Two Isolated Charges, Q And -4q, Are Separated By A Distance R. They Are In Uniform Circular Motion Due To Their Mutual Electrostatic Force. If Each Particle Has A Mass M, What Is Their Angular Speed? (k = 1/(4πε₀))

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Introduction

In the captivating realm of physics, the interplay between charged particles and electrostatic forces gives rise to a mesmerizing dance of orbital motion. Imagine two isolated charges, one carrying a charge of q and the other a charge of -4q, separated by a distance r. These charged entities, bound by their mutual electrostatic attraction, embark on a uniform circular motion, tracing elegant paths dictated by the fundamental laws of physics. Our quest in this exploration is to unravel the secrets of their orbital choreography, specifically, to determine their angular speed, denoted by the symbol ω. The angular speed, a measure of how swiftly an object rotates or revolves relative to a central point, serves as a key descriptor of this celestial ballet at the atomic level.

Delving into the intricacies of this electrostatic interaction, we introduce a pivotal constant, k, which represents the electrostatic constant, mathematically expressed as k = 1/(4πε₀), where ε₀ symbolizes the vacuum permittivity. This constant encapsulates the strength of the electrostatic force in a vacuum, playing a crucial role in dictating the magnitude of the attraction or repulsion between charged particles. Furthermore, we acknowledge that each particle participating in this orbital dance possesses a mass m, a fundamental property that influences its inertia and response to external forces.

To embark on this investigative journey, we will invoke the foundational principles of electrostatics and classical mechanics. We will delve into Coulomb's law, the cornerstone of electrostatics, which quantifies the force between charged particles. We will also employ Newton's second law of motion, a cornerstone of classical mechanics, which relates force, mass, and acceleration. By judiciously applying these laws and principles, we will construct a mathematical framework that allows us to decipher the angular speed of these charged particles as they gracefully orbit one another.

This exploration into the orbital mechanics of charged particles not only provides a deeper understanding of fundamental physical principles but also lays the groundwork for comprehending a myriad of phenomena, from the behavior of atoms and molecules to the intricate workings of plasma physics. The insights gained from this analysis extend far beyond the confines of theoretical physics, finding applications in diverse fields such as materials science, nanotechnology, and astrophysics.

Deciphering the Electrostatic Force and Circular Motion

Electrostatic Force: The cornerstone of our analysis lies in understanding the nature of the electrostatic force that governs the interaction between the two charges. Coulomb's law, a fundamental principle in electrostatics, elegantly quantifies this force. It states that the electrostatic force (F) between two point charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance separating them. Mathematically, this law is expressed as:

F = k * |q₁ * q₂| / r²

where q₁ and q₂ represent the magnitudes of the two charges, r is the distance separating them, and k is the electrostatic constant, as previously defined. In our specific scenario, the charges are q and -4q, so the magnitude of the electrostatic force becomes:

F = k * |q * (-4q)| / r² = 4kq² / r²

The negative sign in the charge -4q indicates that it has an opposite polarity to the charge q, implying that the force between them is attractive. This attraction is the very force that binds the particles together in their orbital dance.

Uniform Circular Motion: The problem statement explicitly states that the charged particles are engaged in uniform circular motion. This means that they are moving in circular paths at a constant speed. A crucial consequence of this motion is the presence of centripetal acceleration, a phenomenon where an object moving in a circle experiences acceleration directed towards the center of the circle. The magnitude of this centripetal acceleration (a_c) is given by:

a_c = v² / r

where v is the tangential speed of the particle and r is the radius of the circular path. Furthermore, the tangential speed (v) is related to the angular speed (ω) by the following equation:

v = rω

where ω represents the rate at which the object rotates, measured in radians per second. By substituting this expression for v into the equation for centripetal acceleration, we obtain:

a_c = (rω)² / r = rω²

This equation reveals a direct proportionality between centripetal acceleration and the square of the angular speed. This relationship is pivotal in our quest to determine the angular speed of the charged particles.

