🧲Electromagnetism I Unit 6 – Current, Resistance, and EMF

Key Concepts and Definitions

• Electric current $I$ quantifies the rate of charge flow through a cross-sectional area, measured in amperes (A)
• Resistance $R$ opposes the flow of electric current, measured in ohms ($\Omega$)
  • Conductors have low resistance, allowing current to flow easily (copper, silver)
  • Insulators have high resistance, preventing current flow (rubber, plastic)
• Electromotive force (EMF) $\mathcal{E}$ is the energy per unit charge provided by a voltage source, measured in volts (V)
• Ohm's law relates current, resistance, and voltage: $V = IR$
• Power $P$ is the rate of energy transfer in an electric circuit, measured in watts (W)
  • Calculated using $P = IV$ or $P = I^2R$
• Kirchhoff's current law (KCL) states that the sum of currents entering a node equals the sum of currents leaving the node
• Kirchhoff's voltage law (KVL) states that the sum of voltages around any closed loop in a circuit is zero

Charge Flow and Electric Current

• Electric current is the flow of electric charge through a conductor
  • Conventional current assumes positive charges flow from positive to negative terminals
  • In reality, electrons (negative charges) flow from negative to positive terminals
• Current is measured in amperes (A), where 1 A = 1 coulomb/second (C/s)
• Direct current (DC) flows in one direction with constant polarity (batteries)
• Alternating current (AC) periodically reverses direction (power outlets)
• Current density $J$ is the current per unit cross-sectional area: $J = I/A$
• Drift velocity $v_d$ is the average velocity of charge carriers in a conductor
  • Relates to current via $I = nAqv_d$, where $n$ is carrier density, $A$ is cross-sectional area, and $q$ is carrier charge
• Continuity equation describes conservation of charge: $\frac{\partial \rho}{\partial t} + \nabla \cdot \vec{J} = 0$

Resistance and Ohm's Law

• Resistance quantifies a material's opposition to electric current flow
  • Measured in ohms ($\Omega$), where 1 $\Omega$ = 1 V/A
• Ohm's law relates voltage $V$, current $I$, and resistance $R$: $V = IR$
  • Applies to ohmic devices, where resistance is constant (resistors)
• Resistivity $\rho$ is an intrinsic property of a material that contributes to resistance: $R = \rho \frac{l}{A}$
  • Depends on temperature, with most materials having higher resistivity at higher temperatures
• Conductance $G$ is the reciprocal of resistance: $G = \frac{1}{R}$, measured in siemens (S)
• Conductivity $\sigma$ is the reciprocal of resistivity: $\sigma = \frac{1}{\rho}$
• Superconductors have zero resistance below a critical temperature (niobium, lead)
• Semiconductors have resistivity between conductors and insulators (silicon, germanium)

Electromotive Force (EMF) and Voltage Sources

• EMF is the energy per unit charge provided by a voltage source
  • Measured in volts (V), where 1 V = 1 joule/coulomb (J/C)
• Ideal voltage sources maintain constant voltage regardless of current (batteries, power supplies)
  • Real voltage sources have internal resistance, causing voltage to drop with increasing current
• EMF is not a force but rather a measure of the energy available to drive current
• Voltage is the potential difference between two points in a circuit
  • Equivalent to EMF for ideal voltage sources
• Kirchhoff's voltage law (KVL) states that the sum of voltages around any closed loop in a circuit is zero
• Voltage dividers create a fraction of the input voltage based on resistor ratios: $V_{out} = V_{in} \frac{R_2}{R_1 + R_2}$
• Thevenin's theorem allows complex circuits to be reduced to an equivalent voltage source and series resistance

