Determine Q² Using K=9×10⁹N.m²/C² Given That An Unknown Charge (q²) And Another Of 7×10⁻⁶C Repel Each Other With 10 N At A Distance Of 0.20 M.

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Introduction: Delving into the Realm of Electrostatic Forces

In the fascinating realm of electromagnetism, the concept of electric charge and the forces it generates plays a pivotal role. Understanding how charged objects interact is crucial for comprehending a wide range of phenomena, from the behavior of atoms and molecules to the workings of electronic devices. This article will delve into a specific scenario involving two charged objects: one with an unknown charge denoted as q² and another with a known charge of 7 × 10⁻⁶ C. These charges, separated by a distance of 0.20 meters, exert a repulsive force of 10 N upon each other. Our primary objective is to unravel the mystery of the unknown charge q² by employing the fundamental principles of electrostatics, specifically Coulomb's Law, and utilizing the given constant k = 9 × 10⁹ N⋅m²/C².

Before we embark on the journey of solving for q², it is essential to lay a solid groundwork by revisiting some key concepts in electrostatics. Electric charge, a fundamental property of matter, can be either positive or negative. Objects with the same type of charge repel each other, while those with opposite charges attract. The magnitude of the electrostatic force between two charged objects is governed by Coulomb's Law, a cornerstone of electromagnetism. This law states that the force is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance separating them. Mathematically, Coulomb's Law is expressed as:

F = k * (|q₁ * q₂|) / r²

where:

  • F represents the magnitude of the electrostatic force
  • k is Coulomb's constant, approximately 9 × 10⁹ N⋅m²/C²
  • q₁ and q₂ denote the magnitudes of the charges
  • r signifies the distance between the charges

Understanding these foundational principles is crucial for effectively tackling the problem at hand. We will meticulously apply Coulomb's Law, carefully considering the given information and employing algebraic manipulation to isolate and determine the value of the unknown charge q². This endeavor will not only provide a numerical solution but also deepen our understanding of the interplay between charge, force, and distance in the electrostatic world.

Applying Coulomb's Law: The Key to Unlocking the Unknown Charge

Now, let's apply the powerful tool of Coulomb's Law to decipher the value of the unknown charge q². We are given the following information:

  • The repulsive force (F) between the charges is 10 N.
  • One charge (q₁) is 7 × 10⁻⁶ C.
  • The distance (r) separating the charges is 0.20 m.
  • Coulomb's constant (k) is 9 × 10⁹ N⋅m²/C².

We need to determine the value of the other charge, which we have denoted as q₂ (which is q² in the original problem statement). To accomplish this, we will strategically rearrange Coulomb's Law equation to isolate q₂. The original equation is:

F = k * (|q₁ * q₂|) / r²

Multiplying both sides by r² and then dividing by k * |q₁|, we obtain:

|q₂| = (F * r²) / (k * |q₁|)

This rearranged equation provides us with a direct pathway to calculate the magnitude of q₂. Now, we simply substitute the known values into the equation:

|q₂| = (10 N * (0.20 m)²) / (9 × 10⁹ N⋅m²/C² * |7 × 10⁻⁶ C|)

Performing the arithmetic calculations, we get:

|q₂| = (10 N * 0.04 m²) / (9 × 10⁹ N⋅m²/C² * 7 × 10⁻⁶ C)

|q₂| = 0.4 N⋅m² / (63 × 10³ N⋅m²/C)

|q₂| ≈ 6.35 × 10⁻⁶ C

Therefore, the magnitude of the unknown charge q₂ (q²) is approximately 6.35 × 10⁻⁶ C. However, we must consider the nature of the force between the charges. Since the problem states that the charges repel each other, we know that they must have the same sign. Given that q₁ is positive (7 × 10⁻⁶ C), q₂ must also be positive. Thus, the unknown charge q² is +6.35 × 10⁻⁶ C. This meticulous application of Coulomb's Law has allowed us to successfully determine the value of the unknown charge, highlighting the law's power in quantifying electrostatic interactions. The solution demonstrates the importance of not only applying the formula correctly but also interpreting the physical context of the problem to ascertain the correct sign of the charge.

