Inductance - AP Physics C: Electricity and Magnetism Study Guide

Getting Started

A current flowing through a wire generates a magnetic field. This field, in turn, creates a magnetic flux through the area enclosed by the circuit itself. What happens if we try to change the current? This action changes the magnetic flux, and according to Faraday's Law, a changing magnetic flux must induce an electromotive force (EMF), which will oppose the very change we are trying to make.

What You Should Be Able to Do

After completing this section, you will be able to:

• Calculate the self-inductance of an ideal solenoid from its geometric and material properties.

• Use the differential relationship between induced EMF and the rate of change of current to analyze the behavior of an inductor.

• Determine the energy stored in an inductor's magnetic field for a given current.

• Derive the governing equation for an inductor, $\mathcal{E}=-L\frac{dI}{dt}$, by applying Faraday's Law of Induction to a current-carrying coil.

Key Concepts & Mechanisms

System & Preconditions

The system under consideration is a current-carrying conductor, typically formed into a coil or solenoid to concentrate the magnetic flux. Our analysis relies on several idealizations:

• Ideal Conductor: The wire has zero electrical resistance. This allows us to isolate the effects of induction from resistive energy dissipation.

• Ideal Solenoid Geometry: For calculations, we assume a long, tightly wound solenoid where the magnetic field is uniform inside and negligible outside. Fringing fields at the ends are ignored.

• Quasi-Static Approximation: We assume the current changes slowly enough that the principles of magnetostatics can be applied at any given instant to determine the magnetic field.

Key Steps / Relations

The phenomenon of self-inductance is a direct consequence of Faraday's Law applied to the device itself. The causal chain proceeds as follows:

1. Current Creates a Field: A current, $I$, flowing through a conductor generates a magnetic field, $\vec{B}$, in the surrounding space. The strength of this field is directly proportional to the current. For an ideal solenoid, the field inside is given by $B=\mu nI$, where $n$ is the number of turns per unit length and $\mu$ is the magnetic permeability of the core material.

2. Field Creates a Flux: The magnetic field produces a magnetic flux, ${\Phi }_B=\int \vec{B}\cdot d\vec{A}$, through the area enclosed by the conductor's loops. For a coil with $N$ turns, the total magnetic flux linkage is $N{\Phi }_B$. Because $B\propto I$, the total flux linkage is also directly proportional to the current.

3. Defining Inductance: We define a new quantity, inductance (L), as the constant of proportionality between the total magnetic flux linkage and the current.

   $L=\frac{N{\Phi }_B}{I}$

   Inductance is a purely geometric property of the conductor, depending on its size, shape, number of turns, and the core material. Its SI unit is the Henry (H), where 1 H = 1 Tesla-meter-squared per Ampere (T·m²/A).

4. Faraday's Law Creates an EMF: According to Faraday's Law of Induction, any change in the magnetic flux over time induces an electromotive force (EMF), $\mathcal{E}$.

   $\mathcal{E}=-\frac{d(N{\Phi }_B)}{dt}$

5. The Inductor Equation: By substituting the definition of inductance ($N{\Phi }_B=LI$) into Faraday's Law, we arrive at the fundamental equation for an inductor. Assuming the inductor's geometry is fixed (so $L$ is constant):

   ${\mathcal{E}}_{induced}=-\frac{d(LI)}{dt}=-L\frac{dI}{dt}$

   This equation is the cornerstone of inductor behavior. It states that a changing current induces an EMF.

Outputs & Effects

• Self-Induced EMF: The primary effect is the creation of a self-induced EMF that opposes the change in current, a direct manifestation of Lenz's Law (indicated by the negative sign). If you try to increase the current ($dI/dt>0$), the inductor generates an opposing EMF. If you try to decrease the current ($dI/dt<0$), the inductor generates a supporting EMF to try to keep the current flowing. This property is often called "electrical inertia."

• Energy Storage: To establish a current in an inductor, the external circuit must do work against this back-EMF. This work is stored as potential energy in the inductor's magnetic field. The energy, $U_L$, stored in an inductor with inductance $L$ carrying a current $I$ is:

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

Regulation & Limits

The models presented are valid for ideal inductors. Real inductors possess internal resistance, which dissipates energy as heat, and parasitic capacitance between coils, which becomes significant at high frequencies. The formula for the inductance of a solenoid, $L=\frac{{\mu }_{core}{N}^{2}A}{\ell }$, is an approximation that is highly accurate for long solenoids ($\ell \gg \sqrt{A/\pi }$) but less so for short, wide coils due to edge effects.

