The Problem Describes Two Electric Charges, Q1 = 7 X 10^-6 C And Q2 = -7 X 10^-6 C, In Equilibrium, Separated By 9 Cm. Determine The Tension In The String, In Newtons, Given G = 10 M/s² And K = 9 X 10^9 Nm²/C².

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In the realm of physics, understanding the interplay between electrostatic forces, gravitational forces, and tension is crucial. This article delves into a classic problem involving charged particles in equilibrium, suspended by a string. We will meticulously analyze the scenario, applying fundamental principles of electrostatics and mechanics to determine the tension in the string. Our specific case involves two charges, q1 = 7 x 10^-6 C and q2 = -7 x 10^-6 C, separated by a distance of 9 cm. Our mission is to calculate the tension in the string that suspends this system in equilibrium, considering the gravitational force acting on the charges and the electrostatic force between them. We'll use g = 10 m/s² for the acceleration due to gravity and K = 9 x 10^9 Nm²/C² for Coulomb's constant. Before diving into the solution, let's first explore the fundamental concepts that underpin this problem: electrostatic force, gravitational force, and the concept of equilibrium.

Fundamental Concepts

Electrostatic Force: The Attraction and Repulsion of Charges

The electrostatic force, a cornerstone of electromagnetism, governs the interaction between charged objects. This force can be either attractive or repulsive, depending on the signs of the charges involved. Objects with like charges (both positive or both negative) experience a repulsive force, pushing them apart. Conversely, objects with opposite charges (one positive and one negative) experience an attractive force, pulling them together. This fundamental interaction is described by Coulomb's Law, which mathematically quantifies the force:

F = K * |q1 * q2| / r²

Where:

  • F represents the magnitude of the electrostatic force.
  • K is Coulomb's constant, a proportionality factor (approximately 9 x 10^9 Nm²/C² in a vacuum).
  • q1 and q2 are the magnitudes of the two charges.
  • r is the distance separating the charges.

This equation reveals that the electrostatic force is directly proportional to the product of the charges' magnitudes. Larger charges exert stronger forces. The force is inversely proportional to the square of the distance between the charges. As the distance increases, the force diminishes rapidly. In our problem, we have two charges, one positive and one negative, meaning they will attract each other with a force dictated by Coulomb's Law.

Gravitational Force: The Earth's Pull

Gravitational force, a ubiquitous force, is the attraction between any two objects with mass. On Earth, we experience this force as the weight of an object, the pull of the Earth towards its center. The magnitude of the gravitational force (weight) is given by:

Fg = m * g

Where:

  • Fg represents the magnitude of the gravitational force.
  • m is the mass of the object.
  • g is the acceleration due to gravity (approximately 9.8 m/s² on Earth, often approximated as 10 m/s² for simplicity). In our problem, the charges will have some mass, and thus, experience a downward gravitational force. This force will be crucial in determining the overall equilibrium of the system.

Equilibrium: A State of Balance

Equilibrium is a state where the net force acting on an object is zero. In simpler terms, it's a state of balance where all forces acting on the object cancel each other out. For an object to be in equilibrium, two conditions must be met:

  1. The vector sum of all forces in the horizontal direction must be zero.
  2. The vector sum of all forces in the vertical direction must be zero.

In our problem, the charges are suspended by a string and are in equilibrium. This means the forces acting on them (electrostatic force, gravitational force, and tension in the string) must balance each other out. The tension in the string provides an upward force, counteracting the downward gravitational force and any vertical component of the electrostatic force.

Problem Setup and Solution

Defining the System and Forces

Let's break down our specific problem. We have two charges:

  • q1 = 7 x 10^-6 C (positive)
  • q2 = -7 x 10^-6 C (negative)

These charges are separated by a distance of 9 cm (0.09 meters). The negative charge indicates an attractive electrostatic force between the charges. We are given g = 10 m/s² and K = 9 x 10^9 Nm²/C². We need to find the tension (T) in the string. To solve this, we'll consider the forces acting on each charge. Let's assume the charges have equal mass 'm'. Each charge experiences:

  1. Gravitational Force (Fg): Acting downward, Fg = m * g
  2. Electrostatic Force (Fe): An attractive force between the charges, calculated using Coulomb's Law.
  3. Tension (T): The force exerted by the string, acting upwards.

