Question

According to Coulomb's law, rank the interactions between charged particles from lowest potential energy to highest potential energy. $$ \begin{array}{l}{\text { a. } a 1+\text { charge and a } 1-\text { charge separated by } 100 \mathrm{pm}} \\ {\text { b. } a2+\text { charge and a } 1-\text { charge separated by } 100 \mathrm{pm}} \\ {\text { c. } a 1+\text { charge and a } 1+\text { charge separated by } 100 \mathrm{pm}} \\ {\text { d. } a1+\text { charge and a } 1-\text { charge separated by } 200 \mathrm{pm}}\end{array} $$

Short answer

From lowest to highest potential energy, the interactions rank as follows: d, a, b, c.

Step-by-step solution

Understanding Coulomb's Law

The potential energy (U) between two charges is given by Coulomb's law: \( U = k \frac{q_1 q_2}{r} \), where \( k \) is Coulomb's constant, \( q_1 \) and \( q_2 \) are the amounts of the charges, and \( r \) is the separation between the charges. The potential energy is lower when opposite charges interact and higher when like charges interact. Also, the potential energy increases with the magnitude of charge and decreases with distance.

Evaluate Charge Interactions

Assign values to the charge interactions in each scenario. For opposite charges, the product will be negative, and for like charges, it will be positive. Higher magnitude charge products result in higher potential energy, and longer separation distances result in lower potential energy. Since the potential energy is inversely proportional to the distance, the effect of doubling the distance will reduce the potential energy by a factor of 2.

Rank the Scenarios by Potential Energy

Rank the scenarios based on the magnitude of charge and distance: a and d have opposite charges; b has one charge twice as large as a and the same distance; c has like charges with the same magnitude and distance as a. For opposite charges, a more negative product means a lower potential energy. For like charges, a more positive product means a higher potential energy.

Determining the Lowest to Highest Potential Energy

Comparing the properties: (d) \( 1+ , 1- \) at 200 pm has the least negative product, implying the lowest potential energy. (a) \( 1+ , 1- \) at 100 pm has a more negative product than (d), so it has higher potential energy than (d), but still lower than (b) and (c). (b) has the largest magnitude product but opposite charges, so it is higher than (a) and (d), but lower than (c). (c) has like charges and will have the most positive product, thus the highest potential energy.

Key concepts

Potential Energy in Charge Interactions

The concept of potential energy in charge interactions is crucial to understanding the forces that charged particles exert on each other. Potential energy, in an electrostatic context, represents the work that would be required to arrange the charges in a particular configuration from an infinite separation. This energy arises due to the position of the charges relative to each other and can be classified as electrostatic potential energy.

According to Coulomb's Law, when two charges are opposite, they attract, and the potential energy of their interaction is negative. This means it would require work to pull them apart. Conversely, if the charges are the same, they repel, resulting in positive potential energy, indicating that work would need to be done to push them closer together.

Understanding this concept is fundamental when solving problems related to charge interactions, as it helps to determine the amount of work associated with moving charges against the electrostatic forces between them. With this knowledge, students can better analyze scenarios like those presented in the textbook exercise, where comparing potential energies is crucial.

Coulomb's Constant

Coulomb's constant, represented by the symbol 'k', plays a significant role in electrostatics and is a part of Coulomb's Law, the formula that calculates the electrostatic force between two point charges. The value of Coulomb's constant is approximately \( 8.9875 \times 10^9 \ N \cdot m^2/C^2 \), and it is derived from the permittivity of free space.

Coulomb's constant essentially quantifies the strength of the electrostatic force for a pair of charges one meter apart in a vacuum. This constancy assures that no matter where we are, given a certain pair of charges and their separation, the force and potential energy between them can be consistently calculated. Having a grasp on Coulomb's constant is important as it allows students to apply Coulomb's Law correctly when determining the force and energy between charges.

Magnitude of Charge

The magnitude of charge in electrostatic interactions refers to the quantity of electric charge present in an object, typically measured in coulombs (C). Charges can be positive or negative and are fundamentally carried by protons and electrons, respectively.

The larger the magnitude of charge, the more significant the force and potential energy between charges become. This is why in the exercise, option (b), which involves a \(2+\) charge, has a higher potential energy with a \(1-\) charge compared to option (a), which has both \(1+\) and \(1-\) charges at the same distance. It's essential for students to understand that doubling the charge does not simply double the potential energy; it actually increases the energy by a factor of four due to the squared relationship in Coulomb's Law.

Separation Distance Effect

The separation distance effect refers to how the distance between two charged objects influences the magnitude of the electrostatic force and potential energy between them. According to Coulomb's Law, the force and potential energy are inversely proportional to the square of the separation distance between the charges, denoted by the variable 'r'.

For example, if the distance between two charges is doubled, the force and potential energy between them would decrease by a factor of four (since \(2^2 = 4\)). This demonstrates the significant impact that distance has on the interactions between charges. In the context of the exercise, scenario (d) has the same charges as (a) but twice the separation distance, hence it has a quarter of the potential energy. Comprehending the separation distance effect allows students to predict how changes in distance will affect the interaction between charges.