Inductance
Help Questions
AP Physics C: Electricity and Magnetism › Inductance
A solenoid with $N$ turns, length $\ell$, and cross-sectional area $A$ produces an approximately uniform internal field $B\approx \mu\frac{N}{\ell}I$, where $\mu=\mu_0\mu_r$ depends on the core material. The flux through each turn is $\Phi=BA$, so the flux linkage is $\lambda=N\Phi$. Using $\lambda=LI$, the solenoid inductance is $L\approx \mu\frac{N^2A}{\ell}$. Faraday’s law connects changing flux linkage to induced EMF: $\varepsilon=-\frac{d\lambda}{dt}=-L\frac{dI}{dt}$, consistent with Lenz’s law opposing current changes. Solenoids and inductors appear in relays and filters, where they resist rapid current variation. Explain the effect of changing current on the inductance of a solenoid.
Inductance increases automatically whenever current increases, regardless of core
Inductance is the same as resistance, so changing current changes both equally
Inductance stays set by geometry and core, while changing current changes induced EMF
Inductance occurs only between two coils, not within one solenoid
Explanation
This question tests AP Physics C understanding of inductance in electromagnetic systems, focusing on the relationship between inductance and changing current in solenoids. Inductance is a geometric property determined by the coil's physical characteristics (L ≈ μN²A/ℓ) that remains constant, while the induced EMF varies with the rate of current change according to Faraday's law. In the passage, the formula L ≈ μN²A/ℓ shows inductance depends on geometry and core material, while ε = -L(dI/dt) shows that changing current produces induced EMF without changing L itself. Choice A is correct because it accurately reflects that inductance is set by geometry and core properties, while the induced EMF changes with current rate of change. Choice B is incorrect because inductance is a fixed property for a given solenoid configuration, not dependent on current magnitude, a common error when students confuse the inductor's response (EMF) with its property (L). To help students: Emphasize that L is like a 'magnetic inertia' constant for the device, while ε = -L(dI/dt) shows the response to changing current. Encourage distinguishing between circuit properties (L, R, C) and circuit variables (I, V). Watch for: Confusing the constant inductance L with the variable induced EMF, or thinking inductance changes with current.
In an AC circuit, an inductor of $L=0.50,\text{H}$ opposes changing current by inducing an EMF. Inductance links current to magnetic flux as $\lambda=N\Phi=LI$, and Faraday’s law gives $\varepsilon=-\frac{d\lambda}{dt}=-L\frac{dI}{dt}$. The inductor stores energy in its magnetic field with $U=\tfrac12 LI^2$, and an ideal inductor does not dissipate energy as heat. Self-inductance refers to a coil’s changing current producing a changing flux through the same coil, while mutual inductance refers to a changing current in one coil producing a changing flux through a nearby coil. Real inductors appear in filters and power supplies, where they smooth current changes by resisting rapid variations. Based on the passage, how does an inductor store energy in a magnetic field?
By building magnetic-field energy while opposing changes in current
By storing energy in an electric field between its turns
By dissipating energy as heat whenever current changes
By storing energy only when current increases, not decreases
Explanation
This question tests AP Physics C understanding of inductance in electromagnetic systems, focusing on how inductors store energy in magnetic fields. Inductance refers to the property of a circuit element to oppose changes in current, storing energy in a magnetic field according to Faraday's law of electromagnetic induction. In the passage, the energy formula U = ½LI² and the statement that 'an ideal inductor does not dissipate energy as heat' illustrate how inductors store energy magnetically rather than dissipating it. Choice C is correct because it accurately reflects the role of inductance as explained in the passage, showing that inductors build magnetic-field energy while opposing current changes through the induced EMF ε = -L(dI/dt). Choice B is incorrect because the passage explicitly states ideal inductors do not dissipate energy as heat, a common error when students confuse inductors with resistors. To help students: Emphasize that inductors store energy in magnetic fields (U = ½LI²), not electric fields, and that this storage occurs during any current change. Encourage visualization of magnetic field lines building around the inductor as current increases. Watch for: Confusing magnetic field energy storage with electric field storage (capacitors) or energy dissipation (resistors).
