Inductance
Inductance measures how strongly a conductor’s own changing current changes linked magnetic flux and therefore generates an emf that opposes that current change. For a linear inductor, flux linkage λ=NΦ_B is proportional to current, λ=LI, giving ε_L=−L dI/dt. A long solenoid provides the central geometry model L=μ_core N²A/ℓ, and an inductor carrying current stores magnetic-field energy U_L=½LI².
Topic sequence · 4 of 6
Learning objectives
Objective 1
Explain self-inductance physically and use flux linkage λ=NΦ_B=LI to connect a changing current to an induced back emf.
Objective 2
Derive and apply the solenoid inductance L=μ_core N²A/ℓ, including correct factor-change reasoning for N, A, ℓ, and core permeability.
Objective 3
Calculate magnetic energy U_L=½LI² and use power/energy conservation to interpret energy transfer into and out of an inductor.
- Topics 12.3–12.4 for magnetic fields of currents and the long-solenoid field B≈μnI.
- Topics 13.1–13.2 for magnetic flux, Faraday’s law, and Lenz’s law.
- Unit 11 current and circuit sign conventions.
- Calculus concepts of derivative, integral, power as dU/dt, and proportionality.
Definitions first
Representations you must read and create
Representation 1
Flux-linkage versus current graph: for an ideal linear inductor λ=LI is a straight line through the origin. Its slope is L. A steeper slope means more linked flux for the same current.
Representation 2
Induced-emf versus dI/dt graph: ε_L=−L dI/dt is linear with slope −L if a fixed sign convention is used. The emf is zero at constant current even when I and stored energy are large.
Representation 3
Solenoid geometry comparison: doubling N with A, ℓ, μ fixed increases L by four; doubling A doubles L; doubling ℓ halves L; increasing μ_core increases L in direct proportion in the ideal model.
Representation 4
Energy-versus-current graph: U_L=½LI² is an upward-opening parabola. Reversing current does not change stored energy because I² is unchanged.
Representation 5
Current-increase sketch: induced polarity is such that the inductor’s emf opposes the increase. During current decrease, the polarity reverses in an attempt to sustain the original current direction.
Representation 6
Field-energy picture: current in a solenoid establishes B inside its volume. The circuit’s electrical work builds the magnetic field; when current falls, that stored energy can be delivered to a resistor or capacitor.
Core relationships and calculus
Variables: λ Wb-turn; N dimensionless; Φ_B Wb; L H; I A
Use: Definition/model relation for linear self-inductance.
Variables: ε V; L H; dI/dt A/s
Use: Back emf of an ideal inductor; the sign encodes opposition to current change.
Variables: L H; μ H/m; A m²; ℓ m
Use: Ideal long-solenoid inductance.
Variables: U J; L H; I A
Use: Energy stored in a linear inductor’s magnetic field.
Variables: P W; L H; I A; dI/dt A/s
Use: Rate at which energy stored in the ideal inductor changes under the passive sign convention.
Variables: dU J
Use: Differential energy relation integrated from zero current to I to obtain U_L=½LI².
Inductance can be read as a derivative as well as a proportionality. In a linear device λ=LI so L=dλ/dI is constant. Faraday gives ε=−dλ/dt=−(dλ/dI)(dI/dt)=−L dI/dt. If L depends on current because of nonlinear material response, the more general derivative dλ/dI replaces a single constant L. AP calculations normally use the linear constant-L model unless told otherwise.
For a long solenoid, B≈μ_core nI=μ_core(N/ℓ)I. Flux through one turn is Φ_B=BA=μ_core(N/ℓ)IA. The total flux linkage is λ=NΦ_B=μ_core N²AI/ℓ. By λ=LI, the inductance is L=μ_core N²A/ℓ. The N² dependence is therefore not arbitrary: one N comes from how turn density strengthens B for a given current, and the other N counts how many turns link that flux.
The current AP framework explicitly requires the physical dependence of solenoid inductance, magnetic-field energy, conservation of energy, and ε_i=−L dI/dt. Straight wires are typically modeled as having zero inductance in this course. Detailed mutual-inductance/transformer calculations, nonlinear hysteresis, saturation, parasitic capacitance, and frequency-dependent real-inductor models go beyond the required ideal self-inductance treatment unless a problem supplies the needed model.
Physical examples
Example 1
A 0.40 H inductor with current increasing at 3.0 A/s has induced-emf magnitude 1.2 V. The polarity is chosen to oppose the increase, not necessarily to oppose the current direction.
Example 2
If current through the same inductor is constant at 5.0 A, ε_L=0 even though stored energy is ½(0.40)(5.0²)=5.0 J.
Example 3
For an ideal solenoid, doubling the number of turns while keeping length and area fixed makes L four times larger because both field production per ampere and flux linkage count increase.
