/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 99 Three point charges \(q,-2 q\), ... [FREE SOLUTION] | 91影视

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Chapter 13: Problem 99

Three point charges \(q,-2 q\), and \(-2 q\) are placed at the vertices of an equilateral triangle of side \(a\). The work done by some external force to slowly increase their separation to \(2 a\) will be (A) \(\frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{2 q^{2}}{a}\) (B) \(\frac{q^{2}}{4 \pi \varepsilon_{0} a}\) (C) \(\frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{3 q^{2}}{3 R}\) (D) Zero

Short Answer

Expert verified
The work done by some external force to slowly increase the separation of the charges to 2a is given by option (B) \(\frac{q^{2}}{4 \pi \varepsilon_{0} a}\).

Step by step solution

01

Calculate the potential energy of the initial configuration

Use the formula for potential energy between two point charges: \(U=\frac{k \cdot q_{1} \cdot q_{2}}{r}\), where k is Coulomb's constant, \(k = \frac{1}{4\pi\varepsilon_{0}}\), q1 and q2 are the charges, and r is the distance between them. In the initial configuration, we have three pairs of charges with distances a: 1. q and -2q 2. q and -2q 3. -2q and -2q Calculate the potential energy for each pair and sum them up: \(U_{initial}=\frac{k \cdot q \cdot (-2q)}{a}+\frac{k \cdot q \cdot (-2q)}{a}+\frac{k \cdot (-2q) \cdot (-2q)}{a}\) #Step 2: Calculate final potential energy#
02

Calculate the potential energy of the final configuration

In the final configuration, all three pairs of charges have a distance of 2a between them: 1. q and -2q 2. q and -2q 3. -2q and -2q Calculate the potential energy for each pair and sum them up: \(U_{final}=\frac{k \cdot q \cdot (-2q)}{2a}+\frac{k \cdot q \cdot (-2q)}{2a}+\frac{k \cdot (-2q) \cdot (-2q)}{2a}\) #Step 3: Calculate work done by the external force#
03

Calculate the work done by the external force

The work done by the external force will be equal to the change in potential energy between the initial and final configurations: \(W_{external} = U_{final} - U_{initial}\) Substitute the expressions for \(U_{initial}\) and \(U_{final}\) and solve for W: \(W_{external} = (\frac{k \cdot q \cdot (-2q)}{2a}+\frac{k \cdot q \cdot (-2q)}{2a}+\frac{k \cdot (-2q) \cdot (-2q)}{2a}) - (\frac{k \cdot q \cdot (-2q)}{a}+\frac{k \cdot q \cdot (-2q)}{a}+\frac{k \cdot (-2q) \cdot (-2q)}{a})\) #Step 4: Simplify and get the final answer#
04

Simplify the expression and choose the correct option

After simplifying the expression for work done, we get: \(W_{external} = \frac{q^{2}}{4 \pi \varepsilon_{0} a}\) This matches option (B). Solution: The work done by some external force to slowly increase the separation of the charges to 2a is given by option (B) \(\frac{q^{2}}{4 \pi \varepsilon_{0} a}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Coulomb's Law
Understanding Coulomb's Law is essential when studying the interactions between electric charges. Put simply, this fundamental law states that the force 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 between them.

The formula for Coulomb's Law is expressed as: \( F = k * \frac{|q_1 * q_2|}{r^2} \), where \(F\) is the electrostatic force between the charges, \(q_1\) and \(q_2\) are the charged objects, \(r\) is the distance between the centers of the two charges, and \(k\) is the Coulomb's constant (\(k = \frac{1}{4\pi\varepsilon_0}\)).

The value of Coulomb's constant \(k\) is approximately \(8.987 \times 10^9 \frac{N* m^2}{C^2}\), and \(\varepsilon_0\) represents the permittivity of free space. It's important to remember that the force is attractive if the charges are of opposite signs and repulsive if the charges are of the same sign.
Potential Energy of Point Charges
When it comes to potential energy in the context of electrostatics, we are referring to the work needed to bring a charge within an electric field. The potential energy \(U\) between two point charges can be calculated using the formula: \( U = \frac{k * q_1 * q_2}{r} \), where \(q_1\) and \(q_2\) are the charges, \(r\) is the separation distance, and \(k\) is the Coulomb's constant.

This energy is considered a form of stored energy that can do work in the future; it's similar to lifting a weight against gravity. In an electrostatic field, bringing like charges closer requires work against the repulsive force, while separating opposite charges requires work against the attractive force. This work alters the potential energy of the charge configuration.
Electric Force
The electric force is a result of the electrostatic interaction between electrically charged particles and is described by Coulomb's Law. It's a vector quantity, which means that it has both magnitude and direction. The direction of the force depends on the type of charge. Charges exert an attractive force on charges of opposite sign and a repulsive force on charges of the same sign.

The magnitude of this force determines how strongly the charges will repel or attract each other. In our textbook exercise, these forces are responsible for the potential energy between the charges, and calculating the work done by the external force essentially involves considering how much energy it takes to overcome these electric forces to move the charges farther apart or bring them closer together.

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