A transformer transfers energy between two coils on the same iron core using mutual inductance. A time-varying current in the primary coil creates a changing magnetic flux in the core, and that flux links the secondary coil. Faraday’s law applies to each coil: $\varepsilon_p=-N_p,\frac{d\Phi}{dt}$ and $\varepsilon_s=-N_s,\frac{d\Phi}{dt}$. For an ideal transformer, the voltage ratio follows the turns ratio: $\frac{V_s}{V_p}=\frac{N_s}{N_p}$. Self-inductance describes a coil’s own changing current producing an opposing EMF, $\varepsilon=-L\frac{dI}{dt}$, while mutual inductance describes flux from one coil inducing EMF in the other. Transformers are used in power transmission to step voltages up or down efficiently. According to the text, what role does mutual inductance play in a transformer?
It affects only the primary coil because the secondary is passive
It transfers changing magnetic flux from primary to secondary to induce EMF
It makes induced EMF directly proportional to the circuit resistance
It stores energy in an electric field between the two coils
Explanation
This question tests AP Physics C understanding of inductance in electromagnetic systems, focusing on mutual inductance in transformers. Inductance refers to the property of a circuit element to oppose changes in current, with mutual inductance describing how changing current in one coil induces EMF in another through shared magnetic flux. In the passage, the transformer description shows how 'time-varying current in the primary coil creates a changing magnetic flux in the core, and that flux links the secondary coil,' illustrating mutual inductance's role in energy transfer. Choice A is correct because it accurately reflects the role of mutual inductance as explained in the passage, showing how changing flux from the primary induces EMF in the secondary according to Faraday's law. Choice C is incorrect because the passage clearly shows both coils are active participants through mutual inductance, a common error when students think of transformers as one-way devices. To help students: Emphasize the distinction between self-inductance (within one coil) and mutual inductance (between coils), and how shared magnetic flux enables energy transfer. Encourage drawing flux lines linking both coils to visualize the coupling. Watch for: Confusing mutual inductance with other energy storage mechanisms or thinking the secondary coil is passive.
A single conducting loop experiences a changing magnetic flux $\Phi_B(t)$ due to a nearby magnet moving. Faraday’s law states $\varepsilon=-\frac{d\Phi_B}{dt}$, and the negative sign encodes Lenz’s law: the induced current produces a magnetic field that opposes the change in flux. In a coil with $N$ turns, the relevant quantity is flux linkage $\lambda=N\Phi_B$, giving $\varepsilon=-\frac{d\lambda}{dt}=-N\frac{d\Phi_B}{dt}$. Self-inductance is a special case where the changing flux is produced by the circuit’s own changing current, leading to $\varepsilon=-L\frac{dI}{dt}$. These ideas explain back EMF in motors and the behavior of inductors in AC circuits. Based on the passage, how is induced EMF calculated in a loop?
Using $\varepsilon=-\frac{d\Phi_B}{dt}$, with sign given by Lenz’s law
Using $\varepsilon=IR$, because resistance sets the induced EMF
Using $\varepsilon=\frac{d\Phi_B}{dt}$, because EMF causes flux to change
Using $\varepsilon=\frac{1}{2}LI^2$, because EMF equals stored energy
Explanation
This question tests AP Physics C understanding of inductance in electromagnetic systems, focusing on calculating induced EMF using Faraday's law. Inductance principles stem from Faraday's law, which states that changing magnetic flux through a circuit induces an EMF that opposes the change according to Lenz's law. In the passage, the formula ε = -dΦB/dt explicitly shows how to calculate induced EMF, with the negative sign encoding Lenz's law that 'the induced current produces a magnetic field that opposes the change in flux.' Choice A is correct because it accurately reflects Faraday's law as stated in the passage, including the crucial negative sign that represents opposition to flux change. Choice C is incorrect because it has the wrong sign and reverses cause and effect - flux change causes EMF, not vice versa, a common error when students forget Lenz's law or confuse temporal sequence. To help students: Emphasize that the negative sign in Faraday's law is essential and represents nature's tendency to oppose change (Lenz's law). Encourage using the right-hand rule to verify the direction of induced effects. Watch for: Forgetting the negative sign, confusing EMF with energy or resistance effects, or reversing cause and effect.
