Ch 25: The Electric Potential

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 25: The Electric Potential

Problem 42

Chapter 25, Problem 42

The four 1.0 g spheres shown in FIGURE P25.42 are released simultaneously and allowed to move away from each other. What is the speed of each sphere when they are very far apart?

Verified step by step guidance

Identify the key concept: The problem involves the conservation of energy. Initially, the system has only electrostatic potential energy due to the charges on the spheres. As the spheres move far apart, this potential energy is converted into the kinetic energy of the spheres.

Write the expression for the initial electrostatic potential energy of the system. For four charges, calculate the potential energy between each pair of charges using the formula: \( U = \frac{k q_1 q_2}{r} \), where \( k \) is Coulomb's constant, \( q_1 \) and \( q_2 \) are the charges, and \( r \) is the distance between them. Sum the potential energy contributions for all unique pairs.

Apply the conservation of energy principle: The total initial potential energy of the system is equal to the total final kinetic energy of the spheres. Use the formula for kinetic energy: \( KE = \frac{1}{2} m v^2 \), where \( m \) is the mass of each sphere and \( v \) is their speed.

Set up the equation: \( U_{initial} = 4 \times \frac{1}{2} m v^2 \), since there are four spheres, each with the same mass and speed. Solve for \( v \) in terms of the known quantities (\( k \), \( q \), \( r \), and \( m \)).

Substitute the given values for the charges, masses, and initial distances into the equation to calculate the speed \( v \). Ensure all units are consistent (e.g., convert grams to kilograms for mass).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Conservation of Energy

The principle of conservation of energy states that the total energy in a closed system remains constant over time. In this scenario, the potential energy stored in the system due to the gravitational or electrostatic forces between the spheres will convert into kinetic energy as they move apart. This relationship allows us to calculate the final speed of the spheres when they are far apart.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion, defined mathematically as KE = 1/2 mv², where m is the mass and v is the velocity of the object. As the spheres move away from each other, their potential energy decreases while their kinetic energy increases, allowing us to determine their speeds at large distances based on the initial conditions.

Electrostatic Force

Electrostatic force is the attractive or repulsive force between charged objects, described by Coulomb's law. In this problem, if the spheres are charged, the electrostatic repulsion will cause them to accelerate away from each other. Understanding this force is crucial for calculating the initial potential energy and how it transforms into kinetic energy as the spheres separate.

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