Circuits with Resistors and Inductors (LR Circuits)
A series LR circuit is governed by a first-order differential equation because the resistor’s voltage drop depends on current while the inductor’s voltage depends on dI/dt. With a battery, ε=IR+L dI/dt and the current approaches ε/R exponentially with time constant τ=L/R. During decay through resistance, I=I₀e^(−t/τ). At one time constant, growth reaches about 63% of its final value and decay falls to about 37% of its initial value.
Topic sequence · 5 of 6
Learning objectives
Objective 1
Derive the LR differential equation from Kirchhoff’s loop rule and solve or interpret current growth and decay with τ=L/R_eq.
Objective 2
Analyze current, resistor voltage, inductor voltage, and magnetic energy at t=0⁺, t≈τ, and t≫τ using continuity and asymptotic reasoning.
Objective 3
Use graphs, energy conservation, and experimental data to distinguish LR behavior from RC behavior and to determine L or R from transients.
- Topic 13.4 inductance, ε_L=−L dI/dt, and U_L=½LI².
- Unit 11 Kirchhoff loop rule, equivalent resistance, electric power, and resistor energy dissipation.
- First-order differential equations and exponential functions.
- RC-circuit time-constant reasoning from Topic 11.8 as a mathematical analogy, while keeping the physical variables distinct.
Definitions first
Representations you must read and create
Representation 1
Growth current graph: I starts at 0 for an initially unenergized inductor and approaches ε/R asymptotically. Initial slope is (dI/dt)_0=ε/L.
Representation 2
Growth voltage graph: |V_L| starts at ε and decays to 0 while V_R starts at 0 and rises to ε; Kirchhoff requires their signed sum to match the source at every time.
Representation 3
Decay current graph: I starts at I₀ and falls exponentially to zero. The tangent at t=0 would hit the time axis at τ, illustrating the time-constant definition.
Representation 4
Energy graph during source-free decay: U_L∝I² therefore decays as e^(−2t/τ), faster in exponent than current. The resistor’s accumulated thermal energy approaches the initial ½LI₀².
Representation 5
Initial/long-time circuit sketches: at t=0⁺ an initially unenergized inductor prevents an instantaneous current jump; at t≫τ under DC it behaves as an ideal wire with V_L≈0.
Representation 6
Semilog representation: ln I versus t for pure decay is a straight line with slope −1/τ, providing an experimental route to L=τR.
Core relationships and calculus
Variables: ε V; I A; R Ω; L H; t s
Use: Series LR growth equation from Kirchhoff’s loop rule with source emf ε.
Variables: τ s; L H; R Ω
Use: Characteristic time for the relevant LR current path. Use the equivalent resistance seen in the inductor’s transient path.
Variables: I A
Use: Initially unenergized series LR circuit switched onto a constant source.
Variables: I A
Use: Source-free decay from initial current I₀ through resistance R.
Variables: V V
Use: Resistor voltage during standard growth; approaches ε.
Variables: V V
Use: Inductor voltage magnitude during standard growth; starts at ε and decays to zero.
Variables: U J
Use: Magnetic energy at any instant after current is known.
Variables: P W
Use: Instantaneous thermal dissipation in the resistor during growth or decay.
For growth, separate variables in L dI/dt=ε−IR: dI/(ε/R−I)=(R/L)dt. Integrating with I(0)=0 gives −ln[(ε/R−I)/(ε/R)]=tR/L=t/τ, so I=(ε/R)(1−e^(−t/τ)). For source-free decay, L dI/dt+IR=0 gives dI/I=−(R/L)dt and I=I₀e^(−t/τ). These are not formulas to memorize in isolation; they are solutions of the loop equation with different sources and initial conditions.
Apply Kirchhoff around a series battery–resistor–inductor loop using the passive sign convention: ε−IR−L dI/dt=0. Rearranged, dI/dt+(R/L)I=ε/L. The homogeneous response is Ce^(−Rt/L), and a constant particular solution is ε/R. Applying I(0)=0 yields I=(ε/R)(1−e^(−Rt/L)). This identifies τ=L/R. For decay, remove the source from the active loop: L dI/dt+IR=0, so I=I₀e^(−Rt/L).
The required Topic 13.5 model is a circuit containing resistors and a single inductor. Use the resistance of the actual transient path, which may require an equivalent-resistance reduction if a problem supplies a more complicated resistor network. Detailed RLC damping, AC impedance/phasors, multiple coupled inductors, nonideal core losses, and switching arcs are beyond this topic unless explicitly modeled. Topic 13.6 treats ideal LC oscillations separately.
Physical examples
Example 1
A 2.0 H inductor in series with 4.0 Ω has τ=0.50 s. After one τ of growth from zero, current is 63.2% of ε/R; after one τ of decay, 36.8% of I₀ remains.
