# Chapter 7: Electricity and Magnetism

## [First Half: Fundamentals of Electricity]

### 7.1: Introduction to Electric Charge

In this sub-chapter, we will explore the foundational concept of electric charge and its role in the physical world.

**Electric Charge**
Electric charge is a fundamental property of matter that can be either positive or negative. Positively charged particles are known as protons, while negatively charged particles are called electrons. These charges are the building blocks of atoms and molecules, and they play a crucial role in various physical and chemical phenomena.

**Charge Conservation**
The principle of charge conservation states that the total amount of electric charge in an isolated system remains constant; charge can neither be created nor destroyed, but it can be transferred or transformed. This means that the net charge of a closed system will always remain the same, even as individual charges may move or change.

**Atomic Structure and Charge**
Atoms, the fundamental units of matter, consist of a positively charged nucleus surrounded by negatively charged electrons. The number of protons in the nucleus determines the element, while the number of electrons determines the atom's overall charge. Neutral atoms have an equal number of protons and electrons, resulting in a net charge of zero.

**Charging by Contact and Induction**
Objects can become charged through two primary mechanisms: contact and induction. Contact charging occurs when two objects with different affinities for electrons come into physical contact, causing the transfer of electrons from one object to the other. Induction, on the other hand, involves the redistribution of charges within an object without direct contact, often due to the presence of a nearby charged object.

**Key Takeaways**

- Electric charge is a fundamental property of matter that can be positive or negative.
- The principle of charge conservation states that the total charge in an isolated system remains constant.
- Atoms consist of a positively charged nucleus and negatively charged electrons, with neutral atoms having an equal number of protons and electrons.
- Objects can become charged through contact or induction, which involve the transfer or redistribution of electrons.

### 7.2: Coulomb's Law and Electric Fields

In this sub-chapter, we will delve into Coulomb's law, which describes the force between stationary electric charges, and the concept of electric fields.

**Coulomb's Law**
Coulomb's law states that the force of interaction between two stationary point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. This relationship is mathematically expressed as:

F = k * (q1 * q2) / r^2

Where:

- F is the force of interaction between the charges
- k is the Coulomb constant, approximately 8.99 × 10^9 N⋅m^2/C^2
- q1 and q2 are the magnitudes of the two point charges
- r is the distance between the two charges

**Electric Fields**
An electric field is a region of space surrounding a charged object, within which other charged objects will experience a force. Electric fields can be visualized using electric field lines, which indicate the direction and strength of the field. The direction of the electric field lines points away from positive charges and toward negative charges.

**Superposition of Electric Fields**
When multiple charged objects are present, the resulting electric field is the vector sum of the individual electric fields created by each charge. This principle of superposition allows us to determine the net electric field at any point in space by adding the contributions from all the charges.

**Equipotential Surfaces**
Equipotential surfaces are regions in space where the electric potential has the same value. These surfaces are perpendicular to the electric field lines and can be used to visualize and understand the behavior of charged particles within an electric field.

**Key Takeaways**

- Coulomb's law describes the force of interaction between two stationary point charges, which is proportional to the product of the charges and inversely proportional to the square of the distance between them.
- Electric fields are regions of space surrounding charged objects, within which other charged objects will experience a force.
- Electric field lines can be used to visualize the direction and strength of the electric field.
- The net electric field at any point is the vector sum of the individual electric fields created by all the charges in the system.
- Equipotential surfaces are regions of constant electric potential, perpendicular to the electric field lines.

### 7.3: Electric Field Calculations

In this sub-chapter, we will explore the mathematical calculations involved in determining electric fields in various scenarios.

**Vector Addition of Electric Fields**
When multiple charges are present, the net electric field at a given point is the vector sum of the individual electric fields created by each charge. This can be calculated by adding the vector components of the individual electric fields.

**Gauss's Law and Symmetric Charge Distributions**
Gauss's law is a powerful tool for calculating electric fields in situations with symmetric charge distributions, such as infinite planes, spheres, and cylinders. By applying Gauss's law, we can determine the electric field without the need for explicit vector addition.

**Examples of Electric Field Calculations**

- Point Charge: The electric field created by a point charge follows an inverse-square relationship, with the field lines radiating outward from the charge.
- Infinite Charged Plane: The electric field above an infinite charged plane is constant and perpendicular to the plane's surface.
- Charged Sphere: The electric field inside a uniformly charged sphere is zero, while the field outside the sphere follows an inverse-square relationship.

**Electric Flux and Gauss's Law**
Electric flux is a measure of the number of electric field lines passing through a given surface. Gauss's law states that the total electric flux through any closed surface is proportional to the net charge enclosed by that surface.

**Key Takeaways**

- The net electric field at a point is the vector sum of the individual electric fields created by all the charges in the system.
- Gauss's law can be used to calculate electric fields in situations with symmetric charge distributions, without the need for explicit vector addition.
- The electric field created by a point charge, an infinite charged plane, and a charged sphere can be calculated using the principles of vector addition and Gauss's law.
- Electric flux is a measure of the number of electric field lines passing through a surface, and Gauss's law relates the electric flux to the net enclosed charge.

