Question

Under the influence of the Coulomb field of charge \(+Q\), a charge \(-q\) is moving around it in an elliptical orbit. Find out the correct statement(s). (A) The angular momentum of the charge \(-q\) is constant (B) The linear momentum of the charge \(-q\) is constant (C) The angular velocity of the charge \(-q\) is constant (D) The linear speed of the charge \(-q\) is constant

Short answer

The correct statement regarding the charge q\text{'}s motion is: (A) The angular momentum of the charge q\text{'} is constant.

Step-by-step solution

Understand the Conservation of Angular Momentum

The angular momentum of a particle in orbit under central forces, like the Coulomb force, is conserved. This is because there is no external torque acting on the system, which means angular momentum must remain constant according to the conservation laws.

Analysing the Linear Momentum

The linear momentum, given by the product of mass and velocity, is not conserved for a particle in elliptical orbit because the direction and magnitude of velocity change continuously as the particle moves around its orbit.

Inspecting the Angular Velocity

The angular velocity of a particle in an elliptical orbit is not constant. It varies according to Kepler's second law, which states that the line segment joining the particle and the focal point sweeps out equal areas during equal intervals of time. Thus, the particle moves faster when it is closer to the focus.

Examining the Linear Speed

The linear speed of the charge is also not constant in an elliptical orbit. According to Kepler's second law, the particle moves faster when it is closer to the central charge and slower when it is farther away, which means that the linear speed changes.

Key concepts

Conservation of Angular Momentum

Understanding the conservation of angular momentum is crucial when analyzing celestial movements, such as those occurring in elliptical orbits. The angular momentum of an object is defined by its tendency to continue rotating around a point or axis. In the context of an elliptical orbit, such as the path traced by a charge (-q) around a central charge (+Q), the angular momentum is given by the product of the object's mass, its velocity, and the perpendicular distance from the axis of rotation.

Due to the absence of external torques in such a system, the angular momentum remains constant. This principle explains option (A) as correct in our exercise. Even as the charge (-q) moves through different positions in its orbit, the product of its mass, the velocity component perpendicular to the radius vector, and the distance from the center of force (in this case, the central charge (+Q)) remains unchanged, assuring the conservation of angular momentum.

Linear Momentum Variability

Linear momentum, unlike angular momentum, is characteristically variable in an elliptical orbit. It is defined as the product of the object's mass and its linear velocity. The key factor in the variability of linear momentum in such orbits is the changing velocity of the object as it travels around the central force.

As the charge (-q) moves along its orbital path, its velocity vector changes both in magnitude and direction. When it is closer to the central charge, it moves faster due to the stronger Coulomb force acting upon it and vice versa. This variation in speed and direction directly influences the linear momentum, causing it to change throughout the orbit. Hence, option (B) is not a correct statement, as the linear momentum is not constant.

Kepler's Second Law

Kepler's Second Law, also known as the Law of Equal Areas, plays a pivotal role in describing the motion of bodies in orbits. It states that a line segment joining a planet (or, in this scenario, a charge (-q)) and the Sun (or the focus of the orbit, which is the central charge (+Q)) sweeps out equal areas during equal intervals of time.

This law imbues the celestial motions with a rhythmic pattern: when the charge (-q) is closer to the focus, it traverses a shorter arc at a higher speed, and when it is farther away, it covers a longer arc at a slower speed, always sweeping out equal areas in equal times. Therefore, the angular velocity is not constant (rejecting option C), and the linear speed of the charge varies (refuting option D). Kepler's Second Law ensures the dynamic balance required for the stability of elliptical orbits.

Elliptical Orbit Dynamics

Elliptical orbit dynamics describe the complex motion of objects under the influence of a gravitational or electrostatic force, such as the Coulomb force in this exercise. An elliptical orbit is shaped by the varying distance between the orbiting object and the central force, leading to changes in speed and momentum as prescribed by both Kepler's laws and the law of conservation of angular momentum.

In such orbits, the body moves the fastest at its closest approach to the center (periapsis) and slowest at the farthest point (apoapsis). This reflects the conservation of energy, as the kinetic and potential energies trade off at different points in the orbit.

The dynamics of an elliptical orbit are fine-tuned by physical laws leading to a stable, albeit variable, motion that can be predicted and analyzed through the concepts covered like the conservation of angular momentum, the variability of linear momentum, and Kepler's Second Law.