Example 2
For a 12 V source, R=6.0 Ω, L=3.0 H: I_final=2.0 A and τ=0.50 s. At t=0⁺, I=0, V_R=0, and the inductor voltage magnitude is 12 V.
Example 3
At t≫τ in the same circuit, dI/dt≈0, so V_L≈0 and I≈12/6=2.0 A. The inductor acts like a wire in the ideal DC steady-state model.
Example 4
During source-free decay with I₀=2.0 A, L=3.0 H, the initial magnetic energy is 6.0 J. With no other storage element, that energy is ultimately dissipated in the resistor.
Example 5
If L doubles with R fixed, τ doubles and the transient is slower. If R doubles with L fixed, τ halves even though the final growth current ε/R also halves.
Example 6
An LR circuit and RC circuit both show first-order exponentials, but the quantities that cannot jump differ: inductor current is continuous, while ideal capacitor voltage is continuous.
Worked reasoning
Exam reasoning patterns
Reasoning pattern 1
Write the loop equation before selecting an exponential formula. This protects against using a growth expression for a decay interval or the wrong resistance path.
Reasoning pattern 2
At t=0⁺, an initially unenergized inductor has I=0 but can have a large voltage across it. At long-time DC, dI/dt=0 and the ideal inductor voltage is zero.
Reasoning pattern 3
Remember τ=L/R, not LR and not R/L. Dimensional analysis helps: H/Ω=s.
Reasoning pattern 4
For one time constant, growth is 63%, decay is 37%. These are complements only for the standard normalized first-order responses.
Reasoning pattern 5
Current through an ideal inductor is continuous across switching. Use the pre-switch current as the post-switch initial current for the next interval.
Reasoning pattern 6
For energy questions, U_L is quadratic in current. If current has decayed to e⁻¹ of I₀, stored energy has decayed to e⁻² of its initial value.
Deeper calculus and limiting-case check
The standard growth solution can be written for any initial current I(0)=I₀ as I(t)=I_∞+[I₀−I_∞]e^(−t/τ), where I_∞ is the DC steady current for the post-switch circuit. This “final plus initial-minus-final times exponential” form unifies many switching problems. It also makes current continuity explicit: at t=0 the exponential equals 1, and at long times it vanishes.
Model-selection workflow: decide what the problem is before choosing an equation
For Circuits with Resistors and Inductors (LR Circuits), the fastest reliable route is not equation hunting. Start by identifying the loop geometry and changing flux, encode it with the flux/emf/current representation, choose the Faraday-Lenz or inductor equation, and then test the orientation, ideal-element, and transient assumption. This sequence prevents a mathematically correct manipulation from being attached to the wrong physical model.
Define the physical object or system
Name what is being modeled, what is outside the system, the coordinate or orientation convention, and the interval or geometry over which the statement applies. Treat LR circuit and time constant τ as physical quantities first and symbols second.
Choose a representation
Build Growth current graph: I starts at 0 for an initially unenergized inductor and approaches ε/R asymptotically. Initial slope is (dI/dt)_0=ε/L.. If the prompt supplies Growth voltage graph: |V_L| starts at ε and decays to 0 while V_R starts at 0 and rises to ε; Kirchhoff requires their signed sum to match the source at every time., translate it into the same sign, axis, orientation, or reference convention before combining information.
Write the governing relationship
A useful starting relationship is ε = IR + L dI/dt. State what the variables mean and why this relation—not merely a familiar formula—matches the physical situation.
Check conditions and limits
The required Topic 13.5 model is a circuit containing resistors and a single inductor. Use the resistance of the actual transient path, which may require an equivalent-resistance reduction if a problem supplies a more complicated resistor network. Detailed RLC damping, AC impedance/phasors, multiple coupled inductors, nonideal core losses, and switching arcs are beyond this topic unless explicitly modeled. Topic 13.6 treats ideal LC oscillations separately. After solving, test units, sign or direction, zero/large-parameter limits, and whether the answer remains consistent with the original representation.
