Circuits with Resistors | AP Physics C: E&M Unit 6 Study Guide

Getting Started

We have previously analyzed circuits with resistors, which respond instantaneously to changes in voltage. Now, we introduce the inductor, a component that resists changes in current. This chapter explores the transient behavior of a simple series circuit containing a resistor and an inductor (an LR circuit) when it is connected to a source of electromotive force. The central question is: How does the current in the circuit evolve from its initial state to its final, steady state?

What You Should Be Able to Do

Upon completing this section, you should be able to perform the following tasks:

Apply Kirchhoff’s loop rule to a series LR circuit to derive its governing first-order linear differential equation.
Solve the differential equation for the current as a function of time, $I (t)$ , for both current growth (energizing) and current decay (de-energizing).
Define and calculate the inductive time constant, $τ$ , and interpret its physical meaning in the context of the circuit's response time.
Analyze the behavior of the inductor and the circuit at two critical moments: immediately after a switch is closed ( $t = 0$ ) and a long time after ( $t \to \infty$ ).

Key Concepts & Mechanisms

The behavior of an LR circuit is fundamentally about the dynamic interplay between components over time. An inductor introduces a kind of "electrical inertia" that smooths out changes in current. We can analyze this using the lens of change over time, focusing on the differential equation that governs the system's evolution.

Assumptions: We will assume ideal components. The battery provides a constant electromotive force ( $E$ ), the wires have zero resistance, the resistor has a constant resistance ( $R$ ), and the inductor has a constant inductance ( $L$ ) with no internal resistance.

Baseline State

Consider two possible starting points. The first is an "off" state, where a switch in the circuit is open, and there is no current flowing ( $I = 0$ ). This is the initial condition for analyzing how the current builds up. The second baseline is a "steady-on" state, where the switch has been closed for a very long time. In this state, the current is constant, meaning its rate of change is zero ( $d I / d t = 0$ ), and the inductor offers no opposition to the current.

Differential Driver(s)

The primary driver of change in an LR circuit is the inductor's response to a changing current. According to Faraday's Law of Induction and Lenz's Law, a changing current through an inductor creates a changing magnetic flux, which in turn induces a "back-EMF" across the inductor. This induced EMF, $V_{L}$ , is given by:

$V_{L} = - L \frac{d I}{d t}$

The term $L$ is the inductance, a measure of how much EMF is induced per unit change in current, measured in Henrys (H). The negative sign signifies that the induced EMF opposes the change in current.

When a switch is closed to connect an LR series circuit to a battery of EMF $E$ , the battery attempts to drive a current. The inductor immediately opposes this change. We can describe the relationship between the potential differences across the components at any instant using Kirchhoff's loop rule, which states that the sum of potential changes around a closed loop is zero ( $\sum Δ V = 0$ ).

Starting from the positive terminal of the battery and traversing the loop in the direction of the current $I$ :

Potential gain from the battery: $+ E$
Potential drop across the resistor: $- I R$
Potential drop across the inductor (opposing the increase in I): $- L \frac{d I}{d t}$

Summing these gives the governing differential equation for the circuit:

$E - I R - L \frac{d I}{d t} = 0$

$E = I R + L \frac{d I}{d t}$

This equation is the mathematical engine of the circuit. It shows that the battery's EMF is split between driving current through the resistor and overcoming the inductor's opposition to the change in current.

Conserved / Invariant Quantities

While the current $I$ and the potential drops $V_{R}$ and $V_{L}$ are functions of time, several quantities are constant in our ideal model. The battery's EMF ( $E$ ), the resistance ( $R$ ), and the inductance ( $L$ ) are fixed parameters of the circuit. Most importantly, Kirchhoff's loop rule (conservation of energy) holds true at every single moment in time as the system evolves.

Key Models & Diagrams

The analysis of an LR circuit's transient behavior can be broken down into two primary scenarios: the growth of current when connected to a power source ("energizing") and the decay of current when the power source is removed ("de-energizing").

Circuit Model & State	Governing Equation (from Kirchhoff's Loop Rule)	Predicted Behavior: Current $I (t)$
Energizing InductorA series LR circuit is connected to a battery $E$ at $t = 0$ .Initial condition: $I (0) = 0$ .	$E - I R - L \frac{d I}{d t} = 0$	The solution to this differential equation is an exponential growth function: $I (t) = \frac{E}{R} (1 - e^{- t / τ})$ where $τ = L / R$ .The current starts at 0 and asymptotically approaches a maximum steady-state value of $I_{ma x} = E / R$ .
De-energizing InductorAn energized LR circuit with initial current $I_{0}$ is shorted (battery removed) at $t = 0$ .Initial condition: $I (0) = I_{0}$ .	$- I R - L \frac{d I}{d t} = 0$	The solution is an exponential decay function: $I (t) = I_{0} e^{- t / τ}$ where $τ = L / R$ .The current starts at its maximum value $I_{0}$ and decays exponentially to zero as the inductor's stored magnetic energy is dissipated by the resistor.