### 7.4: Electric Potential Energy and Potential

In this sub-chapter, we will explore the concepts of electric potential energy and electric potential, and how they are related to electric fields.

**Electric Potential Energy**
Electric potential energy is the potential energy possessed by a charged particle due to its position within an electric field. It is the work done by an external force in moving a charge from an infinite distance to a given point in the electric field.

**Electric Potential**
Electric potential is defined as the electric potential energy per unit charge at a given point in an electric field. It is the amount of work required to bring a unit positive charge from an infinite distance to that point in the field.

**Relationship between Electric Field and Potential**
The electric field and electric potential are closely related. The electric field is the negative gradient of the electric potential, meaning that the electric field points in the direction of the most rapid decrease in electric potential.

**Calculating Electric Potential**
The electric potential due to a point charge can be calculated using the formula:

V = k * (q / r)

Where:

- V is the electric potential
- k is the Coulomb constant
- q is the charge
- r is the distance from the charge

This formula can be extended to calculate the electric potential due to multiple charges or charge distributions.

**Equipotential Surfaces and Potential Differences**
Equipotential surfaces are regions in space where the electric potential has the same value. The potential difference between two points is the change in electric potential between those points.

**Key Takeaways**

- Electric potential energy is the potential energy possessed by a charged particle due to its position within an electric field.
- Electric potential is the electric potential energy per unit charge at a given point in an electric field.
- The electric field is the negative gradient of the electric potential, pointing in the direction of the most rapid decrease in potential.
- The electric potential due to a point charge can be calculated using the formula V = k * (q / r).
- Equipotential surfaces are regions of constant electric potential, and the potential difference between two points is the change in electric potential between those points.

### 7.5: Equipotential Surfaces and Electrical Potential

In this sub-chapter, we will delve deeper into the concept of equipotential surfaces and explore the calculation of electric potential in various scenarios.

**Equipotential Surfaces**
Equipotential surfaces are regions in space where the electric potential has the same value. These surfaces are perpendicular to the electric field lines, and charged particles will not experience a net force when moving along an equipotential surface.

**Calculating Electric Potential**
The electric potential due to a point charge can be calculated using the formula:

V = k * (q / r)

Where:

- V is the electric potential
- k is the Coulomb constant
- q is the charge
- r is the distance from the charge

This formula can be extended to calculate the electric potential due to multiple charges or charge distributions, such as charged conductors and charge configurations with symmetry.

**Examples of Electric Potential Calculations**

- Point Charge: The electric potential due to a point charge decreases inversely with the distance from the charge.
- Charged Sphere: The electric potential outside a uniformly charged sphere is the same as the potential due to a point charge at the center of the sphere.
- Parallel Plate Capacitor: The electric potential between two parallel charged plates is a linear function of the distance between the plates.

**Potential Differences and Electric Work**
The potential difference between two points in an electric field is the change in electric potential between those points. This potential difference can be used to perform work on a charged particle, as the work done is equal to the charge multiplied by the potential difference.

**Key Takeaways**

- Equipotential surfaces are regions in space where the electric potential has the same value, and they are perpendicular to the electric field lines.
- The electric potential due to a point charge can be calculated using the formula V = k * (q / r).
- The electric potential due to more complex charge distributions can be calculated by applying the principles of superposition.
- The potential difference between two points is the change in electric potential, and it can be used to calculate the work done on a charged particle moving between those points.

## [Second Half: Magnetism and Electromagnetism]

### 7.6: Magnetic Fields and Magnetic Forces

In this sub-chapter, we will explore the fundamental concepts of magnetic fields and the forces acting on moving charges within these fields.

**Magnetic Fields**
A magnetic field is a region of space surrounding a magnet or a moving electric charge, within which magnetic forces can be observed. Magnetic fields can be visualized using magnetic field lines, which indicate the direction and strength of the field.

**Magnetic Forces on Moving Charges**
When a charged particle moves through a magnetic field, it experiences a magnetic force that is perpendicular to both the direction of the particle's motion and the direction of the magnetic field. This force, known as the Lorentz force, is given by the formula:

F = q * v * B * sin(θ)

Where:

- F is the magnetic force
- q is the charge of the particle
- v is the velocity of the particle
- B is the magnetic field strength
- θ is the angle between the velocity and the magnetic field

**Charged Particle Motion in Magnetic Fields**
The motion of a charged particle in a magnetic field can take various forms, such as circular, helical, or oscillatory, depending on the initial conditions and the strength of the magnetic field.

**Diamagnetic, Paramagnetic, and Ferromagnetic Materials**
Materials can exhibit different magnetic properties based on the behavior of their atomic or molecular structure. Diamagnetic materials are weakly repelled by magnetic fields, paramagnetic materials are weakly attracted to magnetic fields, and ferromagnetic materials can become strongly magnetized.