Example 4
A 2.0 H inductor carrying 3.0 A stores 9.0 J. Doubling current to 6.0 A increases the stored energy to 36 J, a factor of four.
Example 5
If a core with relative permeability 5 replaces an air/vacuum core in the same idealized linear geometry, the model predicts five times the inductance. Real ferromagnetic cores can be nonlinear or saturate, which is beyond the simple constant-μ model.
Example 6
Two coils can exhibit mutual inductance, but the required 13.4 model centers on the physical/electrical properties of an inductor and its self-induced emf. Treat mutual-inductance calculations only when explicitly introduced by a source or instructor.
Worked reasoning
Exam reasoning patterns
Reasoning pattern 1
Read the sign in ε_L=−L dI/dt as “oppose the change.” If dI/dt reverses sign, the inductor polarity reverses.
Reasoning pattern 2
Do not say an inductor always opposes current. At constant current the ideal inductor has zero induced emf; it opposes changes in current.
Reasoning pattern 3
For solenoid factor changes, remember N². Doubling turns at fixed ℓ and A quadruples L, not doubles it.
Reasoning pattern 4
Keep inductance L separate from solenoid length ℓ. Use clear symbols and units: H for inductance, m for length.
Reasoning pattern 5
Energy is quadratic in current. A current sign reversal changes magnetic-field direction but not U_L=½LI².
Reasoning pattern 6
Use λ=LI and Faraday’s law as a conceptual bridge; it prevents treating the back-emf formula as an unrelated memorized equation.
Deeper calculus and limiting-case check
For a long air-core solenoid, combine U_L=½LI² with L=μ₀N²A/ℓ and B=μ₀(N/ℓ)I. Eliminating N and I gives U=(B²/(2μ₀))(Aℓ). Thus the magnetic energy density is u_B=B²/(2μ₀) in the ideal vacuum/air model. This derivation is valuable for connecting circuit energy to field energy, even when a particular AP question only requires U_L=½LI².
Model-selection workflow: decide what the problem is before choosing an equation
For Inductance, 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 inductance L and self-inductance as physical quantities first and symbols second.
Choose a representation
Build Flux-linkage versus current graph: for an ideal linear inductor λ=LI is a straight line through the origin. Its slope is L. A steeper slope means more linked flux for the same current.. If the prompt supplies Induced-emf versus dI/dt graph: ε_L=−L dI/dt is linear with slope −L if a fixed sign convention is used. The emf is zero at constant current even when I and stored energy are large., translate it into the same sign, axis, orientation, or reference convention before combining information.
Write the governing relationship
A useful starting relationship is λ = NΦ_B = LI. State what the variables mean and why this relation—not merely a familiar formula—matches the physical situation.
Check conditions and limits
The current AP framework explicitly requires the physical dependence of solenoid inductance, magnetic-field energy, conservation of energy, and ε_i=−L dI/dt. Straight wires are typically modeled as having zero inductance in this course. Detailed mutual-inductance/transformer calculations, nonlinear hysteresis, saturation, parasitic capacitance, and frequency-dependent real-inductor models go beyond the required ideal self-inductance treatment unless a problem supplies the needed model. 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 | Flux-linkage versus current graph: for an ideal linear inductor λ=LI is a straight line through the origin. Its slope is L. A steeper slope means more linked flux for the same current. | 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 | λ = NΦ_B = LI | Definition/model relation for linear self-inductance.. Preserve the variables and units long enough to see dependence before inserting numerical values. |
| Second relationship | ε_L = −L dI/dt | Back emf of an ideal inductor; the sign encodes opposition to current change.. Combine relationships only after confirming they describe the same system, interval, frame, or geometry. |
| Calculus operation | Inductance can be read as a derivative as well as a proportionality. In a linear device λ=LI so L=dλ/dI is constant. Faraday gives ε=−dλ/dt=−(dλ/dI)(dI/dt)=−L dI/dt. If L depends on current because of nonlinear material response, the more general derivative dλ/dI replaces a single constant L. AP calculations normally use the linear constant-L model unless told otherwise. | 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
An ideal long solenoid has N=800 turns, length ℓ=0.400 m, cross-sectional area A=3.00×10⁻⁴ m², and an air core so μ_core≈μ₀=4π×10⁻⁷ H/m. Its inductance is L=μ₀N²A/ℓ=(4π×10⁻⁷)(800²)(3.00×10⁻⁴)/(0.400)=6.03×10⁻⁴ H. If the current rises at 250 A/s, the induced-emf magnitude is L dI/dt=0.151 V and its polarity opposes the rise. At I=4.00 A, the stored energy is U=½LI²=4.83×10⁻³ J. The flux linkage is λ=LI=2.41×10⁻³ Wb-turn, so the average flux per turn in the ideal uniform-field model is Φ=λ/N=3.02×10⁻⁶ Wb. 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 13.4
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 λ = NΦ_B = LI should appear only when its physical conditions have been established.
