Question

Imagine two classical charges of \(-q\), each bound to a central charge of \(+4 .\) One \(-q\) charge is in a circular orbit of radius \(R\) about its \(+q\) charge. The other oscillates in an extreme ellipse, essentially a straight line from its \(+q\) charge out to a maximum distance \(r_{\max }\) The two orbits have the same energy. (a) Show that \(r_{\max }=2 R .\) (b) Considering the time spent at each orbit radius, in which orbit is the \(-q\) charge farther from its \(+q\) charge on average?

Short answer

The maximum distance \(r_{\max }\) for the oscillating charge is \(2R\). The \(-q\) charge on the oscillating path spends more time being on average farther from its \(+q\) charge than the \(-q\) charge on the circular orbit.

Step-by-step solution

Establish the given facts

It's clear that both charges have the same energy in both orbits. This implies that the total energy (kinetic energy + potential energy) would be the same for both orbits.

Finding the maximum distance \(r_{\max }\)

For the charge oscillating in a straight line at maximum distance, the kinetic energy would be zero (at the point of return). Therefore, its total energy equates to its potential energy at \(r_{\max }\). We know that the potential energy at any distance \(r\) for a charge \(q\) in a field of charge \(Q\) is given by \( -Qq/r \). Equating this to the energy of the circular orbit (whose potential energy at \(R\) can be written as \( -4q/R \)), we get: \[ -4q/R = -4q/r_{\max} \] Solving for \( r_{\max} \), we get \( r_{\max} = 2R \).

Comparing the average distances of the charges

For the charge on the circular path, the distance from the +q charge is always \(R\), while for the charge on the oscillating path, the distance varies between 0 and \(2R\). However, the \( -q \) charge spends most of the time near the turnaround point, thus on average it's farther from the \( +q \) charge compared to the other \(-q\) charge in the circular orbit.

Key concepts

Orbital Mechanics

Orbital mechanics, also known as celestial mechanics, involves the motions of objects under the influence of gravitational forces.
For the exercise where a charge orbits around a central charge, it involves a balance between centrifugal force and the electrostatic force between the two charges.

• The orbit can be circular or elliptical, with the central force providing the necessary acceleration for the charged particle to follow its path.
• In a stable circular orbit, the electrostatic force, which acts as a centripetal force, must equal the particle's inertia resisting its change in motion.
• For elliptical orbits, the dynamics become more complex as the distance and speed of the orbiting charge change, influencing its kinetic and potential energy.

Application to the Exercise

In our scenario, the '-q' charge in the circular orbit remains at a constant distance, maintaining a steady balance between the electrostatic force and its inertial force. On the other hand, the second '-q' charge experiences more complex dynamics in its oscillating trajectory, akin to the behavior of an object in an elliptical orbit. This illustrates fundamental orbital mechanics principles – the constant interplay of forces dictates the system's stability and motion.

Potential Energy in Electrostatics

Potential energy in electrostatics relates to the energy stored in the configuration of electric charges.

• It is calculated based on the position of a charge within an electric field and is given by the equation \( U = -\frac{Qq}{r} \), where \( Q \) and \( q \) are the magnitudes of two charges and \( r \) is the distance between them.
• This energy is considered 'potential' because it represents the potential work that these charges can do if they were allowed to move freely.
• Since the force between charges is conservative, the potential energy only depends on their relative positions, not on the charges' path history.

Application to the Exercise

The potential energy concepts clearly explain why the two charges in different orbits have the same total energy. For the '-q' charge in the straight line oscillation, the potential energy reaches its maximum at the point of return, \( r_{\max} \), correlating inversely with the distance from the central charge. Likewise, the circular orbiting charge has a fixed distance and thus a constant potential energy. The equation derived from equating their potential energies showcases how their configurations relate and why \( r_{\max} = 2R \) for the charge in the elliptical trajectory.

Energy Conservation in Physics

Energy conservation is a fundamental principle in physics which states that the total energy in a closed system remains constant over time.

• It's important to recognize two main categories of energy: kinetic energy, associated with the motion of objects, and potential energy, related to their position in a force field.
• As an object moves within a force field, these types of energy can transform into one another, but the total energy remains the same if no external work is done or heat is transferred.
• Energy conservation allows us to solve physical problems by recognizing that certain properties, such as total energy, do not change, making it easier to predict a system's behavior over time.

Application to the Exercise

In our exercise, despite the differences in their motion, both charges preserve their total energy throughout their trajectories. For the oscillating '-q' charge, when it momentarily stops at \( r_{\max} \), all of its energy is potential. The kinetic energy becomes zero, highlighting the transformation of energy types while adhering to the law of conservation of energy. Comparing this state to the circular orbit helps us conclude the relationship between their distances, reiterating the indispensability of the energy conservation principle in analyzing and solving physical problems.