Calculating the Electrostatic Force

Using Coulomb's Law, we can calculate the electrostatic force between the charges:

Fe = K * |q1 * q2| / r²

Fe = (9 x 10^9 Nm²/C²) * |(7 x 10^-6 C) * (-7 x 10^-6 C)| / (0.09 m)²

Fe = (9 x 10^9) * (49 x 10^-12) / (0.0081)

Fe = 54.44 N (approximately)

This is the attractive force between the two charges. Since the charges are positioned horizontally, this electrostatic force acts horizontally, pulling the charges towards each other.

Analyzing Forces in Equilibrium

Since the system is in equilibrium, the net force on each charge is zero. Let's consider the vertical forces. The tension in the string (T) must balance the gravitational force (Fg) acting on both charges. Assuming both charges have the same mass (m), the total gravitational force acting downwards is 2 * m * g. Thus:

T = 2 * m * g

However, we don't know the mass 'm' of the charges. This is a crucial point. The electrostatic force calculated above is a horizontal force. It does not directly contribute to the vertical equilibrium equation. The tension in the string primarily counteracts the gravitational force acting on the charges. If the problem statement doesn't provide the mass, we need to consider what the question is truly asking. If the charges are suspended vertically, the tension would simply equal the total weight (2mg). However, the presence of the electrostatic force suggests the configuration might be slightly different. The charges may be hanging at an angle due to the electrostatic attraction. This would introduce a vertical component to the electrostatic force, which would then influence the tension. Without a diagram or further information about the configuration (like the angle of the string), we can only make assumptions.

Two Possible Scenarios and Solutions

Let's consider two plausible scenarios:

Scenario 1: Vertical Suspension (Simplest Case)

If the charges are hanging perfectly vertically, the electrostatic force is purely horizontal and doesn't affect the vertical equilibrium. In this case, the tension in the string simply balances the total weight of the two charges:

T = 2 * m * g

Without the mass, we can't get a numerical answer. However, if the question implied that we should only consider the direct weight, then the problem is potentially flawed as it doesn't provide the mass. However, it's also possible the question is testing conceptual understanding, expecting you to recognize the need for the mass to calculate a numerical answer.

Scenario 2: Angled Suspension (Electrostatic Force with Vertical Component)

If the charges are hanging at an angle due to the electrostatic attraction, the electrostatic force will have both horizontal and vertical components. This makes the problem significantly more complex. We would need to know the angle to resolve the electrostatic force into its components. Let's say the angle between the string and the vertical is θ. The vertical component of the electrostatic force would be Fe * cos(θ). The vertical equilibrium equation would then be:

T = 2 * m * g + Fe * cos(θ)

Again, without knowing the angle θ or the mass m, we cannot obtain a numerical solution. This scenario highlights the importance of understanding the complete physical configuration.

Conclusion and Key Takeaways

Based on the information provided, we can calculate the electrostatic force between the charges, which is approximately 54.44 N. However, determining the tension in the string requires additional information, specifically the mass of the charges or the angle at which they are suspended. Without this information, we can only express the tension in terms of these unknowns.

Key Takeaways:

  • Electrostatic force plays a crucial role in the interaction of charged objects.
  • Equilibrium requires the net force on an object to be zero.
  • Tension in a string counteracts other forces, such as gravity and electrostatic forces.
  • Problem-solving in physics often involves identifying all relevant forces and applying the conditions for equilibrium.
  • Careful consideration of the problem setup and given information is essential for accurate solutions.

This problem serves as an excellent example of how multiple physics concepts intertwine to create a seemingly simple, yet potentially complex, scenario. The key to solving such problems lies in a thorough understanding of the underlying principles and a systematic approach to analyzing the forces involved. Always remember to consider all possible scenarios and identify any missing information that might be necessary for a complete solution. The tension in the string is a fascinating topic, especially when electrostatic forces come into play. Let’s make sure to address the tension in the string properly when dealing with such physics problems. It's crucial to consider both the gravitational pull and the repulsive or attractive electrostatic force that may be present. Understanding Coulomb's law and its implications on electrostatic equilibrium is key here. When we discuss charged particles in a system, particularly when they are hanging or suspended, the tension isn't just about countering the weight of the particles; it also involves balancing the electrical forces pushing or pulling these charges. This is where a strong grasp of physics principles really shines, allowing you to see how different forces interact and how they ultimately define the equilibrium state of the system. The combination of gravitational and electrical forces makes these problems not only interesting but also crucial for a deeper understanding of how our world works on a fundamental level.