Problem 8
Question
What is Coulomb's law? Explain how the potential energy of two charged particles depends on the distance between the charged particles and on the magnitude and sign of their charges.
Step-by-Step Solution
Verified Answer
Coulomb’s law states that the electrostatic force (\( F \) is directly proportional to the product of the electric charges and inversely proportional to the square of the distance between them. The electrostatic potential energy (\( U \) depends similarly on these quantities but lacks the square on the distance, and its sign indicates whether the force is attractive or repulsive.
1Step 1: Defining Coulomb’s Law
Coulomb's Law describes the force of attraction or repulsion between two stationary, electrically charged particles. The law states that the force (\( F \)) is directly proportional to the product of the electrical charges (\( q_1 \) and \( q_2 \) of the particles and inversely proportional to the square of the distance (\( r \) between them. Coulomb’s Law is mathematically given by \( F = k_e \frac{q_1 q_2}{r^2} \), where \( k_e \) is Coulomb's constant (\( k_e ≈ 8.9875 × 10^9 N m^2 C^{-2} \)).
2Step 2: Describing the Potential Energy between Charged Particles
The electrostatic potential energy (\( U \) of two charged particles depends on the same factors as the force between them. It is given by the equation \( U = k_e \frac{q_1 q_2}{r} \). The potential energy is proportional to the product of the charges and inversely proportional to the distance between them. The sign of the charges will determine if the potential energy is positive or negative: like charges (both positive or both negative) imply a positive potential energy (repulsion), while opposite charges (one positive and one negative) imply a negative potential energy (attraction).
3Step 3: Analyzing the Effect of Distance and Magnitude of Charges
As the distance (\( r \)) between the two charged particles increases, the potential energy (\( U \)) and the force between them (\( F \) decrease according to the equations provided. If the magnitude of either charge increases, while everything else remains constant, both the force and the potential energy between the particles will increase. The signs of the charges affect whether the force and energy are considered attractive or repulsive but do not change the magnitude of the force or energy itself.
Key Concepts
Electric ForceElectrostatic Potential EnergyCharge InteractionsForce of Attraction and Repulsion
Electric Force
The concept of electric force is central to understanding the interactions between charged particles. Imagine two particles in space—each carrying an electrical charge. The force that comes into play between these two particles is what we call the electric force. According to Coulomb's Law, this force can be attractive or repulsive and varies based on two crucial factors: the magnitude of the charges and the distance between them.
To put it simply, if you have two charged particles, the electric force (\( F \)) between them can be calculated using the equation \( F = k_e \frac{q_1 q_2}{r^2} \) where \( q_1 \) and \( q_2 \) represent the charges on the particles, \( r \) is the distance separating them, and \( k_e \) is the Coulomb constant. It's fascinating to see that when the charges are increased, the force escalates too, and as the distance grows, the force diminishes. This can be envisioned like magnets—which draw closer or repel based on their orientation—the electric force governs how charged particles interact with each other in a similar manner.
To put it simply, if you have two charged particles, the electric force (\( F \)) between them can be calculated using the equation \( F = k_e \frac{q_1 q_2}{r^2} \) where \( q_1 \) and \( q_2 \) represent the charges on the particles, \( r \) is the distance separating them, and \( k_e \) is the Coulomb constant. It's fascinating to see that when the charges are increased, the force escalates too, and as the distance grows, the force diminishes. This can be envisioned like magnets—which draw closer or repel based on their orientation—the electric force governs how charged particles interact with each other in a similar manner.
Electrostatic Potential Energy
When two charged particles are near each other, they possess electrostatic potential energy (\( U \)) due to their positions. This energy is a form of potential energy, which means it has the potential to do work, such as moving a particle against the electric force. The equation \( U = k_e \frac{q_1 q_2}{r} \) helps us calculate this energy, drawing on the same factors as the electric force—charge magnitudes and distance.
One interesting aspect of electrostatic potential energy is its sign. If the charges are alike (both positive or both negative), the potential energy is positive, and the particles repel each other. Conversely, if the charges are opposite (one positive and one negative), the potential energy is negative, indicating that the particles attract each other. This concept is key to understanding why charged particles move the way they do; like a ball rolling down a hill, charged particles naturally move in a way that reduces their electrostatic potential energy.
One interesting aspect of electrostatic potential energy is its sign. If the charges are alike (both positive or both negative), the potential energy is positive, and the particles repel each other. Conversely, if the charges are opposite (one positive and one negative), the potential energy is negative, indicating that the particles attract each other. This concept is key to understanding why charged particles move the way they do; like a ball rolling down a hill, charged particles naturally move in a way that reduces their electrostatic potential energy.
Charge Interactions
Understanding charge interactions is fundamental in grasping electrostatics. Charges can interact in two primary ways—by attracting or repelling each other. These interactions are governed by the types of charges involved; like charges repel and unlike charges attract.
The implication here is powerful. For example, electrons, which carry a negative charge, will repel one another. The same is true for protons, with their positive charge. However, an electron and a proton would be drawn to each other because of their opposite charges. This basic principle of charge interaction underpins not only Coulomb's Law but the very structure of atoms and molecules, electrical technologies, and many biological processes. The dance of attraction and repulsion at the microscopic level plays out in countless phenomena we observe in the natural world.
The implication here is powerful. For example, electrons, which carry a negative charge, will repel one another. The same is true for protons, with their positive charge. However, an electron and a proton would be drawn to each other because of their opposite charges. This basic principle of charge interaction underpins not only Coulomb's Law but the very structure of atoms and molecules, electrical technologies, and many biological processes. The dance of attraction and repulsion at the microscopic level plays out in countless phenomena we observe in the natural world.
Force of Attraction and Repulsion
The force of attraction and repulsion between charges explains why charged particles move the way they do. If two particles are positively charged or both negatively charged, they repel each other, much like two north poles or two south poles of magnets. On the other hand, a positive and a negative charge will attract each other, similar to opposite poles of a magnet.
One of the beauties of physics is its symmetry, and this principle is evident here. The same formula that we use to calculate the force of repulsion for like charges can also be used to calculate the force of attraction for unlike charges—merely with a switch in the sign. It's this push and pull at the heart of electric interactions that not only keeps electrons in orbit around a nucleus but also allows for the complex interplay of forces that we harness in electronic circuits and other applications of electrostatics.
One of the beauties of physics is its symmetry, and this principle is evident here. The same formula that we use to calculate the force of repulsion for like charges can also be used to calculate the force of attraction for unlike charges—merely with a switch in the sign. It's this push and pull at the heart of electric interactions that not only keeps electrons in orbit around a nucleus but also allows for the complex interplay of forces that we harness in electronic circuits and other applications of electrostatics.
Other exercises in this chapter
Problem 4
Who is credited with arranging the periodic table? How are the elements arranged in the modern periodic table?
View solution Problem 7
What is an electron configuration? Give an example.
View solution Problem 9
What is shielding? In an atom, which electrons tend to do the most shielding (core electrons or valence electrons)?
View solution Problem 10
What is penetration? How does the penetration of an orbital into the region occupied by core electrons affect the energy of an electron in that orbital?
View solution