# Simple harmonic motion of charged particles

• Alexthekid
In summary: Hint: you need something like ##F(x) = -kx##, but with ##k## a function of something else. What can you use that's a function of ##x## in your situation?)In summary, the conversation revolves around the motion of a third particle with charge -Q, which is free to move between two fixed particles with charge +q. The goal is to determine the period of motion for -Q along the perpendicular bisector of the two fixed charges. By considering the case where the distance between -Q and the midpoint of the fixed charges (x) is small compared to the distance between the two fixed charges (d), it can be shown that the motion of -Q is simple harmonic. The period

## 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

Alexthekid said:

## 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?

Alexthekid said:
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?

## 1. What is simple harmonic motion?

Simple harmonic motion is a type of periodic motion in which a particle moves back and forth along a straight line, with its acceleration proportional to its displacement from a fixed point.

## 2. How is simple harmonic motion of charged particles different from regular simple harmonic motion?

In simple harmonic motion of charged particles, the particle experiences a restoring force due to its interaction with an electric or magnetic field, while in regular simple harmonic motion, the restoring force is due to a physical spring or other mechanical device.

## 3. How is the motion of charged particles affected by the strength of the electric or magnetic field?

The strength of the electric or magnetic field directly affects the amplitude and frequency of the simple harmonic motion of charged particles. The stronger the field, the greater the displacement and the shorter the period of the motion.

## 4. Can simple harmonic motion of charged particles occur in more than one dimension?

Yes, simple harmonic motion of charged particles can occur in multiple dimensions, such as in a circular motion or in a plane. In such cases, the restoring force is still proportional to the displacement from a fixed point, but the motion is no longer confined to a straight line.

## 5. What are some real-world applications of simple harmonic motion of charged particles?

Simple harmonic motion of charged particles has many practical applications, including in particle accelerators, mass spectrometers, and oscillating circuits in electronics. It is also used in medical imaging techniques such as magnetic resonance imaging (MRI).