

v_max=?

## The Attempt at a Solution

After following another thread I resolved the Force on -Q in the y direction to be F = -2kqQ/(x2+d2/4) ⋅ x/√(x2+d2/4) I am unsure where to go from here

Ray Vickson
Homework Helper
Dearly Missed

## Homework Statement

Two identical particles, each having charge +q, are fixed in space and separated by a distance d. A third particle with charge -Q is free to move and lies initially at rest on the perpendicular bisector of the two fixed charges a distance x from the midpoint between the two fixed charges. See image.

a) Show that if x is small compared with d, the motion of -Q is simple harmonic along the perpendicular bisector. Determine the period of that motion.

b) determine the period of that motion

c) How fast will the charge -Q be moving when it is at the midpoint between the two fixed charges if initially it is released at a distance a<<d from the midpoint?

v_max=?

## The Attempt at a Solution

After following another thread I resolved the Force on -Q in the y direction to be F = -2kqQ/(x2+d2/4) ⋅ x/√(x2+d2/4) I am unsure where to go from here

The question tells you exactly what to do: look at the case of small ##|x|## (that is, ##|x| \ll d##). Do you know HOW to do that?

Wouldn't the fraction on the right go to zero?

Ray Vickson