Physics by Halliday, Resnick and Krane: Electrical Charge & Coulomb's Law Complete Guide

Chapter 27: Electrical Charge and Coulomb's Law

Complete Physics Notes Based on Halliday, Resnick and Krane - B.Sc. Physics Edition 2015-16

Physics Halliday Resnick Krane Electromagnetism Coulomb's Law Reading Time: 15 min

📋 Table of Contents

• 1. Introduction to Electrical Charge
• 2. Electric Charge and Electrical Forces
• 3. Conductors and Insulators
• 4. Quantization of Charge
• 5. Conservation of Charge
• 6. Coulomb's Law
• 7. Vector Form of Coulomb's Law
• 8. Coulomb vs Newton's Law
• 9. Solved Problems
• Frequently Asked Questions

Introduction to Electrical Charge

⚡ Fundamental Nature of Electromagnetic Forces

Electromagnetic forces are responsible for the structure of atoms and for the binding of atoms in molecules and solids. Many properties of materials that we have studied so far are electromagnetic in their nature. Such as the elasticity of solids and the surface tension of liquids. The spring force, friction, and the normal force all originate with the electromagnetic force between atoms.

🔬 Scope of Electromagnetism

Among the examples of electromagnetism that we shall study are:

• The force between electric charges, such as occurs between an electron and the nucleus in an atom
• The motion of a charged body subject to an external electric force, such as an electron in an oscilloscope beam
• The flow of electric charges through circuits and the behavior of circuit elements
• The force between permanent magnets and the properties of magnetic materials
• Electromagnetic radiation, which ultimately leads to the study of optics, the nature and propagation of light

In this chapter, we begin with a discussion of electric charge, some properties of charged bodies, and the fundamental electric force between two charged bodies.

Electric Charge and Electrical Forces

🔋 What is Electric Charge?

A body is said to be electrical neutral if it contains equal number of positive and negative charges. When two bodies are rubbed together, their neutrality is disturbed due to transfer of electrons from one body to the other. The body which gives electrons becomes electrically positive and the body which gains electrons becomes negative.

⚡ Fundamental Principle of Charge Interaction

"Charges of the same signs repel each other and charges of the oppositely sign attract each other."

These attractive and repulsive forces among the charges are called electrical forces.

🏭 Applications of Electrical Forces

Electrical force between charged bodies has many industrial applications:

• Photocopying or xerography
• Ink-jet printing
• Electrostatic paint spraying
• Powder coating

📄 Xerography Process

The photocopying process is called "Xerography". The main parts of photocopier are:

• Lamp
• Rotating drum
• Toner
• Sheet of paper
• Heated roller

Q # 1. Explain what is meant by the term "a neutral atom." Explain what "a negatively charged atom" means.

A neutral atom is one that has no net charge. This means that it has the same number of electrons orbiting the nucleus as it has protons in the nucleus. A negatively charged atom has one or more excess electrons.

Q # 2. Explain from an atomic viewpoint why charge is usually transferred by electrons.

Electrons are less massive and more mobile than protons. Also, they are more easily detached from atoms than protons.

Q # 3. Would life be different if the electrons were positively charged and the protons were negatively charged? Does the choice of signs have any bearing on physical and chemical interactions? Explain.

No. Life would be no different if electrons were + charged and protons were – charged. Opposite charges would still attract, and like charges would repel. The naming of + and – charge is merely a convention.

Conductors and Insulators

🔌 Classification of Materials

Materials are classified based on their ability to conduct electric charge:

Insulators

The materials through which the charges don't flow are called insulators.

• Glass
• Chemically pure water
• Plastics

If the charge is placed on an insulator, the charges will stay where they are placed.

Conductors

The materials through which charge can flow easily are called conductors.

• Metals in general
• Tap water
• Human body

The copper rod cannot be charged because any charges placed on it easily flow through the rod, through your body (which is also a conductor), and to the ground. The insulating handle, however, blocks the flow and allows charge to build up on the copper.

