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Inductors and Inductance

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Charge, Current, and ContinuityElectric Potential and Voltage+1 more→→Energy Storage in Capacitors and InductorsSeries and Parallel RC, RL, and RLC Networks
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Core Idea

An inductor stores energy in a magnetic field created by current flow. Inductance L depends on coil geometry and material. The voltage-current relationship v = L(di/dt) shows inductors oppose current changes and block AC signals, with impedance Z_L = jωL in AC circuits. Inductors are dual to capacitors in circuit theory.

Explainer

You already know from your prerequisites that current flowing through a conductor creates a magnetic field around it. An inductor — typically a coil of wire — is specifically designed to maximize this magnetic energy storage by concentrating the field through many turns of wire. The measure of how effectively a coil does this is inductance L, measured in henries (H). A larger inductance means more magnetic energy stored per unit of current, and the value depends on the geometry of the coil (number of turns, cross-sectional area, length) and the magnetic permeability of the material in its core.

The defining voltage-current relationship is v = L(di/dt). Read this carefully: the voltage across an inductor is proportional to the *rate of change* of current, not the current itself. If the current through an inductor is constant (DC), di/dt = 0, so the voltage is zero — the inductor looks like a short circuit (a perfect wire) for DC. But if the current is changing rapidly, a large voltage appears. This is the key behavioral rule: inductors resist changes in current. You cannot instantaneously change the current through an inductor; doing so would require infinite voltage. This property is why inductors are used to smooth current in power supplies and why they produce voltage spikes when circuits are abruptly switched.

In AC circuits, the rate of change of current is proportional to frequency. A sinusoidal current i(t) = I₀sin(ωt) has di/dt = ωI₀cos(ωt), so the induced voltage is proportional to ω. This gives the inductor's impedance Z_L = jωL: at low frequencies, the impedance is small (the inductor barely resists current); at high frequencies, the impedance is large (the inductor strongly opposes current). This frequency-dependent behavior makes inductors high-pass filters for current and explains their use in RF circuits, transformers, and filtering applications.

The duality with capacitors is worth internalizing: every property of a capacitor has a mirror image in an inductor. Capacitors store energy in an electric field; inductors in a magnetic field. Capacitors resist voltage changes (i = C·dv/dt); inductors resist current changes (v = L·di/dt). Capacitor impedance Z_C = 1/(jωC) decreases with frequency; inductor impedance Z_L = jωL increases with frequency. The energy stored in a capacitor is ½CV²; in an inductor it is ½LI². Understanding this symmetry lets you transfer your intuition about one element directly to the other, and it underlies the oscillatory behavior of LC circuits — where energy sloshes back and forth between electric and magnetic fields.

Practice Questions 5 questions

Prerequisite Chain

Understanding Zero → The Number Zero → Counting to Five → Counting to 10 → Counting to 20 → Counting a Set of Objects Up to 20 → Cardinality: The Last Number Counted → Matching Numerals to Quantities → Subitizing Small Quantities → Addition Within 10 → Number Bonds to 10 → Addition Within 20 → Doubles and Near Doubles → Doubles Facts Within 10 → Near Doubles Facts Within 20 → Mental Math Strategies for Addition → Mental Math: Adding and Subtracting Tens → Addition Within 100 → Repeated Addition as Multiplication → Multiplication as Equal Groups → Multiplication: Arrays → Basic Multiplication Facts (0s, 1s, 2s, 5s, 10s) → Multiplication Facts Within 100 → Division as Equal Sharing → Division as Grouping (Measurement Division) → Division: Grouping (Repeated Subtraction) Model → Division: Fair Sharing Model → Division as Equal Sharing → Division as Grouping → Basic Division Facts → Division Facts Within 100 → Multiplication and Division Fact Families → Relationship Between Multiplication and Division → Division Facts as Inverse of Multiplication → Remainders and Quotients in Division → Division Word Problems → Multi-Step Word Problems → Solving Multi-Step Word Problems → Multiplication Word Problems → Division Word Problems → Introduction to Long Division → Factors and Multiples → Prime and Composite Numbers → Equivalent Fractions → Relating Fractions and Decimals → Decimal Place Value → Integers and the Number Line → Comparing and Ordering Integers → Absolute Value → Adding Integers → Subtracting Integers → Multiplying Integers → Dividing Integers → Unit Rates → Proportions → Percent Concept → Converting Between Fractions, Decimals, and Percents → Operations with Rational Numbers → Two-Step Equations → Solving Multi-Step Equations → Equations with Variables on Both Sides → Angle Pairs: Complementary, Supplementary, and Vertical → Parallel Lines and Transversals → Corresponding Angles → Alternate Interior Angles → Triangle Angle Sum Theorem → Exterior Angle Theorem → Triangle Inequality Theorem → Similar Triangles: AA Similarity → Similar Triangles: SSS and SAS Similarity → Proportions in Similar Triangles → Right Triangle Trigonometry Introduction → Sine, Cosine, and Tangent Ratios → Trigonometric Ratios Review → Radian Measure → Converting Between Degrees and Radians → The Unit Circle → Graphing Sine and Cosine → Graphing Tangent and Reciprocal Trigonometric Functions → Derivatives of Trigonometric Functions → Antiderivatives → Indefinite Integrals → Basic Integration Rules → Riemann Sums → Definite Integral Definition → Double Integrals: Definition and Setup → Iterated Integrals and Fubini's Theorem → Double Integrals over Rectangular Regions → Double Integrals over General Regions → Applications of Double Integrals: Area, Mass, and Moments → Triple Integrals in Cartesian Coordinates → Triple Integrals in Cylindrical and Spherical Coordinates → Change of Variables and the Jacobian Determinant → Applications of Triple Integrals: Volume and Mass → Vector Fields and Their Representations → Line Integrals of Vector Fields → Work and Circulation → Line Integrals of Scalar and Vector Functions → Fundamental Theorem for Line Integrals → Conservative Vector Fields → Conservative Vector Fields and Potential Functions → Curl and Divergence of Vector Fields → Curl and Divergence → Divergence Theorem → Electric Flux and Divergence Theorem → Gauss's Law: Integral Form and Meaning → Solving Problems with Gauss's Law → Conductors in Electrostatic Equilibrium → Capacitance and Capacitors → Dielectrics → Dielectric Constant and Relative Permittivity → Electric Field Inside Dielectric Materials → Dielectric Materials and Polarization → Dielectric Susceptibility and Permittivity → Energy Density in Electric Fields → Electric Current and Current Density → Electrical Resistance and Resistivity → Ohm's Law and Circuit Elements → Electromotive Force (EMF) and Batteries → Kirchhoff's Circuit Laws: Voltage and Current → DC Circuit Network Analysis Methods → Transient Response in RC Circuits → RC Circuits → LC and RLC Circuits → AC Circuits: Fundamentals → Impedance and Reactance → AC Power and Resonance → Electromagnetic Waves → Frequency-Dependent Permittivity and Dispersion → Electromagnetic Waves in Anisotropic Media → Birefringence and Dichroism → Wave Plates: Quarter-Wave and Half-Wave Plates → Circular and Elliptical Polarization Production → Polarization States: Linear, Circular, and Elliptical → Linear Superposition of Waves → Superposition Principle in Electrostatics → Electric Field Lines and Visualization → Electric Potential and Potential Energy → Electric Potential and Voltage → Inductors and Inductance

Longest path: 140 steps · 971 total prerequisite topics

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