Circuits with Resistors and Inductors (LR Circuits)

Historical Context & Motivation

The story of LR circuits begins with the nineteenth-century race to understand the relationship between electricity and magnetism. When Michael Faraday demonstrated in 1831 that a changing magnetic flux through a loop of wire produces an electromotive force (EMF), he laid the groundwork for understanding how coils of wire—later called inductors—store energy in their magnetic fields and resist sudden changes in current. Joseph Henry independently discovered self-induction around the same time, observing dramatic sparking when circuits carrying large currents through coils were suddenly broken. These observations revealed a new circuit element whose behavior depended not on the magnitude of the current but on its rate of change, fundamentally distinguishing inductors from resistors and capacitors.

The central question that LR circuit analysis addresses is deceptively simple: when a DC voltage source is suddenly connected to a series combination of a resistor and an inductor, how does the current evolve with time? Because the inductor opposes instantaneous changes in current, the circuit cannot jump to its steady-state value immediately—instead, an exponential transient process unfolds, governed by the ratio L/R. Understanding this transient behavior is essential not only for the AP Physics C exam but also for grasping how inductors function in power supplies, relay circuits, electromagnetic actuators, and countless other applications.

Core Principles & Definitions

An LR circuit is any circuit containing an inductor (inductance L, measured in henrys) and a resistor (resistance R, measured in ohms) connected to a source of EMF. The inductor's defining property is that it produces a voltage proportional to the time derivative of the current flowing through it: V_L = −L(dI/dt). This relationship, rooted in Faraday's law applied to a coil's own changing magnetic flux, is what gives LR circuits their characteristic exponential behavior. The following grid summarizes the foundational ideas you must internalize before diving into the mathematics.

Visual Explanation — The Series LR Circuit

The diagram above represents the prototypical series LR circuit that appears on nearly every AP Physics C exam. Before the switch closes, no current flows and no energy is stored in the inductor. The instant the switch closes at t = 0, the full EMF ε appears across the inductor because the current—and therefore the voltage drop IR across the resistor—is initially zero. The inductor's back-EMF exactly matches the source EMF, so dI/dt is at its maximum value of ε/L. As current builds, more voltage appears across R and less across L, causing dI/dt to decrease. This self-regulating feedback loop produces the characteristic exponential approach to the steady-state current I_max = ε/R, at which point dI/dt = 0 and the inductor behaves like an ideal wire.

Mathematical Framework

Deriving the Current Growth Equation

Applying Kirchhoff's voltage law around the single loop of the series LR circuit gives ε − IR − L(dI/dt) = 0. Rearranging, we obtain the first-order linear ODE: L(dI/dt) + IR = ε. This is separable. Dividing both sides by L and using the substitution u = ε/R − I, the equation transforms to du/dt = −(R/L)u. Integration with the initial condition I(0) = 0 yields the exponential growth equation below.

Current Decay (De-Energizing)

When the EMF source is suddenly removed (or the circuit is switched to a path containing only R and L), the current cannot drop to zero instantaneously because the inductor sustains the current via its stored magnetic energy. The loop equation becomes IR + L(dI/dt) = 0, which gives a pure exponential decay.

Current and Voltage Transient Curves

The exponential growth and decay equations produce characteristic curves that you must be able to sketch, interpret, and extract information from on the AP exam. The following diagram plots both the current I(t) and the inductor voltage V_L(t) during the energizing phase, with key time-constant milestones marked.

A crucial feature of the table is that the sum V_R(t) + V_L(t) = ε at every instant—this is simply Kirchhoff's voltage law in action. The practical rule of thumb is that after five time constants, the transient is more than 99% complete, and the circuit is effectively in its DC steady state. For the decay scenario, the same percentages apply in reverse: I(τ) = 0.368 I₀, I(2τ) = 0.135 I₀, and so on.

Worked Example

Comparing LR and RC Circuits

LR circuits and RC circuits are the two fundamental first-order transient circuits in physics. Both exhibit exponential behavior, but the roles of voltage and current are essentially swapped. Recognizing the structural analogy between them is a powerful tool for the AP exam, because if you can solve one type you can solve the other by pattern-matching.

Connection to Advanced Theory — RLC Circuits and AC Analysis

The series LR circuit is the simplest inductive circuit, but it serves as the gateway to richer phenomena. When a capacitor is added to form an RLC circuit, the governing equation becomes a second-order ODE, and the transient response can exhibit oscillatory (underdamped), critically damped, or overdamped behavior depending on the relative magnitudes of R, L, and C. This is analogous to a mass-spring-damper system in mechanics, where L plays the role of mass, 1/C the role of spring constant, and R the role of damping coefficient.

In AC circuit analysis, the inductor's impedance is purely imaginary: Z_L = jωL, where ω is the angular frequency of the AC source. This means the voltage across an inductor leads the current by 90°. Combined with a resistor (Z_R = R), the total impedance magnitude is |Z| = √(R² + ω²L²) and the phase angle is φ = arctan(ωL/R). While full AC phasor analysis extends beyond the typical AP Physics C E&M syllabus, the transient DC analysis you have learned here provides the conceptual foundation—understanding how V_L = L(dI/dt) works in the time domain is essential before extending to the frequency domain.

Practice Problems

Summary — LR Circuits