⚡️College Physics III – Thermodynamics, Electricity, and Magnetism Unit 9 – Current and Resistance in Circuits

Key Concepts and Definitions

• Electric current ($I$) is the flow of electric charge through a conductor, measured in amperes (A)
• Voltage ($V$) is the potential difference between two points in an electrical circuit, measured in volts (V)
  • Voltage is the driving force that causes current to flow in a circuit
• Resistance ($R$) is the opposition to the flow of electric current in a material, measured in ohms ($\Omega$)
  • Conductors have low resistance, while insulators have high resistance
• Conductance ($G$) is the reciprocal of resistance and measures the ease with which electric current flows through a material, measured in siemens (S)
• Ohm's law states that the current through a conductor is directly proportional to the voltage across it and inversely proportional to its resistance
• Power ($P$) is the rate at which electrical energy is converted into other forms of energy, measured in watts (W)
• Energy ($E$) is the capacity to do work, measured in joules (J) or kilowatt-hours (kWh) in electrical systems

Ohm's Law and Its Applications

• Ohm's law is expressed as $V = IR$, where $V$ is voltage, $I$ is current, and $R$ is resistance
• Rearranging Ohm's law allows for calculating current ($I = V/R$) and resistance ($R = V/I$)
• Ohm's law is applicable to many electrical components and circuits, including resistors, batteries, and simple DC circuits
• Ohm's law assumes that the resistance is constant and independent of the applied voltage or current
  • Some materials, such as semiconductors, have a non-linear relationship between voltage and current
• Ohm's law is used to analyze and design electrical circuits, determine the required voltage or current, and select appropriate components
• Kirchhoff's voltage law (KVL) and Kirchhoff's current law (KCL) are used in conjunction with Ohm's law to analyze more complex circuits
  • KVL states that the sum of voltages around any closed loop in a circuit is zero
  • KCL states that the sum of currents entering a node is equal to the sum of currents leaving the node

Types of Electrical Resistance

• Resistance is a property of materials that opposes the flow of electric current
• Resistors are components designed to have a specific resistance value and are used to control current flow in circuits
  • Resistors are made from materials with high resistivity, such as carbon, metal films, or wire-wound elements
• Temperature affects resistance in most materials
  • In metals, resistance increases with temperature due to increased electron scattering
  • In semiconductors, resistance decreases with temperature due to increased charge carrier concentration
• Superconductors are materials that have zero electrical resistance below a critical temperature
• Variable resistors, such as potentiometers and rheostats, allow for adjusting resistance in a circuit
• Non-ohmic resistors, such as diodes and thermistors, have a resistance that depends on the applied voltage or temperature
• Contact resistance occurs at the interface between two conductors and can be minimized by ensuring clean, tight connections

Current Flow in Circuits

• Current is the flow of electric charge carriers, typically electrons in metals, through a conductor
• Conventional current flow assumes positive charges moving from the positive terminal to the negative terminal of a voltage source
  • In reality, electrons flow from the negative terminal to the positive terminal
• Current flow requires a complete path, or circuit, from the voltage source to the load and back to the source
• The direction of current flow is determined by the polarity of the voltage source and the arrangement of components in the circuit
• Current is measured in amperes (A) and can be calculated using Ohm's law ($I = V/R$)
• The current in a circuit is determined by the voltage applied and the total resistance of the circuit
• Kirchhoff's current law (KCL) states that the sum of currents entering a node is equal to the sum of currents leaving the node
  • This law is used to analyze current flow in complex circuits with multiple branches

Series and Parallel Circuits

• Series circuits have components connected end-to-end, forming a single path for current flow
  • In a series circuit, the current is the same through all components
  • The total voltage across a series circuit is the sum of the voltages across each component
  • The total resistance of a series circuit is the sum of the individual resistances: $R_{total} = R_1 + R_2 + ... + R_n$
• Parallel circuits have components connected across the same two nodes, forming multiple paths for current flow
  • In a parallel circuit, the voltage is the same across all components
  • The total current in a parallel circuit is the sum of the currents through each branch
  • The total resistance of a parallel circuit is given by: $\frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + ... + \frac{1}{R_n}$
• Most real-world circuits are a combination of series and parallel connections
  • To analyze these circuits, they are often simplified using equivalent resistance techniques
• Voltage dividers and current dividers are common applications of series and parallel circuits, respectively
  • A voltage divider splits the input voltage into smaller output voltages based on the ratio of resistances
  • A current divider splits the total current into smaller branch currents based on the ratio of conductances

