Circuits with Resistors and Inductors - College Physics

SECTION 1

Historical Context & Motivation

The study of circuits containing both resistors and inductors sits at a pivotal junction in the history of electromagnetism, bridging the gap between static circuit analysis and the full dynamical theory of electromagnetic phenomena. In the early nineteenth century, scientists began to suspect that electricity and magnetism were not independent forces but deeply intertwined aspects of a single physical framework. The discovery that a changing magnetic field could induce a current — and, conversely, that a current-carrying conductor could generate a magnetic field — opened the door to devices that store energy not in electric fields, as capacitors do, but in magnetic fields. The inductor, a coil of wire designed to exploit this principle, became a foundational component in electrical engineering, and its interaction with resistive elements gave rise to the RL circuit — one of the most important building blocks in electronics, power systems, and signal processing.

1820 • Ørsted's Discovery

Hans Christian Ørsted demonstrates that an electric current deflects a compass needle, establishing the first link between electricity and magnetism and inspiring the search for electromagnetic interactions.

1831 • Faraday's Law of Induction

Michael Faraday discovers that a changing magnetic flux through a loop of wire induces an electromotive force (EMF), providing the physical basis for inductance and transformer action.

1834 • Lenz's Law

Heinrich Lenz formulates his law stating that the direction of an induced current opposes the change in flux that produced it, explaining why inductors resist changes in current.

1886 • Self-Inductance Quantified

Oliver Heaviside refines the mathematical treatment of self-inductance and introduces the concept of the time constant τ = L/R for transient analysis of RL circuits.

1893 • Steinmetz and AC Analysis

Charles Proteus Steinmetz develops phasor methods for analyzing AC circuits with inductors and resistors, enabling the rapid growth of alternating-current power systems.

From these historical developments, a central question emerged: when a voltage is suddenly applied to a circuit containing both resistance and inductance, how does the current evolve over time? Unlike purely resistive circuits — where the current instantaneously reaches its steady-state value dictated by Ohm's law — an RL circuit exhibits a gradual, exponential approach to equilibrium. Understanding this transient behavior is essential for designing everything from relay circuits and motor controllers to filters and power supplies, and it serves as a gateway to the broader study of differential equations in physics.

SECTION 2

Core Principles & Definitions

Before analyzing RL circuits quantitatively, it is important to establish the foundational concepts that govern the behavior of inductors and their interaction with resistors. An inductor is a passive circuit element — typically a coil of wire — that stores energy in a magnetic field when current flows through it. Its defining property, inductance (measured in henrys, H), quantifies the proportionality between the magnetic flux linkage through the coil and the current producing that flux. When the current through an inductor changes, the resulting change in magnetic flux induces an EMF that, by Lenz's law, opposes the change — a phenomenon known as self-induction. This opposition to changes in current is what makes inductors fundamentally different from resistors, which simply dissipate energy, and from capacitors, which store energy in electric fields.

Inductance (L)

A measure of an inductor's ability to store energy in a magnetic field. Defined as the ratio of magnetic flux linkage (NΦ) to current (I). The SI unit is the henry (H): 1 H = 1 V·s/A.

Self-Induced EMF

When the current through an inductor changes at rate dI/dt, the inductor produces a back-EMF equal to −L(dI/dt). The negative sign reflects Lenz's law: the induced voltage opposes the change in current.

Time Constant (τ)

The characteristic time scale for transient response in an RL circuit, defined as τ = L/R. After one time constant, the current reaches approximately 63.2% of its final value during energization.

Energy Stored in an Inductor

An inductor carrying current I stores energy U = ½LI² in its magnetic field. This energy is recovered when the current decreases and is dissipated as heat in the resistor during de-energization.

Kirchhoff's Voltage Law (KVL)

The algebraic sum of voltage drops around any closed loop equals zero. In an RL circuit: ε − IR − L(dI/dt) = 0. This first-order ODE governs the transient behavior of the circuit.

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KEY TAKEAWAY

Think of an inductor as the electrical analogue of a heavy flywheel in a mechanical system. Just as a flywheel's rotational inertia resists sudden changes in angular velocity — requiring a torque applied over time to spin up or slow down — an inductor resists sudden changes in current. The inductance L plays the role of moment of inertia, while the resistance R acts like friction that eventually brings the system to steady state. The time constant τ = L/R tells you how long the 'spin-up' takes.