Considerations and Implications: Exploring the Broader Context

Having successfully calculated the value of the unknown charge q², it's crucial to pause and reflect on the implications of this result and the factors that influence electrostatic interactions. We determined that q² is approximately +6.35 × 10⁻⁶ C. This value, in conjunction with the known charge of 7 × 10⁻⁶ C, the separation distance of 0.20 m, and Coulomb's constant, dictates the magnitude of the repulsive force between the charges, which we were given as 10 N. This intricate relationship underscores the sensitivity of electrostatic forces to changes in charge magnitude and distance. For instance, if we were to increase the magnitude of either charge or decrease the distance between them, the repulsive force would increase significantly, as dictated by Coulomb's Law.

Furthermore, it's important to consider the limitations and assumptions inherent in our analysis. We have operated under the assumption that the charges are point charges, meaning that their physical dimensions are negligible compared to the distance separating them. This simplification allows us to apply Coulomb's Law directly without needing to account for charge distributions across extended objects. In reality, charges are often distributed over volumes or surfaces, and calculating the electrostatic force in such scenarios can become significantly more complex, often requiring integration techniques. Another assumption we've made is that the charges are in a vacuum. The presence of a dielectric material between the charges would reduce the electrostatic force due to the polarization of the material. Coulomb's constant, k, is strictly applicable in a vacuum; in a medium, it is replaced by k/εr, where εr is the relative permittivity (dielectric constant) of the medium.

The broader implications of understanding electrostatic forces extend to numerous scientific and technological domains. In chemistry, electrostatic interactions play a critical role in chemical bonding, determining the structure and properties of molecules. In materials science, these forces govern the behavior of charged particles in semiconductors and other materials. In biology, electrostatic interactions are essential for protein folding, enzyme catalysis, and the interactions between biological molecules. Moreover, electrostatic principles underpin the operation of various technologies, including capacitors, electrostatic generators, and particle accelerators. By delving into the intricacies of electrostatic forces, we gain a deeper appreciation for the fundamental interactions that shape the world around us, paving the way for innovation and advancements in diverse fields.

Conclusion: The Power of Coulomb's Law in Unveiling Electrostatic Mysteries

In conclusion, our exploration into the interaction between an unknown charge q² and a known charge of 7 × 10⁻⁶ C, separated by a distance of 0.20 m and repelling each other with a force of 10 N, has been a testament to the power and elegance of Coulomb's Law. By meticulously applying this fundamental principle of electrostatics, we successfully determined the value of the unknown charge q² to be approximately +6.35 × 10⁻⁶ C. This process involved not only the direct application of Coulomb's Law but also a careful consideration of the physical context, particularly the repulsive nature of the force, which allowed us to correctly infer the sign of the unknown charge.

Throughout our analysis, we highlighted the crucial role of Coulomb's Law in quantifying the electrostatic force between charged objects. The law's dependence on the product of the charges and the inverse square of the distance underscores the sensitivity of electrostatic interactions to variations in charge magnitude and separation. We also delved into the assumptions and limitations of our approach, emphasizing the idealized conditions of point charges in a vacuum. While these simplifications allowed us to apply Coulomb's Law directly, we acknowledged that real-world scenarios often involve charge distributions and dielectric media, necessitating more sophisticated analytical techniques.

The broader implications of our investigation extend far beyond the specific problem at hand. Electrostatic forces are ubiquitous in nature, playing a fundamental role in diverse fields such as chemistry, materials science, biology, and various technological applications. From the formation of chemical bonds to the operation of electronic devices, electrostatic interactions govern a vast array of phenomena. By mastering the principles of electrostatics and developing a deep understanding of Coulomb's Law, we equip ourselves with a powerful tool for unraveling the mysteries of the electromagnetic world and paving the way for future scientific discoveries and technological advancements. The journey of exploring the interaction between these charges has not only provided a numerical solution but also illuminated the profound connections between fundamental physical laws and the world we inhabit.