Key Components & Evidence

Electromotive Force ( $E$ ): The work per unit charge supplied by a source, such as a battery. It is the primary driver of current in the circuit. Measured in Volts (V).
Resistance ( $R$ ): A measure of a component's opposition to the flow of steady current, defined by Ohm's Law ( $V_{R} = I R$ ). Measured in Ohms (Ω).
Inductance ( $L$ ): A measure of a component's opposition to a change in current, defined by the back-EMF equation ( $V_{L} = - L d I / d t$ ). Measured in Henrys (H).
Current ( $I$ ): The rate of flow of electric charge. In an LR circuit, it is a function of time, $I (t)$ . Measured in Amperes (A).
Kirchhoff's Loop Rule ( $\sum Δ V = 0$ ): A statement of conservation of energy for an electric circuit. It provides the fundamental relationship that leads to the governing differential equation.
Back-EMF ( $V_{L}$ ): The potential difference induced across the inductor that opposes the change in current. Its magnitude is greatest when the current is changing most rapidly.
Inductive Time Constant ( $τ = L / R$ ): The characteristic time scale for the transient behavior of an LR circuit. It is the time required for the current to complete approximately 63% ( $1 - 1/ e$ ) of its total change. Measured in seconds (s).
Steady State: The condition, achieved after a long time ( $t ≫ τ$ ), where all circuit variables become constant. In this state, $d I / d t = 0$ , and the inductor behaves as a simple conducting wire with zero resistance.

Skill Snapshots

Causation

Driver: A sudden connection to a battery ( $E$ ). Change: A time-varying current $I (t)$ begins to flow, but its growth is opposed by the inductor.
Driver: A changing current ( $d I / d t \neq = 0$ ). Change: The inductor generates a back-EMF ( $V_{L} = - L d I / d t$ ) that counteracts this change.
Driver: The dissipation of energy in the resistor ( $P = I^{2} R$ ). Change: In a de-energizing circuit, this energy loss causes the current to decay exponentially to zero.

Comparison

At $t = 0^{+}$ (energizing): An ideal inductor acts like an open circuit (opposing any initial current, so $I = 0$ ), whereas an ideal capacitor acts like a short circuit (uncharged plates offer no initial opposition to current).
As $t \to \infty$ (steady state): An ideal inductor acts like a short circuit (zero resistance, since $d I / d t = 0$ ), whereas an ideal capacitor acts like an open circuit (fully charged, blocking DC current).
Time Constant: The LR time constant is $τ = L / R$ , while the RC time constant is $τ = RC$ . Increasing resistance $R$ slows down the response of an LR circuit (larger $τ$ ) but speeds up the response of an RC circuit (smaller $τ$ ).

Change, Continuity, and Conservation (CCOT)

Baseline: For $t < 0$ , the switch is open, the current is zero, and no energy is stored in the inductor.
Change: At $t = 0$ , the switch closes. The current begins to rise, and energy begins to be stored in the inductor's magnetic field ( $U_{L} = \frac{1}{2} L I^{2}$ ). The rate of current change, $d I / d t$ , is maximal at this instant.
Change: As $t \to \infty$ , the current approaches its steady-state value $E / R$ , and the rate of change $d I / d t$ approaches zero. The inductor's magnetic field becomes constant.
Continuity: The current $I (t)$ through the inductor is always a continuous function of time; it cannot change instantaneously. Kirchhoff's loop rule, $E = V_{R} (t) + V_{L} (t)$ , holds true at every moment during the transition.

Common Misconceptions & Clarifications

Misconception: Current can change instantly in a circuit with an inductor.
Clarification: The defining property of an inductor is that it generates an EMF to oppose changes in current. An infinite rate of change ( $d I / d t \to \infty$ ) would require an infinite EMF. Therefore, the current through an inductor must always be a continuous function of time.
Misconception: The voltage across an inductor is always proportional to the current, like a resistor.
Clarification: The voltage across an inductor is proportional to the rate of change of the current ( $V_{L} = L d I / d t$ ), not the current itself. A large, steady current produces zero voltage across an ideal inductor, while a rapidly changing but small current can produce a very large voltage.
Misconception: An inductor "stores current" or "stores charge."
Clarification: An inductor stores energy in the magnetic field created by the current. The amount of stored energy is given by $U_{L} = \frac{1}{2} L I^{2}$ . It does not store charge (like a capacitor) or current. This stored energy is returned to the circuit when the current decreases.
Misconception: The time constant $τ$ is the total time it takes for the current to reach its final value.
Clarification: The time constant $τ$ is the time it takes for the current to reach $(1 - 1/ e) \approx 63.2%$ of its final value during energizing, or to decay to $1/ e \approx 36.8%$ of its initial value during de-energizing. The circuit theoretically approaches its final state asymptotically and never truly reaches it, though it is considered to be in a steady state for practical purposes after about five time constants ( $5 τ$ ).

One-Paragraph Summary

A series LR circuit exhibits transient, time-dependent behavior because an inductor generates a back-EMF that opposes any change in the current flowing through it. By applying Kirchhoff's loop rule, we derive a first-order linear differential equation, $E = I R + L (d I / d t)$ , that governs the system. The solution to this equation is an exponential function describing the current's growth or decay, characterized by the inductive time constant $τ = L / R$ . This time constant dictates how quickly the circuit reaches a steady state. Analysis of the circuit's behavior at its temporal limits—acting as an open circuit at $t = 0$ and a short circuit as $t \to \infty$ —provides powerful insights and boundary conditions for solving problems.

Circuits with Resistors and Inductors (LR Circuits) - AP Physics C: Electricity and Magnetism Study Guide