SECTION 3

Visual Explanation: The RL Circuit

A series RL circuit with a battery (ε), switch S, resistor R (shown in pink), and inductor L (shown in violet). When the switch closes at t = 0, the current I(t) rises exponentially toward ε/R. The voltage drops across R and L are labeled, and KVL yields the governing differential equation.

The diagram above illustrates the essential topology of a series RL circuit. When the switch S closes at time t = 0, the battery with EMF ε drives current through the loop. Initially, the inductor strongly opposes any change in current from its initial value of zero, and so virtually the entire source voltage appears across L. As time progresses, the current increases, which increases the voltage drop V_R = IR across the resistor and simultaneously decreases V_L = L(dI/dt). Eventually, the current asymptotically approaches I_max = ε/R, at which point dI/dt = 0 and the inductor behaves as a short circuit (assuming ideal, zero-resistance wire). The transient lasts roughly 5τ, where τ = L/R is the time constant of the circuit.

SECTION 4

Mathematical Framework

The transient behavior of an RL circuit is governed by a first-order linear ordinary differential equation that emerges directly from Kirchhoff's voltage law. By solving this ODE with the appropriate initial conditions, we obtain exponential expressions for both the energizing (current growth) and de-energizing (current decay) phases of the circuit. The mathematics is elegant and physically transparent, offering a clear connection between the circuit parameters R and L and the observable time-domain behavior.

Energizing the RL Circuit (Switch Closes at t = 0)

GOVERNING ODE — ENERGIZING

ε − IR − L(dI/dt) = 0

Where ε is the EMF of the source, I is the instantaneous current, R is the resistance, L is the inductance, and dI/dt is the rate of change of current. Rearranging: dI/dt + (R/L)I = ε/L.

CURRENT DURING ENERGIZING

I(t) = (ε/R)(1 − e^(−t/τ)) where τ = L/R

The current rises from 0 toward ε/R with time constant τ = L/R. At t = τ, I reaches about 63.2% of its maximum value. At t = 5τ, I is within 0.7% of ε/R — effectively at steady state.

Voltage Across Components During Energizing

VOLTAGE ACROSS RESISTOR

V_R(t) = ε(1 − e^(−t/τ))

The resistor voltage mirrors the current: it starts at zero and grows to ε as the current reaches steady state.

VOLTAGE ACROSS INDUCTOR

V_L(t) = ε · e^(−t/τ)

The inductor voltage starts at the full EMF ε (opposing the change) and decays exponentially to zero. At all times, $V_R$ (t) + $V_L$ (t) = ε, consistent with KVL.

De-energizing the RL Circuit (Source Removed at t = 0)

CURRENT DURING DE-ENERGIZING

I(t) = I₀ · e^(−t/τ) where I₀ = ε/R

When the source is removed and the circuit is short-circuited through the resistor, the current decays from I₀ toward zero. The inductor now acts as the source, driving current through R and dissipating the stored magnetic energy as heat.

ENERGY STORED IN INDUCTOR

U = ½LI²

The energy stored in the magnetic field of an inductor carrying current I. This is analogous to the kinetic energy ½mv² of a moving mass — the inductance L plays the role of mass, and the current I plays the role of velocity.

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Derivation Sketch

To solve dI/dt + (R/L)I = ε/L, multiply both sides by the integrating factor e^(R/L)t. The left side becomes d/dt[I · e^(R/L)t], which integrates directly to I · e^(R/L)t = (ε/R)e^(R/L)t + C. Applying the initial condition I(0) = 0 gives C = −ε/R, yielding I(t) = (ε/R)(1 − e^−t/τ).

SECTION 5

Transient Response in Detail

The transient response of an RL circuit is best understood visually by plotting the current and voltage waveforms as functions of time. The exponential growth and decay curves reveal how energy is exchanged between the source, the magnetic field of the inductor, and the thermal dissipation in the resistor. Crucially, the time constant τ = L/R governs the pace of this exchange: large inductance or small resistance leads to slow transients (large τ), while small inductance or large resistance produces fast transients (small τ). This is physically intuitive — more 'inertia' (larger L) means the system takes longer to reach equilibrium, while more 'friction' (larger R) dissipates energy more quickly, hastening the approach to steady state.

