Harmonic Motion of a Charged Particle

As ##r = l \pm x##, ##r^2 = l^2 \pm 2lx + x^2##. When ##x \ll l##, ##r^2 \approx l^2##. Thus, ##F = \frac{kqQ}{r^2} \approx \frac{kqQ}{l^2}##. This is a constant value, and therefore the particle will undergo simple harmonic motion with frequency ##\omega = \sqrt{\frac{kqQ}{ml^3}}##.In summary, for a particle with positive charge q and mass m initially located between two positive point charges Q at points (±l, 0) and constrained
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Homework Statement



Two positive point charges Q are located at points (±l, 0). A particle with positive charge q and mass m is initially located midway between them and is then given a tiny kick. If it is constrained to move along the line joining the two charges Q, show that it undergoes simple harmonic motion (for small oscillations), and find the frequency.

Homework Equations



F = (kqQ)/r^2

The Attempt at a Solution


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I believe the force on the center particle should be F = (kqQ)/(l-x)^2 - (kqQ)/(l+x)^2

I think I need to find that the force is equal to some constant multiplied by x to show that there is simple harmonic motion, but I’m not sure how to do it.

 
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1. What is harmonic motion of a charged particle?

Harmonic motion of a charged particle refers to the oscillatory movement of a charged particle in an electric or magnetic field. The motion is periodic and is characterized by a constant frequency and amplitude.

2. What factors affect the harmonic motion of a charged particle?

The harmonic motion of a charged particle is affected by the strength of the electric or magnetic field, the mass and charge of the particle, and the initial conditions of the motion.

3. How is harmonic motion of a charged particle described mathematically?

The motion of a charged particle in an electric or magnetic field can be described using the equations of motion, such as Newton's second law and Lorentz force law. The resulting equation is a second-order differential equation, which can be solved to obtain the position and velocity of the particle at any given time.

4. What is the significance of harmonic motion of a charged particle?

Harmonic motion of a charged particle has many practical applications, such as in particle accelerators, mass spectrometers, and particle detectors. It is also an important concept in understanding the behavior of charged particles in various physical systems.

5. Can harmonic motion of a charged particle be observed in everyday life?

Yes, harmonic motion of a charged particle can be observed in everyday life. For example, the motion of electrons in a television screen, the vibrations of a guitar string, and the motion of charged particles in a plasma ball are all examples of harmonic motion of charged particles.

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