13.4 Inductance

TLDR

Inductance (L) measures how strongly a conductor opposes a change in the current through it. An inductor like a solenoid stores energy in its magnetic field, and any change in current produces a back emf given by $\mathcal{E}_i = -L\frac{dI}{dt}$. For AP Physics C: E&M, you need to connect an inductor's physical build to its inductance, energy storage, and induced emf.

Why This Matters for the AP Physics C: E&M Exam

Inductance sits inside Unit 13, Electromagnetic Induction, which carries a 13-17% weighting on the multiple-choice section. This topic builds directly on Faraday's law and Lenz's law, then sets up the LR and LC circuit topics that come next. On both multiple-choice and free-response questions, you may be asked to predict how an induced emf or stored energy changes when current, geometry, or core material changes, and to justify those claims using functional relationships between variables. Lab work with solenoids in circuits is also a way these abstract ideas show up concretely.

Key Takeaways

• Inductance is a conductor's tendency to oppose a change in current; straight wires are modeled as having zero inductance, while solenoids have significant inductance.
• A solenoid's inductance is $L_{sol} = \frac{\mu_{core} N^2 A}{\ell}$, so it grows with more turns, larger area, and higher core permeability, and shrinks with greater length.
• Inductance depends on the square of the number of turns, so $N$ is the strongest lever in inductor design.
• Stored energy is $U_L = \frac{1}{2}LI^2$, so doubling the current quadruples the energy in the magnetic field.
• Induced emf is $\mathcal{E}_i = -L\frac{dI}{dt}$; the negative sign shows the emf opposes the change in current (Lenz's law).
• Energy stored in an inductor can be dissipated through a resistor or used to charge a capacitor, always following conservation of energy.

Physical and Electrical Properties of Inductors

Inductance and Current Opposition

Inductance quantifies how strongly a conductor opposes changes in the current flowing through it. When current changes, the magnetic field it creates also changes, and that changing field induces an emf that pushes back against the change. This property depends on the conductor's physical build and its surroundings.

• Straight wires typically have negligible inductance and are modeled as having zero inductance in most circuit analyses.
• Inductors, such as solenoids, are circuit components specifically built to have significant inductance.
• The inductance of a solenoid depends on:
  • Number of turns (N): more turns means higher inductance
  • Length of the solenoid (ℓ): shorter solenoids have higher inductance
  • Cross-sectional area (A): larger area means higher inductance
  • Magnetic permeability of the core material (μ_core): ferromagnetic cores greatly increase inductance

The inductance of a solenoid can be calculated using:

$L_{sol}=\frac{\mu_{core} N^{2} A}{\ell}$

Inductance increases with the square of the number of turns, which makes $N$ the most influential factor in inductor design.

Energy Storage in Magnetic Fields

Inductors store energy in the magnetic field generated by the current flowing through them. This is why inductors matter in so many electronic applications.

$U_{L}=\frac{1}{2} L I^{2}$

This equation shows several important features:

• Energy storage increases with the square of the current.
• Doubling the current quadruples the stored energy.
• Stored energy is directly proportional to inductance.
• This energy can be transferred to other forms while following conservation laws.
• Energy can be dissipated as heat through a resistor.
• Energy can charge a capacitor, converting magnetic energy into electric potential energy.

Induced EMF in Inductors

When the current through an inductor changes, it creates a changing magnetic field that induces an emf according to Faraday's law of induction. This induced emf opposes the change in current, which is Lenz's law in action.

The induced emf in an inductor is given by:

$\mathcal{E}_{i}=-L \frac{d I}{d t}$

This equation tells you:

• The induced emf is proportional to the inductance.
• The induced emf is proportional to how quickly the current is changing.
• The negative sign indicates the emf opposes the change in current.
• Rapid current changes produce larger induced emfs.

Because of this, inductors resist sudden changes in current and produce a delayed response in circuits. When current is abruptly interrupted, an inductor can generate a large voltage spike as it tries to keep the current flowing, which is why circuit design has to manage these spikes carefully.

How to Use This on the AP Physics C: E&M Exam

Problem Solving

• Identify which of the three core relationships fits the question: geometry to inductance ($L_{sol}$), inductance and current to energy ($U_L$), or inductance and current rate to emf ($\mathcal{E}_i$).
• Convert all quantities to SI units before substituting. Centimeters to meters and cm² to m² are common slip points.
• For core material, remember $\mu_{core} = \mu_0 \mu_r$, where $\mu_r$ is the relative permeability.