Transient response of a series RL circuit during energizing. The current I(t) (cyan) rises exponentially toward ε/R, reaching 63.2% at one time constant τ. The inductor voltage

V_L

(t) (violet) decays from ε to zero as the current stabilizes. The amber marker highlights the time constant τ = L/R.

Time (multiples of τ)	I(t) / I_max	$V_L$ (t) / ε	$V_R$ (t) / ε
0	0.000	1.000	0.000
1τ	0.632	0.368	0.632
2τ	0.865	0.135	0.865
3τ	0.950	0.050	0.950
5τ	0.993	0.007	0.993

Fraction of steady-state values at key multiples of the time constant τ = L/R

The table above reveals the practical engineering convention that an RL circuit is considered to have reached steady state after approximately 5 time constants, at which point the current is within 0.7% of its final value. Note the complementary nature of V_R and V_L: at every instant, their sum equals ε, as required by Kirchhoff's voltage law. The energy initially stored entirely in the magnetic field redistributes: during energizing, the inductor absorbs energy from the source while the resistor continuously dissipates it; at steady state, all power from the source is dissipated in the resistor since dI/dt = 0 and V_L = 0.

SECTION 6

Worked Example

Let us work through a complete example that illustrates the energizing transient of an RL circuit, calculating the time constant, instantaneous current, voltages, and energy stored at a specific time.

RL Circuit Energizing Transient

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Step 1 — Identify Given Values

A series RL circuit has R = 12 Ω, L = 60 mH = 0.060 H, and is connected to a DC source with ε = 24 V. The switch closes at t = 0, and we wish to find the current, voltages, and stored energy at t = 3.0 ms.
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Step 2 — Calculate the Time Constant

The time constant is τ = L/R = 0.060 H / 12 Ω = 0.005 s = 5.0 ms. This tells us that at t = 3.0 ms, we are at t/τ = 3.0/5.0 = 0.60 time constants — still in the early portion of the transient.

τ = 5.0 ms
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Step 3 — Calculate the Current at t = 3.0 ms

Using I(t) = (ε/R)(1 − e^−t/τ), the maximum current is I_max = ε/R = 24/12 = 2.0 A. At t = 3.0 ms: I = 2.0 × (1 − e^−0.60) = 2.0 × (1 − 0.5488) = 2.0 × 0.4512 = 0.902 A.

I(3.0 ms) ≈ 0.90 A
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Step 4 — Calculate Voltages Across R and L

The resistor voltage is V_R = IR = 0.902 × 12 = 10.8 V. The inductor voltage is V_L = ε − V_R = 24 − 10.8 = 13.2 V. Alternatively, V_L = εe^−0.60 = 24 × 0.5488 = 13.2 V. ✓ Consistent.

$V_R$ ≈ 10.8 V, $V_L$ ≈ 13.2 V
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Step 5 — Calculate Energy Stored in the Inductor

The energy stored in the magnetic field is U = ½LI² = ½ × 0.060 × (0.902)² = 0.030 × 0.8136 = 0.0244 J ≈ 24.4 mJ. At steady state, the maximum stored energy would be U_max = ½ × 0.060 × (2.0)² = 0.120 J = 120 mJ. So at t = 3.0 ms, the inductor has accumulated about 20.3% of its maximum energy.

U(3.0 ms) ≈ 24.4 mJ (20.3% of U_max)

SECTION 7

RL vs. RC Circuits: A Comparison

RL circuits are often taught alongside RC circuits (resistor-capacitor circuits) because they share the same mathematical structure — both are first-order systems governed by exponential transients — yet they differ fundamentally in the form of energy storage and the physical role of each component. Understanding their parallels and differences provides deeper insight into both circuit types and prepares the groundwork for second-order RLC circuits.

