Simple harmonic motion of charged particles

[Alexthekid]

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=?

Homework Equations

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 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=?

Homework Equations

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?

 

[Alexthekid]

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

 

[Ray Vickson]

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

Small $|x|$, not zero!

Think of it this way: what sort of force equation would you need in order to have Newton's laws give you simple harmonic motion? That is, if $\text{Force} = F(x),$ for some function $F(x)$, what type of function $F$ do you need? Can you get such a function of $x$ in your "electrical" case, at least if $|x|$ is small enough?