Electromagnetism I

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

Current, resistance, and EMF form the backbone of electrical circuits. These concepts describe how charges flow, encounter opposition, and are driven by energy sources. Understanding their relationships is crucial for analyzing and designing electrical systems. Ohm's law connects voltage, current, and resistance, while power equations quantify energy transfer. Kirchhoff's laws govern current and voltage in complex circuits. These principles apply to various devices, from simple resistors to advanced electronics, enabling us to harness electricity effectively.

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Key Concepts and Definitions

  • Electric current II quantifies the rate of charge flow through a cross-sectional area, measured in amperes (A)
  • Resistance RR opposes the flow of electric current, measured in ohms (Ω\Omega)
    • Conductors have low resistance, allowing current to flow easily (copper, silver)
    • Insulators have high resistance, preventing current flow (rubber, plastic)
  • Electromotive force (EMF) E\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=IRV = IR
  • Power PP is the rate of energy transfer in an electric circuit, measured in watts (W)
    • Calculated using P=IVP = IV or P=I2RP = 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 JJ is the current per unit cross-sectional area: J=I/AJ = I/A
  • Drift velocity vdv_d is the average velocity of charge carriers in a conductor
    • Relates to current via I=nAqvdI = nAqv_d, where nn is carrier density, AA is cross-sectional area, and qq is carrier charge
  • Continuity equation describes conservation of charge: ρt+J=0\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 (Ω\Omega), where 1 Ω\Omega = 1 V/A
  • Ohm's law relates voltage VV, current II, and resistance RR: V=IRV = IR
    • Applies to ohmic devices, where resistance is constant (resistors)
  • Resistivity ρ\rho is an intrinsic property of a material that contributes to resistance: R=ρlAR = \rho \frac{l}{A}
    • Depends on temperature, with most materials having higher resistivity at higher temperatures
  • Conductance GG is the reciprocal of resistance: G=1RG = \frac{1}{R}, measured in siemens (S)
  • Conductivity σ\sigma is the reciprocal of resistivity: σ=1ρ\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: Vout=VinR2R1+R2V_{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: Req=R1+R2+...+RnR_{eq} = R_1 + R_2 + ... + R_n
    • Parallel resistors: 1Req=1R1+1R2+...+1Rn\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 CC relates charge QQ and voltage VV: C=QVC = \frac{Q}{V}, measured in farads (F)
    • Series capacitors: 1Ceq=1C1+1C2+...+1Cn\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + ... + \frac{1}{C_n}
    • Parallel capacitors: Ceq=C1+C2+...+CnC_{eq} = C_1 + C_2 + ... + C_n
  • Inductors store energy in magnetic fields and oppose changes in current (air core, iron core)
    • Inductance LL relates voltage VV and rate of current change dIdt\frac{dI}{dt}: V=LdIdtV = L \frac{dI}{dt}, measured in henries (H)
    • Series inductors: Leq=L1+L2+...+LnL_{eq} = L_1 + L_2 + ... + L_n
    • Parallel inductors: 1Leq=1L1+1L2+...+1Ln\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=IVP = IV or P=I2RP = I^2R
    • In resistors, power is dissipated as heat: P=V2RP = \frac{V^2}{R}
  • Energy is the capacity to do work, measured in joules (J)
    • Electrical energy is calculated using E=PtE = Pt, where tt 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 η\eta is the ratio of output power to input power: η=PoutPin\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=IRV = 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: Req=R1+R2+...+RnR_{eq} = R_1 + R_2 + ... + R_n
    • Parallel resistors: 1Req=1R1+1R2+...+1Rn\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + ... + \frac{1}{R_n}
  • Apply the appropriate power equations (P=IVP = IV, P=I2RP = I^2R, P=V2RP = \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


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© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.