🔬 Distinction Among Conductors, Insulators and Semi-Conductors

An experiment, called the Hall Effect, shows that it is the negative charges (electrons) that are free to move in metal. In metals, the atoms are so close to each other, their outermost shell is overlapped. The electrons in the outermost shells are already loosely bound; they are attracted by the neighboring nuclei and become free to move among the lattice atoms. These electrons are called free electrons or conduction electrons.

The distinction between conductors and insulators can be made on the basis of number of conduction electrons:

• In conductors: Each atom contributes one conduction electron. Therefore, there will be on the average about \(10^{23}\) conduction electrons per \(cm^3\).
• In insulators: At room temperature, it is very difficult to find even one conduction electrons per \(cm^3\).
• Intermediate between conductor and insulators are the semi-conductors e.g. Ge and Si, which might contain \(10^{10} - 10^{12}\) conduction electrons per \(cm^3\).

📌 Point Charges

The charge bodies whose sizes are much smaller than the distance between them are called point charges.

Quantization of Charge

🔢 Discrete Nature of Charge

When the two bodies are rubbed together, transfer of electrons from one body to the other takes place and they are said to be electrified. The magnitude of charge q that can be detected and measured on any object is given by:

Charge Quantization Formula

\[ q = ne \]

where \( n = 0, \pm 1, \pm 2, \ldots \) and \( e \) is the elementary unit of charge:

\[ e = 1.6 \times 10^{-19} \, C \]

🧩 Quantized Physical Quantity

When a physical quantity is discrete values, it is called quantized quantity.

This shows that charge is also a quantized quantity like matter, energy, angular momentum etc. It means that we can find a body that can have a charge of \( 10e \) or \(-5e \) but it is not possible to find a body with fractional charge such as \( +3.57e \) or \(-2.35e \).

⚛️ Quarks and Fractional Charges

According to the theory of elementary particles, protons and neutrons are not the elementary particles like electrons. They are consider to be composite particles made up of more fundamental particles called "QUARKS", which have fractional charges of magnitude \( +\frac{2}{3}e \) and \( -\frac{1}{3}e \).

Proton with the positive charge composed of:

\[ \frac{2}{3}e + \frac{2}{3}e - \frac{1}{3}e = \frac{3e}{3} = e \]

Neutron with the zero charge is made up of:

\[ -\frac{1}{3}e - \frac{1}{3}e + \frac{2}{3}e = 0 \]

Although there is a strong evidence of existence of quarks within the proton and neutrons, but yet it is impossible to create free quark.

Problem 24: Total Charge of Electrons

Question: Find total charge in coulombs of 75 kg of electron.

Solution:

Total mass of electrons \( m = 75 \, \text{kg} \)

Mass of one electron \( m_e = 9.1 \times 10^{-31} \, \text{kg} \)

Number of electrons \( n = \frac{m}{m_e} = \frac{75}{9.1 \times 10^{-31}} \)

\( n = 8.23 \times 10^{31} \)

Total Charge \( q = ne = 8.23 \times 10^{31} \times 1.6 \times 10^{-19} \)

\( q = 13.17 \times 10^{12} \, \text{C} \)

Conservation of Charge

⚖️ Fundamental Principle

When the two bodies are rubbed together, they are electrified. The process of rubbing does not create charge but only transfer it from one body to the other. Thus the charges can neither be created nor destroyed. This hypothesis is called conservation of charge.

Pair Production

When a high energy \( \gamma \)-ray photon strike the heavy nucleus:

\[ \gamma \rightarrow e^- + e^+ \]

The net charge is zero on both sides.

Pair Annihilation

When an electron and positron meet:

\[ e^- + e^+ \rightarrow \gamma + \gamma \]

The net charge is zero on both sides.

🧪 Nuclear Reactions and Charge Conservation

Decay of \(\pi^0\)-Meson

\[ \pi^0 \rightarrow \gamma + \gamma \]

Neutron Decay

\[ n \rightarrow p + e^- + \bar{\nu}_e \]

In all these processes, the total charge before and after the reaction remains the same.