**Key Takeaways**

- Magnetic fields are regions of space surrounding magnets or moving electric charges, where magnetic forces can be observed.
- The Lorentz force describes the magnetic force acting on a moving charged particle in a magnetic field, and it is perpendicular to both the particle's velocity and the magnetic field.
- The motion of a charged particle in a magnetic field can take various forms, such as circular, helical, or oscillatory.
- Materials can exhibit different magnetic properties, such as diamagnetism, paramagnetism, and ferromagnetism, based on the behavior of their atomic or molecular structure.

### 7.7: Sources of Magnetic Fields

In this sub-chapter, we will explore the various sources of magnetic fields and how they can be described using the right-hand rule.

**Permanent Magnets**
Permanent magnets are objects that exhibit a persistent magnetic field, even in the absence of an external field. This is due to the alignment of the magnetic dipoles within the material, which can be either naturally occurring (e.g., lodestone) or artificially created (e.g., rare-earth magnets).

**Current-Carrying Wires**
When an electric current flows through a wire, it creates a magnetic field around the wire. The direction and strength of this magnetic field can be determined using the right-hand rule, which states that if the thumb points in the direction of the current flow, the fingers will curl in the direction of the magnetic field lines.

**Solenoids and Electromagnets**
A solenoid is a coil of wire through which an electric current flows, creating a uniform magnetic field inside the coil. When a soft iron core is placed inside the solenoid, it becomes an electromagnet, which can produce a much stronger magnetic field.

**Magnetic Dipoles and the Right-Hand Rule**
Magnetic dipoles, such as those found in permanent magnets or current-carrying loops, have a north and a south pole. The direction of the magnetic field lines around these dipoles can be determined using the right-hand rule, where the fingers point in the direction of the field lines when the thumb points in the direction of the current flow or the north-to-south orientation of the dipole.

**Key Takeaways**

- Permanent magnets exhibit a persistent magnetic field due to the alignment of magnetic dipoles within the material.
- The magnetic field around a current-carrying wire can be determined using the right-hand rule, where the fingers curl in the direction of the magnetic field lines when the thumb points in the direction of the current flow.
- Solenoids and electromagnets can create strong, uniform magnetic fields by using coils of wire and soft iron cores, respectively.
- The right-hand rule can be used to determine the direction of the magnetic field lines around a magnetic dipole, with the fingers curling in the direction of the field lines and the thumb pointing in the direction of the north-to-south orientation of the dipole.

### 7.8: Electromagnetic Induction

In this sub-chapter, we will explore the phenomenon of electromagnetic induction, which describes the generation of an electromotive force (EMF) due to a changing magnetic field.

**Faraday's Law of Induction**
Faraday's law of induction states that a changing magnetic field induces an electromotive force (EMF) in a conducting loop or coil. The magnitude of the induced EMF is proportional to the rate of change of the magnetic flux through the loop or coil.

**Lenz's Law and the Direction of Induced EMF**
Lenz's law states that the direction of the induced EMF is such that it opposes the change in the magnetic flux that caused it. This means that the induced current will create a magnetic field that opposes the original change in the magnetic flux.

**Examples of Electromagnetic Induction**

- Moving a Magnet through a Coil: When a magnet is moved in and out of a coil of wire, the changing magnetic flux induces an EMF in the coil, causing a current to flow.
- Rotating a Coil in a Magnetic Field: When a coil is rotated in a uniform magnetic field, the changing magnetic flux through the coil induces an EMF, which can be used to generate alternating current (AC) electricity.
- Transformer Action: When a changing current in one coil induces a changing magnetic field, which in turn induces a voltage in a nearby coil, the device is functioning as a transformer.

**Eddy Currents and Their Effects**
Eddy currents are induced currents that flow in circular paths within a conducting material when it is exposed to a changing magnetic field. These currents can have both beneficial and detrimental effects, such as power losses in transformers and the creation of eddy current brakes.

**Key Takeaways**

- Faraday's law of induction states that a changing magnetic field induces an electromotive force (EMF) in a conducting loop or coil.
- Lenz's law describes the direction of the induced EMF, which opposes the change in the magnetic flux that caused it.
- Examples of electromagnetic induction include moving a magnet through a coil, rotating a coil in a magnetic field, and transformer action.
- Eddy currents are induced currents that flow in circular paths within a conducting material exposed to a changing magnetic field, and they can have both beneficial and detrimental effects.

### 7.9: Transformers and Generators

In this sub-chapter, we will explore two important applications of electromagnetic induction: transformers and generators.

**Transformers**
A transformer is a device that uses the principle of electromagnetic induction to change the voltage level of an alternating current (AC) electrical signal. It consists of two or more coils wound around a common magnetic core. The primary coil carries the input voltage, and the changing magnetic field induces a