Circuit Elements and Configurations

• Resistors oppose current flow and dissipate power as heat (carbon, metal film)
  • Series resistors: $R_{eq} = R_1 + R_2 + ... + R_n$
  • Parallel resistors: $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + ... + \frac{1}{R_n}$
• Capacitors store energy in electric fields and oppose changes in voltage (ceramic, electrolytic)
  • Capacitance $C$ relates charge $Q$ and voltage $V$: $C = \frac{Q}{V}$, measured in farads (F)
  • Series capacitors: $\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + ... + \frac{1}{C_n}$
  • Parallel capacitors: $C_{eq} = C_1 + C_2 + ... + C_n$
• Inductors store energy in magnetic fields and oppose changes in current (air core, iron core)
  • Inductance $L$ relates voltage $V$ and rate of current change $\frac{dI}{dt}$: $V = L \frac{dI}{dt}$, measured in henries (H)
  • Series inductors: $L_{eq} = L_1 + L_2 + ... + L_n$
  • Parallel inductors: $\frac{1}{L_{eq}} = \frac{1}{L_1} + \frac{1}{L_2} + ... + \frac{1}{L_n}$
• Switches control current flow by opening or closing a circuit (toggle, pushbutton)
• Fuses and circuit breakers protect circuits from excessive current (glass tube, bimetallic strip)

Power and Energy in Electric Circuits

• Power is the rate of energy transfer in an electric circuit
  • Measured in watts (W), where 1 W = 1 joule/second (J/s)
• Power can be calculated using $P = IV$ or $P = I^2R$
  • In resistors, power is dissipated as heat: $P = \frac{V^2}{R}$
• Energy is the capacity to do work, measured in joules (J)
  • Electrical energy is calculated using $E = Pt$, where $t$ is time
• Kirchhoff's current law (KCL) states that the sum of powers entering a node equals the sum of powers leaving the node
• Maximum power transfer theorem states that maximum power is delivered to a load when its resistance equals the source's internal resistance
• Efficiency $\eta$ is the ratio of output power to input power: $\eta = \frac{P_{out}}{P_{in}}$
  • Ideal circuits have 100% efficiency, but real circuits always have losses (heat, radiation)
• Energy conservation requires that the total energy in a closed system remains constant

Applications and Real-World Examples

• Electric power distribution systems transmit AC power at high voltages to minimize losses (transformers, power lines)
  • Residential outlets provide 120 V AC at 60 Hz in North America
• Batteries convert chemical energy into electrical energy (alkaline, lithium-ion)
  • Rechargeable batteries can be recharged by applying an external voltage
• Solar cells convert light energy into electrical energy through the photovoltaic effect (silicon, perovskites)
  • Series-parallel configurations optimize voltage and current output
• Electric motors convert electrical energy into mechanical energy (DC motors, AC motors)
  • Used in a wide range of applications (electric vehicles, robotics, home appliances)
• Sensors convert physical quantities into electrical signals (thermistors, strain gauges)
  • Enable measurement and control of various systems (temperature, pressure, motion)
• Electrochemical cells use redox reactions to generate electricity (fuel cells, electrolysis)
  • Promising for clean energy storage and conversion (hydrogen fuel cells)

Problem-Solving Strategies

• Identify the given information and the quantity to be calculated
  • Determine which equations and principles apply to the problem
• Draw a schematic diagram of the circuit, labeling all components and variables
  • Use standard symbols for circuit elements (resistors, capacitors, voltage sources)
• Apply Kirchhoff's laws (KCL and KVL) to analyze the circuit
  • KCL: Sum of currents entering a node equals sum of currents leaving the node
  • KVL: Sum of voltages around any closed loop in a circuit is zero
• Use Ohm's law ($V = IR$) to relate voltage, current, and resistance
  • Rearrange the equation as needed to solve for the desired quantity
• Simplify the circuit by combining series or parallel components
  • Series resistors: $R_{eq} = R_1 + R_2 + ... + R_n$
  • Parallel resistors: $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + ... + \frac{1}{R_n}$
• Apply the appropriate power equations ($P = IV$, $P = I^2R$, $P = \frac{V^2}{R}$)
  • Determine the power dissipated by each component and the total power in the circuit
• Check the solution for reasonableness and verify that it satisfies the given conditions
  • Perform a unit analysis to ensure the answer has the correct units
  • Compare the result to known values or expected ranges