Two nearby coils form a simple transformer. The primary coil carries an AC current, producing a changing magnetic flux in a shared core. The secondary coil experiences that changing flux, so Faraday’s law gives $\varepsilon_s=-N_s\frac{d\Phi}{dt}$. The coupling between coils is described by mutual inductance $M$, often written as $\varepsilon_s=-M\frac{dI_p}{dt}$ when the flux is produced by the primary current. Self-inductance in each coil still produces back EMF, $\varepsilon=-L\frac{dI}{dt}$, opposing changes in that coil’s current. Transformers in power adapters rely on this mutual induction to transfer energy efficiently. According to the text, what role does mutual inductance play in a transformer?
It occurs only within one coil and is the same as self-inductance
It links changing primary current to induced EMF in the secondary
It makes the secondary current independent of changing magnetic flux
It makes induced EMF proportional to resistance rather than flux change
Explanation
This question tests AP Physics C understanding of inductance in electromagnetic systems, focusing on mutual inductance between transformer coils. Inductance includes both self-inductance (within one coil) and mutual inductance (between coils), with mutual inductance enabling energy transfer through shared magnetic flux according to Faraday's law. In the passage, the relationship εs = -M(dIp/dt) shows how mutual inductance M links changing primary current to induced secondary EMF, with 'transformers in power adapters rely on this mutual induction to transfer energy efficiently.' Choice A is correct because it accurately reflects mutual inductance's role in linking primary current changes to secondary EMF through shared magnetic flux. Choice C is incorrect because the passage clearly distinguishes mutual inductance from self-inductance, showing they are different phenomena, a common error when students don't recognize the distinction between intra-coil and inter-coil effects. To help students: Emphasize that self-inductance acts within one coil (ε = -L dI/dt) while mutual inductance acts between coils (εs = -M dIp/dt). Encourage drawing separate flux paths for self and mutual effects. Watch for: Confusing mutual inductance with self-inductance or thinking they are the same phenomenon.
A conducting loop is placed in a region where the magnetic field changes with time, so the magnetic flux $\Phi_B(t)=\int \vec{B}\cdot d\vec{A}$ through the loop changes. Faraday’s law states that the induced EMF is $\varepsilon=-\frac{d\Phi_B}{dt}$, and Lenz’s law explains the sign: the induced current creates a magnetic field that opposes the flux change. In a coil with $N$ turns, $\varepsilon=-N\frac{d\Phi_B}{dt}$. Self-inductance is the case where the circuit’s own changing current changes the flux, giving $\varepsilon=-L\frac{dI}{dt}$. These principles explain generator operation and back EMF in motors. Based on the passage, how is induced EMF calculated in a loop?
By $\varepsilon=\frac{d\Phi_B}{dt}$, because EMF drives flux changes
By $\varepsilon=\tfrac12 LI^2$, because EMF equals magnetic energy
By $\varepsilon=-\frac{d\Phi_B}{dt}$, opposing the change in magnetic flux
By $\varepsilon=IR$, because induced EMF is set by resistance
Explanation
This question tests AP Physics C understanding of inductance in electromagnetic systems, focusing on applying Faraday's law to calculate induced EMF. Inductance principles originate from Faraday's law of electromagnetic induction, which quantifies how changing magnetic flux induces an EMF that opposes the change according to Lenz's law. In the passage, the fundamental equation ε = -dΦB/dt appears with clear explanation that 'Lenz's law explains the sign: the induced current creates a magnetic field that opposes the flux change,' showing both the calculation method and physical meaning. Choice A is correct because it accurately states Faraday's law with the proper negative sign indicating opposition to flux change, as emphasized throughout the passage. Choice B is incorrect because it omits the crucial negative sign that encodes Lenz's law, a common error when students focus on magnitude without considering the opposition principle. To help students: Emphasize that Faraday's law always includes the negative sign - it's not optional but represents a fundamental principle of nature opposing change. Encourage always checking induced current direction using Lenz's law. Watch for: Omitting the negative sign, confusing induced EMF with other electrical quantities, or not understanding the physical meaning of the opposition.
