Physics: Electricity & Magnetism Summary

This document provides a concise summary of all the key concepts, formulas, and principles from our discussions on electricity and magnetism.

Electrical Fundamentals

These concepts are the building blocks of electricity:

Electric Charge ($q$): The fundamental property of matter that causes it to experience a force in an electric field.
- Unit: Coulomb (C). A base SI unit derived from Ampere and second: $1 \, \text{C} = 1 \, \text{A} \cdot \text{s}$.
Electric Current ($I$): The rate of flow of electric charge.
- Unit: Ampere (A). This is a base SI unit.
Voltage ($V$): The electric potential energy per unit charge. It's the "push" that makes current flow.
- Unit: Volt (V). A derived unit: $1 \, \text{V} = 1 \, \text{J/C} = 1 \, \frac{\text{kg} \cdot \text{m}^2}{\text{A} \cdot \text{s}^3}$.
Resistance ($R$): The opposition to the flow of electric current.
- Unit: Ohm ($\Omega$). A derived unit: $1 \, \Omega = 1 \, \text{V/A} = 1 \, \frac{\text{kg} \cdot \text{m}^2}{\text{A}^2 \cdot \text{s}^3}$.

Key Formulas

These formulas describe the fundamental relationships between the electrical quantities.

Ohm's Law:

$$V = IR$$

Electrical Power & Energy

Electrical Power ($P$): The rate at which electrical energy is transferred or used.
- Unit: Watt (W). A derived unit: $1 \, \text{W} = 1 \, \text{J/s} = 1 \, \frac{\text{kg} \cdot \text{m}^2}{\text{s}^3}$.
Electrical Energy ($E$): The total work done by an electric current.
- Unit: Joule (J). A derived unit: $1 \, \text{J} = 1 \, \text{N} \cdot \text{m} = 1 \, \frac{\text{kg} \cdot \text{m}^2}{\text{s}^2}$.

Power Formulas:

$$P = VI$$

$$P = I^2 R$$

$$P = \frac{V^2}{R}$$

Energy Formula:

$$E = Pt$$

Magnetism

This is the force that arises from moving electric charges.

Magnetic Field ($B$): The area of force surrounding a magnet or a moving charge.
- Unit: Tesla (T). A derived unit: $1 \, \text{T} = 1 \, \frac{\text{N}}{\text{A} \cdot \text{m}} = 1 \, \frac{\text{kg}}{\text{A} \cdot \text{s}^2}$.
Right-Hand Rules: Mnemonic devices to determine the direction of magnetic fields and forces.
Electromagnetism: The unified theory linking electricity and magnetism.

Key Formulas

Magnetic Field of a Long, Straight Wire:

$$B = \frac{\mu_0 I}{2\pi r}$$

where $\mu_0$ is the permeability of free space ($4\pi \times 10^{-7} \, \text{T} \cdot \text{m/A}$).

Lorentz Force on a Moving Charge:

$$F = |q|vB \sin\theta$$

Maxwell's Equations

These four fundamental equations describe the behavior of electric and magnetic fields:

$$\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}$$

$$\nabla \cdot \mathbf{B} = 0$$

$$\nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t}$$

$$\nabla \times \mathbf{B} = \mu_0 \left( \mathbf{J} + \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} \right)$$

Unit Transformations to Base SI Units

Here are the step-by-step proofs for how each derived unit is expressed in terms of the four base units: kilogram (kg), meter (m), second (s), and ampere (A).

Volt (V)

1 V is defined as 1 Joule per Coulomb ($1 \, \text{J/C}$).

$$1 \, \text{V} = \frac{1 \, \text{J}}{1 \, \text{C}}$$

Since $1 \, \text{J} = 1 \, \text{N} \cdot \text{m}$ and $1 \, \text{C} = 1 \, \text{A} \cdot \text{s}$, we can substitute:

$$1 \, \text{V} = \frac{1 \, \text{N} \cdot \text{m}}{1 \, \text{A} \cdot \text{s}}$$

Since $1 \, \text{N} = 1 \, \frac{\text{kg} \cdot \text{m}}{\text{s}^2}$:

$$1 \, \text{V} = \frac{\left( \frac{\text{kg} \cdot \text{m}}{\text{s}^2} \right) \cdot \text{m}}{\text{A} \cdot \text{s}} = \frac{\text{kg} \cdot \text{m}^2}{\text{A} \cdot \text{s}^3}$$

Ohm ($\Omega$)

1 $\Omega$ is defined as 1 Volt per Ampere ($1 \, \text{V/A}$).