Applying Newton's Second Law to Decipher Angular Speed

Newton's Second Law of Motion: To bridge the gap between electrostatic forces and circular motion, we invoke Newton's second law of motion, a cornerstone of classical mechanics. This law states that the net force acting on an object is equal to the product of its mass and its acceleration. Mathematically, this is expressed as:

F = ma

where F is the net force, m is the mass of the object, and a is its acceleration. In our scenario, the electrostatic force we calculated earlier acts as the net force causing the centripetal acceleration of the charged particles. Therefore, we can equate the electrostatic force to the centripetal force, which is the force required to keep an object moving in a circular path:

F_electrostatic = F_centripetal

Using Newton's second law, we can express the centripetal force as:

F_centripetal = ma_c = mrω²

Equating Forces and Solving for Angular Speed: Now, we have two expressions for the force acting on the charged particles: the electrostatic force (F_electrostatic = 4kq²/r²) and the centripetal force (F_centripetal = mrω²). By equating these two expressions, we establish a direct link between the electrostatic interaction and the circular motion of the particles:

4kq² / r² = mrω²

Our goal is to isolate the angular speed (ω) and solve for it. To do this, we perform a series of algebraic manipulations:

  1. Divide both sides of the equation by mr:

4kq² / (mr³) = ω²

  1. Take the square root of both sides:

ω = √(4kq² / (mr³))

  1. Simplify the expression:

ω = 2q√(k / (mr³))

This final expression elegantly reveals the angular speed (ω) of the charged particles in terms of fundamental physical quantities: the charge q, the electrostatic constant k, the mass m, and the separation distance r. This equation serves as the culmination of our analytical journey, providing a quantitative answer to our initial question.

Unveiling the Significance of the Angular Speed Equation

The angular speed equation, ω = 2q√(k / (mr³)), encapsulates a wealth of information about the orbital dynamics of the charged particles. It reveals how the angular speed is intricately intertwined with the fundamental properties of the system, such as the charges, mass, and separation distance.

  • Charge (q): The equation highlights a direct proportionality between the angular speed (ω) and the magnitude of the charge q. This means that as the magnitude of the charges increases, the angular speed of the particles also increases. The stronger electrostatic attraction between larger charges leads to a more vigorous orbital motion.
  • Electrostatic Constant (k): The electrostatic constant k appears under the square root, indicating a positive correlation with angular speed. A larger value of k signifies a stronger electrostatic force, resulting in a higher angular speed. This aligns with our understanding that stronger forces lead to more rapid motion.
  • Mass (m): The mass m resides in the denominator under the square root, signifying an inverse relationship with angular speed. Heavier particles, possessing greater inertia, exhibit lower angular speeds. This is consistent with our intuition that heavier objects are more resistant to changes in their motion.
  • Separation Distance (r): The separation distance r appears in the denominator raised to the power of 3/2, indicating a strong inverse relationship with angular speed. As the separation between the charges increases, the angular speed diminishes rapidly. This is because the electrostatic force weakens significantly with increasing distance, leading to a slower orbital motion.

Implications and Applications: The angular speed equation transcends mere theoretical significance, offering profound insights into various physical phenomena and finding practical applications in diverse fields.

  • Atomic Physics: At the atomic level, electrons orbit the nucleus due to electrostatic attraction. The angular speed equation provides a framework for understanding the motion of electrons in atoms, contributing to our understanding of atomic structure and behavior.
  • Plasma Physics: Plasmas, composed of ionized gases, involve interactions between charged particles. The equation aids in comprehending the dynamics of charged particles within plasmas, crucial for applications such as fusion energy research and plasma processing.
  • Astrophysics: In astrophysical settings, charged particles can interact through electrostatic forces. The equation helps in analyzing the motion of charged particles in cosmic environments, such as planetary magnetospheres and accretion disks around black holes.

Conclusion: A Symphony of Physics in Orbital Motion

Our journey into the orbital dance of charged particles has culminated in a profound understanding of the interplay between electrostatic forces and circular motion. By invoking the fundamental principles of Coulomb's law and Newton's second law, we have successfully derived an equation that elegantly quantifies the angular speed of these particles: ω = 2q√(k / (mr³)). This equation serves as a testament to the power of physics in deciphering the intricacies of the natural world.

The angular speed equation not only provides a numerical answer but also illuminates the intricate relationships between physical quantities. It reveals how the angular speed is influenced by the charges, mass, separation distance, and the electrostatic constant, offering a deeper appreciation for the fundamental forces that govern our universe.

The implications of this analysis extend far beyond the specific scenario of two charged particles. The principles and insights gained find applications in diverse fields, from atomic physics to astrophysics, underscoring the universality and power of fundamental physical laws.

As we conclude this exploration, we are reminded of the beauty and elegance inherent in the laws of physics. The orbital dance of charged particles, governed by the fundamental forces of nature, serves as a captivating example of the intricate symphony that unfolds at the heart of the cosmos. This exploration not only enriches our understanding of the physical world but also inspires further inquiry into the mysteries that await our discovery.