PHYS 121: Electricity and Magnetism – Introduction, Vectors, and Electric Charge

Electricity and Magnetism: Introduction

Overview and Applications

This section introduces the fundamental concepts of electricity and magnetism, highlighting their importance in modern technology and daily life. Applications include:

• Circuits: The foundation of electronic devices, enabling the flow and control of electric current.

• Power Plants: Facilities that generate electricity for widespread distribution and use.

• Electric Engines: Devices that convert electrical energy into mechanical motion, used in vehicles and machinery.

Electricity and magnetism are deeply interconnected, forming the basis for much of modern physics and engineering.

A Short History of Electricity

Key Historical Developments

• Static Electricity and Natural Phenomena: Early observations included static electricity (e.g., rubbing amber) and natural magnets (lodestone).

• Benjamin Franklin: Proposed the concept of positive and negative electric "fluid" (now understood as charge).

• André-Marie Ampère: Discovered that electric currents create magnetic fields.

• Michael Faraday: Demonstrated that changing magnetic fields produce electric fields (electromagnetic induction).

• James Clerk Maxwell: Unified electricity and magnetism into a single theory, predicting electromagnetic waves.

• Electromagnetic Spectrum: Includes radio, infrared, visible light, ultraviolet, X-rays, and more.

Key Point: Electric and magnetic effects are interconnected and form the basis of electromagnetic theory.

Vectors in Physics

Definition and Representation

Vectors are quantities that have both magnitude and direction. They are essential for describing physical quantities such as force, velocity, and electric field.

• Unit Vectors: Indicate direction along coordinate axes (e.g., i, j, k).

• Vector Addition: Vectors can be added graphically (tip-to-tail method) or algebraically by components.

Example: If A = 2i + 2j and B = 4i - j, then the sum is:

• $\vec{A} + \vec{B} = (2+4)\hat{i} + (2-1)\hat{j} = 6\hat{i} + 1\hat{j}$

Vector Multiplications

• Dot Product (Scalar Product): The product of two vectors resulting in a scalar. It measures the projection of one vector onto another.

  • $\vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos\theta$

• Cross Product (Vector Product): The product of two vectors resulting in a new vector perpendicular to both.

  • $\vec{A} \times \vec{B} = (A_x B_y - A_y B_x)\hat{k}$ (for 2D vectors)

  • For unit vectors: $\hat{i} \times \hat{j} = \hat{k}$

Fields in Physics

Scalar and Vector Fields

A field assigns a value to every point in space. Fields can be:

• Scalar Fields: Assign a single value (e.g., temperature, potential energy) to each point.

• Vector Fields: Assign a vector (magnitude and direction) to each point (e.g., electric field, gravitational field).

Fields are used to describe how physical quantities vary in space and time.

Shell Theorem

Gravitational Field of a Spherical Shell

The shell theorem states:

• For a uniform spherical shell of mass m:

  • The gravitational force inside the shell is zero.

  • Outside the shell, the shell acts as if all its mass were concentrated at its center.

• This theorem is analogous to the behavior of electric fields in spherical charge distributions.

Fundamental Interactions

The Four Fundamental Forces

• Gravitational Force: Attractive force between masses; weakest but acts over long distances.

• Electromagnetic Force: Acts between electrically charged particles; can be attractive or repulsive.

• Strong Nuclear Force: Holds protons and neutrons together in the nucleus; strongest but short-ranged.

• Weak Nuclear Force: Responsible for radioactive decay; short-ranged and weaker than the strong force.

Electric Charge

Properties of Electric Charge

• Charge is a fundamental property of matter, similar to mass.

• There are two types: positive (protons) and negative (electrons).

• Like charges repel; opposite charges attract.

• Atoms are electrically neutral when they have equal numbers of protons and electrons.

Conductors, Insulators, and Semiconductors

Classification and Properties

Type	Examples	Properties
Conductor	Metals (Copper, Iron)	Electrons move easily; allows electric current to flow.
Insulator	Wood, Dry Air	Electrons do not move easily; resists electric current.
Semiconductor	Silicon, Germanium	Intermediate conductivity; can be modified by doping.

Electric Polarization: In insulators, electrons can be slightly displaced by external electric fields, causing polarization (separation of charge within molecules).

Dielectric Constant: Measures a material's ability to be polarized by an electric field.

Coulomb’s Law

Electrostatic Force Between Point Charges

Coulomb’s Law quantifies the force between two point charges:

• The force is proportional to the product of the charges and inversely proportional to the square of the distance between them.

Formula:

• $F = k_e \frac{|q_1 q_2|}{r^2}$

• Where $k_e = 8.99 \times 10^9 \ \mathrm{N \cdot m^2 / C^2}$ is Coulomb’s constant.

The force acts along the line joining the two charges. Like charges repel; unlike charges attract.

Superposition Principle

• The net force on a charge due to multiple other charges is the vector sum of the individual forces from each charge.

• There are no three-body (or higher) force terms; only pairwise interactions are considered.

Worked Examples

• Two Charges: Calculate the force using Coulomb’s law and vector addition.

• Three Charges: Place charges along a line, calculate individual forces, and sum vectorially to find net force on each charge.

• Equilibrium Position: For a system of charges, determine the position where the net force on a charge is zero by setting the sum of forces to zero and solving for position.

Example Calculation:

• Given $q_1 = +3.2 \times 10^{-7} \ \mathrm{C}$, $q_2 = +3.2 \times 10^{-7} \ \mathrm{C}$, $r = 0.02 \ \mathrm{m}$:

• $F = k_e \frac{q_1 q_2}{r^2} = 8.99 \times 10^9 \frac{(3.2 \times 10^{-7})^2}{(0.02)^2}$

• Calculate to find the magnitude of the force.

Atomic Structure and Electric Forces

Hydrogen Atom Example

• A hydrogen atom consists of one proton and one electron.

• Proton charge: $+1.6 \times 10^{-19} \ \mathrm{C}$; Electron charge: $-1.6 \times 10^{-19} \ \mathrm{C}$

• Typical orbital radius: $r \approx 10^{-10} \ \mathrm{m}$

• Use Coulomb’s law to calculate the electrostatic force between the proton and electron.

Comparison: The electrostatic force is much stronger than the gravitational force between the same two particles.

Additional info: The above notes provide foundational concepts for further study in electricity and magnetism, including vector mathematics, field theory, and the behavior of electric charges in various materials.