Power and Energy in Electrical Systems

• Power is the rate at which electrical energy is converted into other forms of energy, such as heat, light, or mechanical work
• Power is measured in watts (W) and is calculated using the formula $P = VI$, where $V$ is voltage and $I$ is current
  • Power can also be calculated using $P = I^2R$ or $P = \frac{V^2}{R}$, derived from Ohm's law
• Energy is the capacity to do work and is measured in joules (J) or kilowatt-hours (kWh) in electrical systems
  • Energy is calculated using the formula $E = Pt$, where $P$ is power and $t$ is time
• The power dissipated by a resistor is given by $P = I^2R$, which is known as Joule's first law
  • This power dissipation results in the generation of heat, which can be useful (heating elements) or undesirable (energy loss)
• The efficiency of an electrical system is the ratio of the useful output power to the total input power, expressed as a percentage
  • Efficiency = $\frac{P_{output}}{P_{input}} \times 100\%$
• Energy conservation is an important consideration in the design and operation of electrical systems
  • Minimizing power losses and optimizing efficiency can reduce energy consumption and costs

Measuring Current and Resistance

• Ammeters are used to measure electric current in a circuit
  • Ammeters are connected in series with the component or branch being measured
  • Ideal ammeters have zero resistance to minimize their impact on the circuit
• Voltmeters are used to measure voltage between two points in a circuit
  • Voltmeters are connected in parallel with the component or section being measured
  • Ideal voltmeters have infinite resistance to minimize their impact on the circuit
• Ohmmeters are used to measure resistance directly
  • Ohmmeters apply a known current and measure the resulting voltage to calculate resistance using Ohm's law
  • Resistance measurements are typically performed with the component disconnected from the circuit to avoid interference
• Multimeters are versatile instruments that combine the functions of ammeters, voltmeters, and ohmmeters
  • Digital multimeters (DMMs) are the most common type and provide accurate, easy-to-read measurements
• Shunt resistors are used to measure high currents by converting the current to a smaller voltage drop across a known resistance
• Kelvin sensing, or four-wire resistance measurement, is used to accurately measure low resistances by minimizing the impact of lead and contact resistance

Real-World Applications and Examples

• Electrical wiring in buildings and homes uses series and parallel circuits to distribute power and control devices
  • Wall outlets are typically wired in parallel to maintain a constant voltage
  • Light switches and fixtures are often wired in series to control power flow
• Electronic devices, such as smartphones, computers, and televisions, rely on complex circuits with many resistors, capacitors, and semiconductors
  • These components work together to process signals, store data, and display information
• Temperature sensors, such as thermistors and resistance temperature detectors (RTDs), use the temperature-dependent resistance of materials to measure temperature
  • Thermistors have a negative temperature coefficient (NTC), meaning their resistance decreases with increasing temperature
  • RTDs, typically made from platinum, have a positive temperature coefficient (PTC) and are known for their linearity and accuracy
• Strain gauges are resistive sensors that measure mechanical strain by detecting changes in resistance due to deformation
  • Strain gauges are used in load cells, pressure sensors, and structural health monitoring systems
• Potentiometers are variable resistors used in volume controls, dimmer switches, and position sensors
  • The resistance of a potentiometer changes as the wiper moves along the resistive element
• Heating elements, such as those found in electric stoves, ovens, and water heaters, use the power dissipation of resistors to generate heat
  • The power dissipated by the heating element is controlled by adjusting the voltage or current supplied to it
• Fuses and circuit breakers are safety devices that protect electrical systems from overcurrent conditions
  • Fuses contain a thin wire that melts and breaks the circuit when the current exceeds a specific value
  • Circuit breakers use electromagnetic or thermal mechanisms to trip and open the circuit when an overcurrent is detected