Representation → equation → calculus → physics chain
| Stage | What to write or inspect | What proves the step is physically valid |
|---|---|---|
| Representation | Growth current graph: I starts at 0 for an initially unenergized inductor and approaches ε/R asymptotically. Initial slope is (dI/dt)_0=ε/L. | Every arrow, sign, axis, node, surface, path, or reference has an explicit meaning. The diagram must contain enough information to determine the relevant components or constraints. |
| Primary model | ε = IR + L dI/dt | Series LR growth equation from Kirchhoff’s loop rule with source emf ε.. Preserve the variables and units long enough to see dependence before inserting numerical values. |
| Second relationship | τ = L/R_eq | Characteristic time for the relevant LR current path. Use the equivalent resistance seen in the inductor’s transient path.. Combine relationships only after confirming they describe the same system, interval, frame, or geometry. |
| Calculus operation | For growth, separate variables in L dI/dt=ε−IR: dI/(ε/R−I)=(R/L)dt. Integrating with I(0)=0 gives −ln[(ε/R−I)/(ε/R)]=tR/L=t/τ, so I=(ε/R)(1−e^(−t/τ)). For source-free decay, L dI/dt+IR=0 gives dI/I=−(R/L)dt and I=I₀e^(−t/τ). These are not formulas to memorize in isolation; they are solutions of the loop equation with different sources and initial conditions. | State what the derivative, integral, differential equation, or limiting process represents physically; do not treat calculus notation as an algebraic decoration. |
| Physical check | Return to the original words and representation. | The final sign, direction, units, endpoint behavior, and functional dependence must all agree. A correct-looking expression that violates one of these checks needs repair. |
Advanced synthesis: how to handle an unfamiliar version of this topic
A 18.0 V battery is connected at t=0 to R=9.00 Ω and L=3.60 H in series, with zero initial current. The time constant is τ=L/R=0.400 s and the final current is I_∞=ε/R=2.00 A. At t=0.400 s, I=2.00(1−e⁻¹)=1.264 A. The resistor voltage is V_R=IR=11.38 V. The inductor voltage magnitude is ε−V_R=6.62 V, equal to εe⁻¹. The current-change rate is dI/dt=V_L/L=6.62/3.60=1.84 A/s. The stored magnetic energy is U_L=½(3.60)(1.264²)=2.88 J. At long times I→2.00 A and U_L→7.20 J. Every value is mutually consistent with Kirchhoff’s rule and the exponential solution. To deepen this example, change one assumption or parameter, predict the effect before recomputing, and then use the symbolic dependence to verify the prediction.
FRQ-quality response architecture for Topic 13.5
Mathematical routines
Start from the governing law or definition, keep the derivation symbolic, identify the condition that permits any simplification, and finish with units plus a limiting-case check. For this topic, the relationship ε = IR + L dI/dt should appear only when its physical conditions have been established.
Translation between representations
Move deliberately between Growth current graph: I starts at 0 for an initially unenergized inductor and approaches ε/R asymptotically. Initial slope is (dI/dt)_0=ε/L. and Growth voltage graph: |V_L| starts at ε and decays to 0 while V_R starts at 0 and rises to ε; Kirchhoff requires their signed sum to match the source at every time.. Explain what a slope, area, direction, curvature, intercept, node value, field orientation, or other feature means instead of merely redrawing it.
Experimental design and analysis
Turn the topic into a measurable relationship. State the independent and dependent variables, important controls, repeated/ranged trials, and the graph or transformed graph whose slope/intercept would test the model or extract a physical parameter.
Qualitative/quantitative translation
Make the comparison or direction claim first, justify it through the model, derive the symbolic dependence independently, and then reconcile the derivation with the original qualitative claim.
Do not let an equation stand in for an explanation. A complete response names what changes, identifies the governing physical relationship, follows the causal chain, and states the requested outcome explicitly.
Experiment and data-analysis upgrade
Construct a series LR circuit using a known resistor and coil/inductor. Use a voltage or current sensor to record the switch-on and/or decay transient. Fit I(t) to an exponential or plot ln I versus t during decay to obtain τ. Compute L=τR using the measured total resistance, including coil resistance when relevant. Compare the experimental L with a manufacturer value or a geometry-based estimate. Repeat with a different R and test the inverse τ∝1/R dependence. Discuss sampling rate, switch bounce, internal resistance, coil resistance, and sensor loading.
Operationalize the variables
Say exactly what instrument or observable supplies each measured quantity. Distinguish a directly measured variable from a quantity calculated later from those measurements.
Predict the graph before collecting data
Use the governing relationship to decide what should be plotted. If the raw relationship is nonlinear, choose a transformation that produces a line whose slope or intercept has a defined physical meaning.
Control and repeat for a reason
Hold variables fixed when they would change the predicted relationship. Repeat trials to estimate scatter; vary the independent quantity over a useful range to test functional dependence. These solve different experimental problems.
Evaluate the model, not only the number
Compare the best-fit result with the theoretical prediction, inspect residual or endpoint behavior when appropriate, and identify whether a discrepancy is random scatter, instrument resolution, or a systematic violation of an ideal-model assumption.
