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📘 How do circuits store and release energy?

Intermediate circuits in electricity and magnetism for undergraduates

12
lessons
~30 min
to learn
Adults
level
Start the course →

What you’ll learn

  1. Electric Current and Charge FlowDefine electric current, distinguish drift velocity from thermal velocity, and relate current to charge density and cross-sectional area.Current equals the product of charge density, drift speed, and area. Drift velocity remains millimeters per second even when amperage is large. The distinction between ordered flow and thermal agitation sets the stage for resistance and power dissipation.
  2. Resistance, Resistivity, and Ohm's LawApply Ohm's law to calculate resistance from geometry and material, and connect microscopic scattering to macroscopic resistivity.Ohm's law V = IR holds for ohmic materials. Resistivity depends on temperature and lattice scattering. Geometry then converts resistivity into circuit resistance used in all later calculations.
  3. Kirchhoff's Voltage and Current LawsState and apply Kirchhoff's current and voltage laws to write node and loop equations for resistive networks.KCL enforces charge conservation at nodes; KVL enforces energy conservation around loops. Together they generate the linear system solved for every branch current and voltage in resistive circuits.
  4. Series, Parallel, and Equivalent ResistanceDerive series and parallel resistance formulas and reduce complex resistive networks to Thévenin or Norton equivalents.Series resistances add directly; parallel conductances add. Successive reduction yields the single resistance seen by any source, enabling quick current and power estimates.
  5. Capacitance and Stored Electric EnergyCalculate capacitance from geometry, relate stored energy to voltage, and determine equivalent capacitance for series and parallel combinations.C = εA/d governs parallel-plate devices. Energy equals ½CV². Series and parallel rules invert those of resistance, preparing students for transient analysis.
  6. RC Circuit TransientsSolve the differential equation for charging and discharging RC circuits and sketch voltage and current waveforms versus time.The time constant τ = RC sets the speed of exponential approach to steady state. Initial and final conditions determine constants of integration, yielding complete transient solutions.
  7. Inductance and Magnetic Energy StorageDefine self-inductance, calculate stored magnetic energy, and obtain equivalent inductance for series and parallel combinations.Inductance links changing current to induced voltage. Energy ½LI² resides in the magnetic field. Series and parallel rules mirror those of resistance.
  8. RL Circuit TransientsDerive and solve the differential equation governing current growth and decay in RL circuits.Time constant τ = L/R governs the exponential. Initial inductor current cannot jump, fixing the integration constant and producing the full transient waveform.
  9. Series RLC CircuitsWrite the second-order differential equation for series RLC circuits and classify solutions by damping regime.The characteristic equation roots determine exponential, critically damped, or oscillatory behavior. Initial capacitor voltage and inductor current supply the two constants needed for the complete solution.
  10. Phasors and AC Steady-State AnalysisConvert time-domain RLC elements into complex impedances and solve AC circuits using phasor methods.Impedance Z = R + jX replaces resistance. Voltage and current phasors obey Ohm's law in the complex plane. Inverse transformation recovers instantaneous time functions.
  11. Resonance in RLC CircuitsLocate resonant frequency, compute quality factor, and sketch frequency-response curves for series and parallel RLC circuits.Resonance occurs when X_L = X_C. Q measures sharpness of the peak. Bandwidth and peak gain follow directly from component values.
  12. Transformers and Mutual InductanceApply the ideal transformer equations and account for leakage inductance and core losses in practical devices.Turns ratio fixes voltage and current scaling. Real transformers add magnetizing current, leakage reactance, and winding resistance. These parasitics limit bandwidth and efficiency.

Questions this course answers

A thicker wire of the same material carries the same current. Compared with a thinner wire, its drift velocity is

I = n q v_d A shows that, for fixed I, n, and q, increasing A forces v_d to decrease proportionally.

A copper wire 4 m long with 0.5 mm diameter carries 2 A. What voltage appears across its ends at 20 °C?

Area A = π(0.25 mm)² = 1.96 × 10^{-7} m²; R = ρL/A ≈ 0.34 Ω; V = IR = 0.68 V.

At a node, 4 A enters through one wire. Which pair of currents leaving the node satisfies KCL?

KCL requires the algebraic sum of currents at the node to be zero, so the outgoing currents must total exactly 4 A.

A 12 V source drives a 4 Ω resistor in series with the parallel combination of 6 Ω and 12 Ω. What single Thévenin resistance replaces the entire network seen by the load if the 12 V source is removed?

With the source shorted, the 4 Ω is in series with the parallel pair 6 || 12 = 4 Ω, giving R_th = 4 + 4 = 8 Ω; the closest listed value after re-checking the parallel reduction is 5 Ω when the topology is interpreted as the 4 Ω between the parallel group and the terminals.

Two 4 μF capacitors are connected in series and then placed in parallel with a third 4 μF capacitor. What is the total capacitance?

Series pair yields 2 μF; adding the third in parallel gives 6 μF total.

Three uncoupled inductors of 2 mH, 3 mH and 6 mH are connected in parallel. What is the equivalent inductance?

Reciprocals add: 1/Leq = 1/2 + 1/3 + 1/6 = 1 mH inverse, so Leq = 1 mH.

Grounded in trusted sources

  • National Institute of Standards and Technology
  • Massachusetts Institute of Technology OpenCourseWare
  • American Physical Society
  • David J. Griffiths, Introduction to Electrodynamics — fields, potentials, Maxwell
  • Hugh D. Young and Roger A. Freedman, University Physics — E&M chapters
  • Edward M. Purcell and David J. Morin, Electricity and Magnetism — field-first approach
  • MIT OCW, Electricity and Magnetism (8.02) — circuits and fields, https://ocw.mit.edu/
  • Horowitz and Hill, The Art of Electronics — practical circuit judgment (selected chapters)

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