A transformer has primary turns $N_p$ and secondary turns $N_s$ on a shared core. The same time-varying core flux $\Phi(t)$ links both coils, so Faraday’s law gives $V_p\approx N_p\left|\frac{d\Phi}{dt}\right|$ and $V_s\approx N_s\left|\frac{d\Phi}{dt}\right|$. For an ideal transformer, $\frac{V_s}{V_p}=\frac{N_s}{N_p}$. Self-inductance still acts in each coil as $\varepsilon=-L\frac{dI}{dt}$, opposing abrupt current changes, while mutual inductance enables energy transfer between coils. Transformers are used to match voltages in power distribution and electronics. Based on the passage, what is the significance of the turns ratio in a transformer?
It removes mutual inductance, so coils cannot exchange energy
It makes flux change rate depend only on wire resistance
It ensures inductance is identical in AC and DC for all circuits
It sets the voltage ratio between secondary and primary in an ideal transformer
Explanation
This question tests AP Physics C understanding of inductance in electromagnetic systems, focusing on the turns ratio's role in transformer voltage transformation. Inductance principles, particularly mutual inductance through shared flux, enable transformers to change voltages according to the ratio of primary to secondary turns. In the passage, the derivation showing Vp ≈ Np|dΦ/dt| and Vs ≈ Ns|dΦ/dt| leads directly to Vs/Vp = Ns/Np, demonstrating how turns ratio determines voltage ratio for 'match voltages in power distribution and electronics.' Choice A is correct because it accurately reflects the fundamental transformer equation, showing the turns ratio sets the voltage transformation ratio. Choice B is incorrect because the passage shows flux change rate cancels out in the voltage ratio, depending only on turns, a common error when students think resistance affects ideal transformer ratios. To help students: Emphasize that the same flux links both coils, so dΦ/dt cancels when taking the ratio, leaving only the turns ratio. Encourage working through the math to see this cancellation explicitly. Watch for: Thinking resistance or other factors affect the ideal transformer voltage ratio, or confusing turns ratio with current ratio.
In an ideal transformer, two coils share the same changing core flux $\Phi(t)$. The induced EMFs follow Faraday’s law: $\varepsilon_p=-N_p,\frac{d\Phi}{dt}$ and $\varepsilon_s=-N_s,\frac{d\Phi}{dt}$. This leads to the turns-ratio relation $\frac{V_s}{V_p}=\frac{N_s}{N_p}$, which engineers use to step voltage up or down. Self-inductance, $\varepsilon=-L\frac{dI}{dt}$, describes how each coil opposes changes in its own current, while mutual inductance explains how the primary’s changing current produces a changing flux that links the secondary. Transformers appear in chargers and power grids because they transfer energy efficiently without moving parts. Based on the passage, what is the significance of the turns ratio in a transformer?
It determines the electric-field energy stored between windings
It makes magnetic flux depend only on the load resistance
It sets the voltage ratio between coils in an ideal transformer
It eliminates self-inductance so current changes instantly
Explanation
This question tests AP Physics C understanding of inductance in electromagnetic systems, focusing on the turns ratio relationship in ideal transformers. Inductance principles, specifically mutual inductance through shared magnetic flux, enable transformers to transfer energy between coils with voltage transformation determined by the turns ratio. In the passage, the relationship Vs/Vp = Ns/Np directly shows how the turns ratio determines voltage transformation, with 'engineers use to step voltage up or down' illustrating its practical significance. Choice B is correct because it accurately reflects the role of the turns ratio as explained in the passage, showing it sets the voltage ratio between coils in an ideal transformer. Choice A is incorrect because the passage shows voltage ratio depends on turns ratio, not load resistance, a common error when students confuse transformer operation with resistive circuits. To help students: Emphasize that the turns ratio is a geometric property that determines voltage transformation through Faraday's law applied to both coils. Encourage working through the derivation from εp = -Np(dΦ/dt) and εs = -Ns(dΦ/dt) to see why the ratio emerges. Watch for: Confusing the turns ratio effect with resistance-based voltage division or energy storage mechanisms.