Translation between representations
Move deliberately between Flux-linkage versus current graph: for an ideal linear inductor λ=LI is a straight line through the origin. Its slope is L. A steeper slope means more linked flux for the same current. and Induced-emf versus dI/dt graph: ε_L=−L dI/dt is linear with slope −L if a fixed sign convention is used. The emf is zero at constant current even when I and stored energy are large.. 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
Investigate inductance with a coil, resistor, switch, voltage/current sensors, and optionally interchangeable cores. First determine L from the initial/current-transient behavior in a known series circuit or from a measured ε_L versus dI/dt slope. Then vary one geometric factor such as number of turns or core material while controlling others. Test the predicted scaling with N² or μ_core qualitatively/quantitatively. Measure current and voltage simultaneously to estimate energy delivered to the inductor and compare with ½LI². Include coil resistance as a systematic limitation rather than pretending the physical coil is ideal.
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 |
|---|---|---|
| Saying back emf opposes current rather than opposing the change in current. | 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 L=μN A/ℓ instead of the required N² dependence for the ideal solenoid. | 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 to convert cross-sectional area into m² or length into meters. | 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 ε_L is nonzero whenever current is nonzero; ε_L depends on dI/dt. | 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? |
| Treating ½LI² as energy dissipated by the inductor. An ideal inductor stores and can return energy; resistance dissipates energy. | 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? |
| Confusing flux Φ through one turn with flux linkage NΦ for the coil. | 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 12.4 gives the long-solenoid magnetic field used to derive L_sol. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Transfer 2
Topic 13.2 Faraday’s law becomes ε_L=−L dI/dt when changing flux is produced by the circuit’s own changing current. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Transfer 3
Unit 10 capacitor energy provides a useful comparison: capacitors store electric-field energy while inductors store magnetic-field energy. 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 λ = NΦ_B = LI 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 13.4 mastery ladder
Interactive solenoid inductance and energy check
Use the ideal long-solenoid model to calculate L, flux linkage, back emf, stored energy, and the corresponding interior B field.
Experimental connection
Investigate inductance with a coil, resistor, switch, voltage/current sensors, and optionally interchangeable cores. First determine L from the initial/current-transient behavior in a known series circuit or from a measured ε_L versus dI/dt slope. Then vary one geometric factor such as number of turns or core material while controlling others. Test the predicted scaling with N² or μ_core qualitatively/quantitatively. Measure current and voltage simultaneously to estimate energy delivered to the inductor and compare with ½LI². Include coil resistance as a systematic limitation rather than pretending the physical coil is ideal.
Connections across Physics C
Connection 1
Topic 12.4 gives the long-solenoid magnetic field used to derive L_sol.
Connection 2
Topic 13.2 Faraday’s law becomes ε_L=−L dI/dt when changing flux is produced by the circuit’s own changing current.
Connection 3
Unit 10 capacitor energy provides a useful comparison: capacitors store electric-field energy while inductors store magnetic-field energy.
Connection 4
Topic 13.5 places the inductor in a resistive circuit and turns ε_L into a first-order differential equation for current.
Connection 5
Topic 13.6 will exchange energy between capacitor and inductor, using the same ½LI² magnetic-energy model.
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
Saying back emf opposes current rather than opposing the change in current.
Avoid this
Using L=μN A/ℓ instead of the required N² dependence for the ideal solenoid.
Avoid this
Forgetting to convert cross-sectional area into m² or length into meters.
Avoid this
Assuming ε_L is nonzero whenever current is nonzero; ε_L depends on dI/dt.
Avoid this
Treating ½LI² as energy dissipated by the inductor. An ideal inductor stores and can return energy; resistance dissipates energy.
Avoid this
Confusing flux Φ through one turn with flux linkage NΦ for the coil.
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 inductance physically?
It measures how much magnetic flux linkage changes with current and therefore how strongly a device generates an emf when its current changes.
Why does an inductor oppose current changes?
A current change changes its magnetic flux. Faraday–Lenz induction creates an emf whose polarity opposes that change.
What does one henry mean?
An inductance of 1 H produces 1 V of induced-emf magnitude for a current-change rate of 1 A/s.
Why does solenoid inductance depend on N squared?
More turns increase the magnetic field per ampere and also increase the number of turns linked by that field, producing two factors of N.
Can an inductor have energy with zero voltage across it?
Yes. At constant nonzero current, an ideal inductor can store ½LI² while dI/dt=0 and therefore ε_L=0.
Do straight wires really have zero inductance?
Real wires have some inductance, but the AP course typically models straight wires as having zero inductance unless another model is specified.