Key Models & Diagrams

The causal relationships governing an inductor can be visualized as a process flowchart:

graph TD

    subgraph "Physical System & Properties"

        A[Current I(t)] --> B{Inductor Geometry & Core (L)};

    end


    subgraph "Causal Chain (Faraday's Law)"

        C(Rate of Change of Current<br>dI/dt) --> D(Changing Magnetic Field<br>dB/dt);

        D --> E(Changing Magnetic Flux<br>dΦ_B/dt);

        E --> F(Induced EMF<br>ε = -L dI/dt);

    end


    subgraph "Energy State"

        A --> G(Stored Magnetic Energy<br>U_L = 1/2 LI²);

    end


    C --> F;

Key Components & Evidence

Skill Snapshots

Causation

• Driver → Change: A non-zero rate of change of current ($dI/dt\ne 0$) through a coil → causes a time-varying magnetic flux ($d{\Phi }_B/dt\ne 0$) through that same coil.

• Driver → Change: A time-varying magnetic flux ($d{\Phi }_B/dt$) → causes a self-induced EMF ($\mathcal{E}$) across the coil, according to Faraday's Law.

• Driver → Change: An induced EMF ($\mathcal{E}=-LdI/dt$) → opposes the change in current, creating a potential difference that counteracts the external voltage source's attempt to change the current.

Comparison

• Inductor vs. Resistor: The potential difference across an inductor depends on the rate of change of current ($\Delta V=L\frac{dI}{dt}$), while for a resistor it depends on the instantaneous value of the current ($\Delta V=IR$).

• Inductor vs. Capacitor: An inductor stores energy in a magnetic field ($U_L=\frac{1}{2}L{I}^2$) and resists changes in current. A capacitor stores energy in an electric field ($U_C=\frac{Q^2}{2C}$) and resists changes in voltage.

• Inductance vs. Resistance: Inductance is a property based on geometry and magnetic effects that causes opposition to changes in current. Resistance is a property based on material and geometry that causes opposition to the flow of current itself.

Change Over Time

• Baseline: A constant, non-zero current $I_0$ flows through an ideal inductor. A constant magnetic field exists, and a constant energy $U_L=\frac{1}{2}L{I}_0^2$ is stored. The potential difference across the inductor is zero.

• Change: The external voltage is increased, causing the current to begin to rise ($dI/dt>0$). A back-EMF is immediately induced, opposing this increase.

• Change: The external voltage is removed, causing the current to begin to fall ($dI/dt<0$). A forward-EMF is immediately induced, attempting to sustain the current and prevent it from stopping abruptly.

• Continuity: The current through an inductor cannot change instantaneously. An instantaneous change would imply $dI/dt\to \infty$, which would require an infinite induced EMF—a physical impossibility.

Common Misconceptions & Clarifications

1. Misconception: "Inductors stop DC current."

   Clarification: Inductors only oppose a change in current. In a DC circuit, once the current reaches a steady state ($dI/dt=0$), an ideal inductor has zero voltage drop across it and behaves like a simple connecting wire (a short circuit). Its effect is only felt when the circuit is turned on or off.

2. Misconception: "The voltage across an inductor is always negative."

   Clarification: The negative sign in $\mathcal{E}=-L\frac{dI}{dt}$ represents Lenz's Law—the EMF opposes the change. The potential difference across the inductor terminals, $\Delta V_L$, is typically defined in the direction of current flow. Thus, $\Delta V_L=-\mathcal{E}=+L\frac{dI}{dt}$. If current is increasing, $\Delta V_L$ is positive (a voltage drop). If current is decreasing, $\Delta V_L$ is negative (a voltage gain).

3. Misconception: "Inductors store current or charge."

   Clarification: Inductors do not store charge; capacitors do. Inductors store energy in the magnetic field that is generated by the moving charges (the current). This energy is released back into the circuit when the current decreases.

One-Paragraph Summary

Inductance is the property of an electrical circuit that causes it to oppose any change in the current flowing through it. This "electrical inertia" arises from Faraday's Law of Induction: a changing current creates a changing magnetic field, which in turn induces a self-EMF that counteracts the original change. The magnitude of this effect is quantified by the inductance, $L$, a value determined by the physical geometry of the conductor, as seen in the solenoid equation $L=\mu N^{2}A/\ell$. The relationship between the induced EMF and the changing current is given by the fundamental differential equation $\mathcal{E}=-L\frac{dI}{dt}$. Consequently, establishing a current requires doing work against this back-EMF, with the energy being stored in the inductor's magnetic field according to $U_L=\frac{1}{2}L{I}^2$.