Property	RL Circuit	RC Circuit
Energy storage element	Inductor (L) — magnetic field	Capacitor (C) — electric field
Stored energy	U = ½LI²	U = ½CV²
Time constant τ	τ = L/R	τ = RC
Opposes changes in	Current (dI/dt)	Voltage (dV/dt)
Larger R effect on τ	Decreases τ (faster response)	Increases τ (slower response)
At t = 0 (energizing)	Inductor acts as open circuit	Capacitor acts as short circuit
At steady state	Inductor acts as short circuit	Capacitor acts as open circuit
Mechanical analogue	Mass (inertia) with friction	Spring with friction (dashpot)

Comparison of first-order RL and RC transient circuits

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KEY TAKEAWAY

A useful mnemonic: in an RL circuit, increasing R decreases the time constant (τ = L/R), so more resistance means faster transients. In an RC circuit, increasing R increases the time constant (τ = RC), so more resistance means slower transients. This opposite behavior arises because resistance limits current flow: in RL circuits this accelerates energy dissipation from the inductor, while in RC circuits it throttles the rate of charge delivery to the capacitor.

SECTION 8

Connection to Advanced Theory

The first-order RL circuit is a stepping stone to more sophisticated analyses in both physics and electrical engineering. When an inductor and capacitor are combined in the same circuit — forming an RLC circuit — the resulting second-order differential equation produces oscillatory behavior, including phenomena such as resonance, underdamping, overdamping, and critical damping. These dynamics are directly analogous to the driven harmonic oscillator in mechanics. Furthermore, in AC (alternating current) analysis, inductors introduce a frequency-dependent opposition to current known as inductive reactance (X_L = ωL), which plays a central role in impedance, phasor analysis, and filter design.

Concept	First-Order RL (This Lesson)	Advanced Extension
Governing equation	First-order ODE: L(dI/dt) + RI = ε	Second-order ODE (RLC): Ld²q/dt² + Rdq/dt + q/C = ε
Response type	Exponential growth/decay only	Oscillatory (underdamped), exponential (overdamped), or critical
Source type	DC source	AC source → phasor/impedance methods
Opposition to current	Resistance R (frequency-independent)	Impedance Z = R + jωL (frequency-dependent)
Key phenomenon	Exponential transient with τ = L/R	Resonance at ω₀ = 1/√(LC)

From first-order RL circuits to advanced electromagnetic theory

Looking ahead, you will encounter RL circuits in the context of mutual inductance and transformers, where two inductively coupled coils enable voltage transformation and impedance matching — the operating principle behind power grids. The concept of inductive reactance becomes essential when analyzing AC power transmission, where inductors introduce a phase shift between voltage and current. Mastery of the DC transient analysis presented here provides the physical and mathematical foundation for all of these advanced topics.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

A series RL circuit is connected to a DC battery. Immediately after the switch is closed, the inductor voltage equals the battery EMF and the current is zero. Explain, using Lenz's law, why the current cannot jump instantaneously to its steady-state value ε/R.

Show answer

By Lenz's law, an inductor opposes any change in the current flowing through it by generating a back-EMF proportional to dI/dt. At the instant the switch closes, the current attempts to change from zero to a nonzero value, so the inductor generates a voltage V_L = −L(dI/dt) that opposes the full source EMF. This means the net voltage available to drive current through the resistor is initially zero, and the current starts at zero. The current can only increase gradually as the inductor's opposition diminishes with the decreasing rate of change of current. An instantaneous jump to ε/R would require dI/dt = ∞, which would demand infinite energy — a physical impossibility.

Question 2: BASIC CALCULATION

A 200 mH inductor is connected in series with a 50 Ω resistor and a 10 V DC source. Calculate (a) the time constant τ, (b) the steady-state current, and (c) the current at t = 2τ.

Show answer

(a) τ = L/R = 0.200 H / 50 Ω = 4.0 × 10⁻³ s = 4.0 ms. (b) The steady-state current is I_max = ε/R = 10 V / 50 Ω = 0.20 A = 200 mA. (c) At t = 2τ, I = I_max(1 − e⁻²) = 0.20 × (1 − 0.1353) = 0.20 × 0.8647 = 0.173 A ≈ 173 mA. The current has reached approximately 86.5% of its final value after two time constants.

Question 3: INTERMEDIATE

In a series RL circuit with ε = 36 V, R = 18 Ω, and L = 90 mH, find the time at which the voltage across the inductor has dropped to 25% of the source EMF.