Coulomb's Law

📏 Fundamental Law of Electrostatics

Charles Augustin Coulomb (1736-1806) measured electrical attractions and repulsions quantitatively and deduced the law that governs them.

📝 Statement of Coulomb's Law

The magnitude of electrical force between two point charges is directly proportional to the product of magnitude of charges and inversely proportional to the square of the distance between their centers.

Mathematical Formulation

Suppose two point charges \(q_1\) and \(q_2\) separated by distance \(r\):

\[ F \propto q_1 q_2 \]

\[ F \propto \frac{1}{r^2} \]

Combining Proportionalities

\[ F \propto \frac{q_1 q_2}{r^2} \]

\[ F = k \frac{q_1 q_2}{r^2} \]

Coulomb's Law Formula

\[ F = k \frac{q_1 q_2}{r^2} \]

where \(k\) is Coulomb's constant:

\[ k = \frac{1}{4\pi \epsilon_0} \]

\(\epsilon_0\) is permittivity of free space:

\[ \epsilon_0 = 8.85 \times 10^{-12} \, \text{C}^2 \text{N}^{-1} \text{m}^{-2} \]

Sample Problem 5: Nuclear Repulsion

Question: Nucleus of an iron atom has radius \( 4 \times 10^{-15} \, \text{m} \) and contains 26 protons. What electric repulsive force acts between them when separated by a distance of one radius?

Solution:

Radius \( r = 4 \times 10^{-15} \, \text{m} \)

Charge of proton \( q_1 = q_2 = 1.6 \times 10^{-19} \, \text{C} \)

From Coulomb's law:

\[ F = k \frac{q_1 q_2}{r^2} \]

\[ F = 9 \times 10^9 \times \frac{1.6 \times 10^{-19} \times 1.6 \times 10^{-19}}{(4 \times 10^{-15})^2} \]

\[ F = 14 \, \text{N} \]

Vector Form of Coulomb's Law

🧭 Directional Nature of Force

Force, being a vector, has directional properties as well. The direction of the force is determined by the relative sign of the two electric charges.

Vector Form of Coulomb's Law

\[ \vec{F}_{12} = k \frac{q_1 q_2}{r_{12}^2} \hat{r}_{12} \]

where:

• \( \vec{F}_{12} \) = force on charge 1 due to charge 2
• \( \hat{r}_{12} \) = unit vector from charge 2 to charge 1
• \( r_{12} \) = distance between charges

🔄 Mutual Force Property

Coulomb's force is a mutual force:

\[ \vec{F}_{12} = - \vec{F}_{21} \]

This satisfies Newton's Third Law of Motion.

🧮 Principle of Superposition

When multiple charges are present, the total force on any one charge is the vector sum of the forces due to each of the other charges.

\[ \vec{F}_1 = \vec{F}_{12} + \vec{F}_{13} + \vec{F}_{14} + \ldots + \vec{F}_{1n} \]

🔢 Coulomb Force Due to Many Point Charges

For charges \( q_1, q_2, q_3, \ldots, q_n \):

\[ \vec{F}_1 = \frac{1}{4\pi\epsilon_0} \sum_{i=1}^n \frac{q_1 q_i}{r_{1i}^2} \hat{r}_{1i} \]

This expression gives the electrical force on a point charge due to many point charges.

Coulomb vs Newton's Law

⚖️ Two Inverse-Square Laws

Coulomb's Law for electrical forces and Newton's Law of Gravitation both follow inverse-square relationships.

Comparison of Formulas

\[ F_{\text{elec}} = k \frac{q_1 q_2}{r^2} \]

\[ F_{\text{grav}} = G \frac{m_1 m_2}{r^2} \]

Aspect	Coulomb's Law	Newton's Law
Nature	Conservative force	Conservative force
Force Type	Attractive or repulsive	Only attractive
Constant	\( k \approx 9 \times 10^9 \)	\( G \approx 6.67 \times 10^{-11} \)
Medium	Depends on medium	Independent of medium

Solved Problems

Problem 4: Two Equally Charged Particles

Question: Two equally charged particles held 3.3 mm apart when released from rest. Initial acceleration of 1st particle is 7.22 m/s² and 2nd is 9.16 m/s². Mass of 1st particle is \( 6.31 \times 10^{-7} \, \text{kg} \). Find mass of 2nd particle and common charge.