$$1 \, \Omega = \frac{1 \, \text{V}}{1 \, \text{A}}$$

Using the derived base units for the Volt ($1 \, \text{V} = 1 \, \frac{\text{kg} \cdot \text{m}^2}{\text{A} \cdot \text{s}^3}$):

$$1 \, \Omega = \frac{\left( \frac{\text{kg} \cdot \text{m}^2}{\text{A} \cdot \text{s}^3} \right)}{\text{A}} = \frac{\text{kg} \cdot \text{m}^2}{\text{A}^2 \cdot \text{s}^3}$$

Watt (W)

1 W is defined as 1 Joule per second ($1 \, \text{J/s}$).

$$1 \, \text{W} = \frac{1 \, \text{J}}{1 \, \text{s}}$$

Using the base units for the Joule ($1 \, \text{J} = 1 \, \frac{\text{kg} \cdot \text{m}^2}{\text{s}^2}$):

$$1 \, \text{W} = \frac{\left( \frac{\text{kg} \cdot \text{m}^2}{\text{s}^2} \right)}{\text{s}} = \frac{\text{kg} \cdot \text{m}^2}{\text{s}^3}$$

Joule (J)

1 J is defined as 1 Newton-meter ($1 \, \text{N} \cdot \text{m}$).

$$1 \, \text{J} = 1 \, \text{N} \cdot \text{m}$$

Using the base units for the Newton ($1 \, \text{N} = 1 \, \frac{\text{kg} \cdot \text{m}}{\text{s}^2}$):

$$1 \, \text{J} = \left( \frac{\text{kg} \cdot \text{m}}{\text{s}^2} \right) \cdot \text{m} = \frac{\text{kg} \cdot \text{m}^2}{\text{s}^2}$$

Tesla (T)

1 T is defined as 1 Newton per Ampere-meter ($1 \, \frac{\text{N}}{\text{A} \cdot \text{m}}$).

$$1 \, \text{T} = \frac{1 \, \text{N}}{1 \, \text{A} \cdot \text{m}}$$

Using the base units for the Newton ($1 \, \text{N} = 1 \, \frac{\text{kg} \cdot \text{m}}{\text{s}^2}$):

$$1 \, \text{T} = \frac{\left( \frac{\text{kg} \cdot \text{m}}{\text{s}^2} \right)}{\text{A} \cdot \text{m}} = \frac{\text{kg}}{\text{A} \cdot \text{s}^2}$$

Interactive Implementations & Problems

Interactive Simulations

The presentations included several interactive visualizations to help understand these concepts:

Electric Current Animation: Shows electrons flowing through a wire.
V-I Graph: Demonstrates the linear relationship in Ohm's Law.
Power & Energy Meter: Visualizes energy accumulation over time.
Magnetic Induction: A simulation showing how a moving magnet creates a current.
Lorentz Force Simulation: Visualizes the path of a charged particle in a magnetic field.
Earth's Magnetic Field: An interactive 3D model of the Earth's protective magnetosphere.

Practice Problems

Ohm's Law: A light bulb with a resistance of 150 $\Omega$ is connected to a 120 V power source. What is the current flowing through it?
Circuit Analysis (Parallel): Three resistors, $R_1=20 \, \Omega$, $R_2=30 \, \Omega$, and $R_3=60 \, \Omega$, are connected in parallel. What is the total resistance? If a voltage of $12 \, \text{V}$ is applied across the combination, what is the total current and the current through each resistor?
Circuit Analysis (Combined): A circuit has a $10 \, \Omega$ resistor in series with a parallel combination of a $20 \, \Omega$ and a $30 \, \Omega$ resistor. If the total voltage is $24 \, \text{V}$, what is the total resistance and the current from the source?
Electrical Power: An electric heater draws a current of 12.5 A from a 240 V outlet. What is its power consumption in Watts? How much energy does it consume in 30 minutes (in Joules and kWh)?
Magnetism: An electron moves at a velocity of $5 \times 10^5 \, \text{m/s}$ to the east. It enters a magnetic field of $2 \, \text{T}$ directed vertically downward. What is the magnitude and direction of the magnetic force on the electron? (The charge of an electron is $-1.6 \times 10^{-19} \, \text{C}$.)
Advanced Combined Problem: A wire of length $0.75 \, \text{m}$ carries a current of $8 \, \text{A}$. If the wire is placed in a uniform magnetic field and experiences a maximum force of $2.4 \, \text{N}$, what is the strength of the magnetic field? How much power is consumed if the wire has a resistance of $0.5 \, \Omega$?