An inductor in a circuit is characterized by inductance $L$, defined by flux linkage $\lambda=N\Phi=LI$. When the current changes, Faraday’s law gives a self-induced EMF $\varepsilon=-\frac{d\lambda}{dt}=-L\frac{dI}{dt}$ that opposes the change. This opposition explains why inductors resist sudden current changes in both DC transients and AC circuits. The energy stored in the magnetic field is $U=\tfrac12 LI^2$, which can be returned to the circuit later. Inductors appear in speaker crossovers and switching power supplies to shape current and voltage waveforms. Based on the passage, how does an inductor store energy in a magnetic field?
By forcing current to change instantly, so no energy is stored
By opposing changes in current while energy accumulates in the magnetic field
By storing energy in an electric field like a capacitor does
By dissipating energy as heat whenever flux changes through the coil
Explanation
This question tests AP Physics C understanding of inductance in electromagnetic systems, focusing on the fundamental mechanism of magnetic energy storage. Inductance enables energy storage through the buildup of magnetic fields when current flows, with the inductor opposing current changes while accumulating energy according to U = ½LI². In the passage, the self-induced EMF ε = -L(dI/dt) shows how inductors oppose current changes, while U = ½LI² reveals energy storage in the magnetic field that 'can be returned to the circuit later.' Choice A is correct because it accurately captures both aspects: opposition to current change (through back EMF) and energy accumulation in the magnetic field. Choice B is incorrect because inductors store energy in magnetic fields, not electric fields like capacitors do, a common error when students confuse the two energy storage mechanisms. To help students: Emphasize the connection between opposing current change (Lenz's law) and building magnetic energy - they are two aspects of the same phenomenon. Encourage visualizing magnetic field lines strengthening as current increases. Watch for: Confusing magnetic and electric field energy storage, or missing the connection between EMF opposition and energy storage.
A long air-core solenoid has $N=800$ turns, length $\ell=0.40,\text{m}$, and cross-sectional area $A=2.0\times10^{-3},\text{m}^2$. Inside, $B\approx \mu_0\dfrac{N}{\ell}I$, so flux is $\Phi_B=BA$ and self-inductance follows $L=\dfrac{N\Phi_B}{I}\approx \mu_0\dfrac{N^2A}{\ell}$. Faraday’s law gives $\mathcal{E}=-L,\dfrac{dI}{dt}$. Based on the passage, explain the effect of changing current on the inductance of a solenoid.
Changing current changes $L$ because $L$ is proportional to $I$.
Changing current changes $L$ only when current increases.
Changing current changes $L$ because $L$ equals the circuit resistance.
Changing current does not change $L$ for fixed geometry and core.
Explanation
This question tests AP Physics C understanding of inductance in electromagnetic systems, focusing on the relationship between inductance and current in a solenoid. Inductance is a geometric property determined by the physical configuration of a conductor, specifically L≈μ₀(N²A/ℓ) for a solenoid, independent of the current flowing through it. In the passage, the air-core solenoid formula shows L depends only on turns N, area A, length ℓ, and permeability μ₀, not on current I. Choice B is correct because inductance is a fixed property for given geometry and core material - changing current changes the magnetic field strength and stored energy, but not the inductance itself. Choice A is incorrect because it confuses the induced EMF (which depends on dI/dt) with inductance L, a common error when students misinterpret ε=-L(dI/dt). To help students: Emphasize that L is like capacitance C - a geometric property. Use the analogy that changing water flow doesn't change pipe diameter.