Error-repair matrix
| Failure pattern | Repair move | Self-check |
|---|---|---|
| Using τ=LR or τ=R/L instead of τ=L/R. | Return to the flux/emf/current representation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Treating an initially unenergized inductor as a short circuit at t=0⁺. It becomes short-like only after long-time DC steady state in the ideal model. | Return to the Faraday-Lenz or inductor equation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Allowing inductor current to jump instantly when a switch changes position. | Return to the flux/emf/current representation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Using ε/R as the current at all times immediately after closing the switch; that is the long-time asymptote. | Return to the Faraday-Lenz or inductor equation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Forgetting coil or network resistance when the problem specifies it as part of the transient path. | Return to the flux/emf/current representation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Assuming magnetic energy decays with the same exponential factor as current; because U∝I², pure-decay energy has e^(−2t/τ). | Return to the Faraday-Lenz or inductor equation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
Spiral review: make this topic transferable
A topic is exam-ready only when it can be recognized inside a mixed problem. Use the connections below as transfer tests rather than as optional reading.
Transfer 1
Topic 13.4 supplies ε_L=−L dI/dt and magnetic energy; Topic 13.5 combines them with Kirchhoff’s loop rule and resistance. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Transfer 2
Topic 11.8 RC circuits have the same first-order exponential mathematics but with τ=RC; comparing them helps separate mathematical structure from physical mechanism. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Transfer 3
Unit 11 electric power explains I²R thermal dissipation during inductor discharge. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Formula-condition retrieval
From memory, write ε = IR + L dI/dt and one condition that must be true before using it. Then state one situation where a more general derivative, integral, conservation law, or field/force model is required.
Topic 13.5 mastery ladder
Interactive LR growth and decay check
Compare source-driven growth from zero with source-free decay. The tool reports τ=L/R, current, resistor/inductor voltages, current slope, and magnetic energy.
Experimental connection
Construct a series LR circuit using a known resistor and coil/inductor. Use a voltage or current sensor to record the switch-on and/or decay transient. Fit I(t) to an exponential or plot ln I versus t during decay to obtain τ. Compute L=τR using the measured total resistance, including coil resistance when relevant. Compare the experimental L with a manufacturer value or a geometry-based estimate. Repeat with a different R and test the inverse τ∝1/R dependence. Discuss sampling rate, switch bounce, internal resistance, coil resistance, and sensor loading.
Connections across Physics C
Connection 1
Topic 13.4 supplies ε_L=−L dI/dt and magnetic energy; Topic 13.5 combines them with Kirchhoff’s loop rule and resistance.
Connection 2
Topic 11.8 RC circuits have the same first-order exponential mathematics but with τ=RC; comparing them helps separate mathematical structure from physical mechanism.
Connection 3
Unit 11 electric power explains I²R thermal dissipation during inductor discharge.
Connection 4
Topic 13.6 removes resistance and adds a capacitor, producing a second-order oscillatory differential equation instead of first-order exponential relaxation.
Connection 5
Mechanics first-order drag equations provide another mathematical analogy: the system approaches an asymptote because the rate of change depends on the distance from that asymptote.
Related lessons and resources
Use these links when the current model depends on a prerequisite or becomes the foundation for the next model.
Common mistakes
Avoid this
Using τ=LR or τ=R/L instead of τ=L/R.
Avoid this
Treating an initially unenergized inductor as a short circuit at t=0⁺. It becomes short-like only after long-time DC steady state in the ideal model.
Avoid this
Allowing inductor current to jump instantly when a switch changes position.
Avoid this
Using ε/R as the current at all times immediately after closing the switch; that is the long-time asymptote.
Avoid this
Forgetting coil or network resistance when the problem specifies it as part of the transient path.
Avoid this
Assuming magnetic energy decays with the same exponential factor as current; because U∝I², pure-decay energy has e^(−2t/τ).
Mastery check
- Define the main quantity without relying on an equation.
- Choose the correct representation and sign convention.
- Explain why the calculus relationship matches the physics.
- Design or interpret an experiment that could test the model.
Frequently asked questions
What is the time constant of an LR circuit?
τ=L/R_eq for the active transient path. It measures how quickly the current approaches its new steady state.
What happens at one time constant?
Growth from zero reaches about 63.2% of its final current; pure decay leaves about 36.8% of the initial current.
Why can’t inductor current change instantly?
A finite instantaneous jump would require an unbounded dI/dt and therefore an unbounded induced emf in the ideal model.
What does an inductor do after a long time on DC?
Once current is constant, dI/dt=0 and the ideal inductor has zero voltage drop, so it behaves like a conducting wire.
Where does stored inductor energy go during decay?
In a simple LR decay it is dissipated as thermal energy in the resistance.
How do LR circuits compare with RC circuits?
Both are first-order exponential transients, but LR circuits have τ=L/R and continuous inductor current, while RC circuits have τ=RC and continuous capacitor voltage.