Solution:

\( a_1 = 7.22 \, \text{m/s}^2 \)

\( a_2 = 9.16 \, \text{m/s}^2 \)

\( r = 3.3 \, \text{mm} = 3.3 \times 10^{-3} \, \text{m} \)

\( m_1 = 6.31 \times 10^{-7} \, \text{kg} \)

As both particles exert equal force:

\[ m_1 a_1 = m_2 a_2 \]

\[ m_2 = \frac{m_1 a_1}{a_2} \]

\[ m_2 = \frac{6.31 \times 10^{-7} \times 7.22}{9.16} \]

\[ m_2 = 4.97 \times 10^{-7} \, \text{kg} \]

Now for charge:

\[ F = m_1 a_1 = k \frac{q^2}{r^2} \]

\[ q^2 = \frac{m_1 a_1 r^2}{k} \]

\[ q^2 = \frac{6.31 \times 10^{-7} \times 7.22 \times (3.3 \times 10^{-3})^2}{9 \times 10^9} \]

\[ q^2 = 55.055 \times 10^{-4} \]

\[ q = 7.41 \times 10^{-2} \, \text{C} \]

Problem 36: Radioactive Decay Force

Question: In radioactive decay of \( U^{238} \), center of emerging \( He^4 \) particle is at distance \( 12 \times 10^{-15} \, \text{m} \) from center of residual \( Th^{234} \) nucleus. Find force on helium atom and its acceleration.

Solution:

\( U^{238} \rightarrow He^4 + Th^{234} \)

\( r = 12 \times 10^{-15} \, \text{m} \)

Charge of helium: \( q_1 = 2e \)

Charge of thorium: \( q_2 = 90e \)

\[ F = k \frac{q_1 q_2}{r^2} \]

\[ F = 9 \times 10^9 \times \frac{(2e) \times (90e)}{(12 \times 10^{-15})^2} \]

\[ F = 9 \times 10^9 \times \frac{180 \times (1.6 \times 10^{-19})^2}{(12 \times 10^{-15})^2} \]

\[ F = 288 \, \text{N} \]

Mass of helium atom:

\( m = 4 \, \text{amu} = 4 \times 1.67 \times 10^{-27} \, \text{kg} \)

\( m = 6.67 \times 10^{-27} \, \text{kg} \)

From Newton's 2nd law:

\[ F = ma \]

\[ a = \frac{F}{m} = \frac{288}{6.67 \times 10^{-27}} \]

\[ a = 43.18 \times 10^{27} \, \text{m/s}^2 \]

Frequently Asked Questions

❓ What is meant by a "neutral atom"?

A neutral atom is one that has no net charge. This means it has the same number of electrons orbiting the nucleus as it has protons in the nucleus.

❓ Why is charge usually transferred by electrons?

Electrons are less massive and more mobile than protons. They are also more easily detached from atoms than protons, making them the primary carriers of charge in most situations.

❓ Would physics be different if electrons were positively charged and protons negatively charged?

No, the physical and chemical interactions would be the same. Opposite charges would still attract, and like charges would still repel. The naming of positive and negative charge is merely a convention.

📚 Continue Your Physics Journey

Mastering electrical charge and Coulomb's Law is fundamental to understanding electromagnetism. These comprehensive notes based on Halliday, Resnick and Krane provide a solid foundation for further studies in physics.

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© 2025 Physics Education Initiative | Chapter 27: Electrical Charge and Coulomb's Law

These comprehensive notes are designed to help B.Sc. Physics students understand fundamental concepts of electrical charge and Coulomb's Law based on Halliday, Resnick and Krane

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