Free Response

• When asked to compare two setups, use functional dependence. State how the changed variable scales the result, for example "doubling N quadruples L because L depends on N²."
• For emf direction, apply Lenz's law and explain that the induced emf opposes the change in flux, then back it up with the sign in $\mathcal{E}_i = -L\frac{dI}{dt}$.
• When energy transfers between an inductor and a resistor or capacitor, justify your answer using conservation of energy.

Common Trap

• The minus sign in $\mathcal{E}_i = -L\frac{dI}{dt}$ describes direction, not a negative size. Report magnitude separately when the question asks for it.

Practice Problem 1: Solenoid Inductance

Solution

Use the solenoid inductance formula:

$L_{sol}=\frac{\mu_{core} N^{2} A}{\ell}$

First convert values to SI units:

• Number of turns: N = 500 turns
• Length: ℓ = 25 cm = 0.25 m
• Cross-sectional area: A = 4.0 cm² = 4.0 × 10⁻⁴ m²
• Core permeability: μ_core = μ₀μᵣ = (4π × 10⁻⁷ H/m)(5000) = 6.28 × 10⁻³ H/m

Substitute into the formula:

$L_{sol}=\frac{(6.28 \times 10^{-3}\,\text{H/m})(500)^2(4.0 \times 10^{-4}\,\text{m}^2)}{0.25\,\text{m}}$

$L_{sol}=\frac{(6.28 \times 10^{-3})(250{,}000)(4.0 \times 10^{-4})}{0.25}$

$L_{sol}=\frac{(6.28 \times 10^{-3})(100)}{0.25}$

$L_{sol}=\frac{0.628}{0.25}=2.51\,\text{H}$

The inductance of the solenoid is 2.51 H.

Practice Problem 2: Energy Storage in an Inductor

Solution

Use the energy storage formula:

$U_{L}=\frac{1}{2} L I^{2}$

Given:

• Inductance: L = 30 mH = 30 × 10⁻³ H
• Current: I = 2.0 A

Substitute:

$U_{L}=\frac{1}{2} (30 \times 10^{-3} \text{ H}) (2.0 \text{ A})^{2}$

$U_{L}=\frac{1}{2} (30 \times 10^{-3}) (4.0)$

$U_{L}=\frac{1}{2} (120 \times 10^{-3})$

$U_{L}= 60 \times 10^{-3} \text{ J} = 60 \text{ mJ}$

The energy stored in the inductor's magnetic field is 60 mJ.

Practice Problem 3: Induced EMF in an Inductor

Solution

Use the induced emf formula:

$\mathcal{E}_{i}=-L \frac{d I}{d t}$

Given:

• Inductance: L = 50 mH = 50 × 10⁻³ H
• Rate of current change: dI/dt = 100 A/s

Substitute:

$\mathcal{E}_{i}=-(50 \times 10^{-3} \text{ H})(100 \text{ A/s})$

$\mathcal{E}_{i}=-(5.0) \text{ V}$

The negative sign indicates the induced emf opposes the change in current (Lenz's law). The magnitude of the induced emf is 5.0 V.

Common Misconceptions

• Inductance is not the same as resistance. A resistor opposes current itself, while an inductor opposes a change in current. A steady, unchanging current produces no induced emf in an ideal inductor.
• The negative sign in $\mathcal{E}_i = -L\frac{dI}{dt}$ does not mean negative energy or a smaller value. It only tells you the direction of the induced emf relative to the current change.
• Inductance depends on geometry and core material, not on the current. Changing the current changes the stored energy and the induced emf, but not L itself.
• More turns help, but the effect is not linear. Inductance scales with N², so doubling the turns multiplies inductance by four, not two.
• An inductor does not block current forever. After current stops changing, an ideal inductor behaves like a plain wire with no induced emf.
• Stored energy lives in the magnetic field, not in the wire as charge. That energy can move to a resistor or capacitor, but it is always conserved.

Related AP Physics C: E&M Guides

• 13.2 Electromagnetic Induction
• 13.3 Induced Currents and Magnetic Forces
• 13.5 Circuits with Resistors and Inductors (LR Circuits)
• 13.6 Circuits with Capacitors and Inductors (LC Circuits)
• 13.1 Magnetic Flux