Show answer

The inductor voltage is V_L(t) = εe⁻ᵗ/τ. We need V_L = 0.25ε, so 0.25ε = εe⁻ᵗ/τ, giving e⁻ᵗ/τ = 0.25. Taking the natural log: −t/τ = ln(0.25) = −1.386. Therefore t = 1.386τ. With τ = L/R = 0.090/18 = 5.0 × 10⁻³ s = 5.0 ms, we get t = 1.386 × 5.0 ms = 6.93 ms ≈ 6.9 ms. At this instant, the current is I = (ε/R)(1 − 0.25) = (36/18)(0.75) = 1.5 A, and V_R = 1.5 × 18 = 27 V — confirmed by V_R + V_L = 27 + 9 = 36 V = ε. ✓

Question 4: APPLIED

An electromagnetic relay coil has L = 0.50 H and R = 100 Ω and requires a minimum current of 40 mA to close the relay contacts. If the coil is driven by a 12 V DC supply, how long after the switch closes will the relay activate? How much energy is stored in the magnetic field at the moment of activation?

Show answer

First, τ = L/R = 0.50/100 = 5.0 ms. The steady-state current is I_max = 12/100 = 0.120 A. Setting I(t) = 0.040 A: 0.040 = 0.120(1 − e⁻ᵗ/τ). Solving: 1 − e⁻ᵗ/τ = 0.333, so e⁻ᵗ/τ = 0.667, giving −t/τ = ln(0.667) = −0.405. Therefore t = 0.405 × 5.0 ms = 2.03 ms ≈ 2.0 ms. The energy stored at activation is U = ½LI² = ½ × 0.50 × (0.040)² = 0.25 × 1.6 × 10⁻³ = 4.0 × 10⁻⁴ J = 0.40 mJ. This represents only about 11% of the maximum stored energy (½ × 0.50 × 0.120² = 3.6 mJ), since the relay activates early in the transient.

Question 5: CRITICAL THINKING

An RL circuit with L = 0.30 H and R = 15 Ω has been connected to a 30 V source for a long time and has reached steady state. At t = 0, the source is suddenly removed and the circuit is short-circuited through the resistor. (a) Derive an expression for the total energy dissipated in the resistor from t = 0 to t = ∞. (b) Show that this equals the energy initially stored in the inductor. (c) At what time has exactly half the stored energy been dissipated?

Show answer

(a) During de-energizing, I(t) = I₀e⁻ᵗ/τ where I₀ = ε/R = 2.0 A and τ = 0.30/15 = 0.020 s. The power dissipated in R is P(t) = I²R = I₀²Re⁻²ᵗ/τ. Integrating from 0 to ∞: W = ∫₀^∞ I₀²Re⁻²ᵗ/τ dt = I₀²R × (τ/2) = I₀²R × L/(2R) = ½LI₀². (b) The initial stored energy is U₀ = ½LI₀² = ½ × 0.30 × 4.0 = 0.60 J. The total dissipated energy is W = ½LI₀² = 0.60 J — identical to U₀, confirming energy conservation: all magnetic field energy is converted to heat. (c) Half the energy is dissipated when the remaining stored energy equals ½U₀. Since U(t) = ½LI(t)² = ½LI₀²e⁻²ᵗ/τ, setting ½LI₀²e⁻²ᵗ/τ = ½ × ½LI₀² gives e⁻²ᵗ/τ = 0.5, so t = (τ/2)ln(2) = (0.020/2) × 0.693 = 6.93 ms ≈ 6.9 ms.

SUMMARY

Summary & Key Concepts

Summary

Circuits with resistors and inductors form first-order systems whose transient behavior is governed by exponential functions and a single time constant τ = L/R. An inductor stores energy in its magnetic field (U = ½LI²) and opposes changes in current through self-induced EMF (V_L = −L dI/dt), a direct consequence of Faraday's law and Lenz's law. During energizing, current grows as I(t) = (ε/R)(1 − e⁻ᵗ/τ), reaching 63.2% of its maximum at t = τ and effectively reaching steady state after ~5τ.

The KVL equation ε − IR − L(dI/dt) = 0 is the foundation for all RL transient analysis. During de-energizing, the current decays as I(t) = I₀e⁻ᵗ/τ, with all stored magnetic energy dissipated as heat in the resistor. RL circuits are complementary to RC circuits — sharing the same exponential mathematical form but differing in which quantity (current vs. voltage) experiences the transient. Mastery of RL circuit analysis prepares you for RLC oscillations, AC impedance methods, and the broader electromagnetic theory that underpins modern technology.