#AP Physics C: E&M - Inductors: The Ultimate Study Guide ⚡

Hey there, future physicist! Let's dive into inductors and circuits, making sure you're totally prepped for the AP exam. This guide is designed to be your go-to resource, especially the night before the test. Let's make this click!

#What is an Inductor? 🤔

An inductor is essentially a coil of wire, often wrapped around a core, that resists changes in current. Think of it as a current smoother. It stores energy in a magnetic field, kind of like how a capacitor stores energy in an electric field. 💡

Image from wikipedia.org - Various types of inductors

We define inductance (L) as the proportionality constant between magnetic flux and current:

$\Phi_B = LI$

Where:

$\Phi_B$ is the magnetic flux
L is the inductance (measured in Henries (H))
I is the current

Key Concept

Inductance L depends on the coil's geometry (number of loops, wire gauge) and the core material. More loops = more inductance! A larger core also increases inductance.

Applying Faraday's Law, we find the induced EMF in an inductor:

$\varepsilon = -L \frac{dI}{dt}$

This induced EMF opposes any change in current. It's like the inductor is saying, "Hey, hold on! Don't change the current so fast!" This is Lenz's Law in action.

#Energy Stored in an Inductor 🔋

Consider a simple circuit with a battery, resistor, and inductor:

When the switch closes, current flows, and the inductor stores energy in its magnetic field. Using Kirchhoff's Voltage Law, we can derive the energy stored:

$U = \frac{1}{2}LI^2$

Quick Fact

This formula is analogous to the energy stored in a capacitor ( $\frac{1}{2}CV^2$ ). Inductors store energy in the magnetic field, capacitors in the electric field.

#LR Circuit Behavior 🔍

Now, let's look at the current and voltage behavior in a circuit with both a resistor and an inductor (LR circuit):

The current in an LR circuit changes exponentially, described by:

$I(t) = \frac{\varepsilon}{R}(1 - e^{-t/\tau})$

Where the time constant is:

$\tau = \frac{L}{R}$

Memory Aid

Remember, the time constant for an LR circuit is L over R ( $\tau = L/R$ ). Think 'L' comes before 'R' in the alphabet!

The voltage across the inductor is given by:

$V_L(t) = \varepsilon e^{-t/\tau}$

Here's what the graphs look like:

Image from electronics-tutorials.ws - Current and voltage behavior in LR circuit

Exam Tip

Initially, an inductor acts like an open circuit (resists current flow), and over time, it acts like a wire (allowing current to flow freely). This is crucial for analyzing circuit behavior!

#LC Circuit Behavior 🧲📸

Now, let's explore a circuit with only an inductor and a capacitor (LC circuit):

Image from phys.libretexts.org - LC circuit oscillations

In an LC circuit, energy oscillates between the capacitor's electric field and the inductor's magnetic field. This leads to sinusoidal oscillations in charge and current.

LC circuits are a classic example of simple harmonic motion in electrical circuits. This connection is a frequent topic on the exam.

The total energy in the circuit remains constant (assuming no resistance):

$U_{total} = U_C + U_L = \frac{Q^2}{2C} + \frac{1}{2}LI^2$

The charge and current vary sinusoidally with time:

$Q(t) = Q_{max} \cos(\omega t)$

$I(t) = -\omega Q_{max} \sin(\omega t)$

Where the angular frequency is:

$\omega = \frac{1}{\sqrt{LC}}$

Memory Aid

Remember the angular frequency of an LC circuit: $\omega = \frac{1}{\sqrt{LC}}$ . It's like the reciprocal of the square root of 'LC', which is the opposite of the time constant for LR circuits.

#Final Exam Focus 🎯

Alright, let's get down to the nitty-gritty. Here's what you absolutely need to nail for the exam:

Inductor Basics: Understand what an inductor is, how it stores energy, and how it resists changes in current. [See Inductor Definition]
Inductance (L): Be able to calculate inductance and understand how it is affected by the geometry of the coil and the core material. [See Inductor Definition]
Energy in Inductors: Know the formula for energy stored in an inductor and how it relates to the current. [See Inductor Energy]
LR Circuits: Understand the time constant, and how current and voltage change over time. [See LR Circuits]
LC Circuits: Be familiar with the oscillations in charge and current and the angular frequency. [See LC Circuits]
Circuit Analysis: Practice applying Kirchhoff's laws to analyze circuits with inductors, capacitors, and resistors.

Exam Tip

Pay close attention to the initial and final states of circuits. Inductors act like open circuits initially and like wires after a long time. Use this to simplify the circuit analysis.

Common Question Types:

Conceptual Questions: Understanding the behavior of inductors in circuits.
Circuit Analysis Problems: Calculating currents, voltages, and energy in LR and LC circuits.
Graphing: Interpreting and sketching graphs of current and voltage vs. time.
Differential Equations: Understanding the relationship between the differential equations and the solutions for LC circuits.

Last-Minute Tips:

Time Management: Don't spend too much time on a single question. Move on and come back if you have time.
Units: Always include units in your calculations and answers.
Draw Diagrams: Drawing circuit diagrams can help you visualize the problem and avoid mistakes.
Review Formulas: Make sure you know all the key formulas and relationships.
Stay Calm: Take deep breaths and believe in yourself. You've got this!

#Practice Problems ✅

Let's try some practice problems to solidify your understanding.

Practice Question

Multiple Choice Questions

An inductor with inductance L carries a current I. If the current is doubled, the energy stored in the inductor is: (A) Halved (B) Doubled (C) Quadrupled (D) Remains the same
In an LR circuit, the time constant is given by: (A) R/L (B) L/R (C) LR (D) 1/(LR)
In an LC circuit, the angular frequency of oscillation is given by: (A) \sqrt{LC} (B) 1/\sqrt{LC} (C) LC (D) 1/(LC)

Free Response Question

Consider the circuit below, where A and B are terminals to which different circuit components can be connected.

(a) Calculate the potential difference across R2 immediately after the switch S is closed in each of the following cases.

i. A 50Ω resistor connects A and B.
ii. A 40mH inductor connects A and B.
iii. An initially uncharged 0.80μF capacitor connects A and B.

(b) The switch gets closed at time t = 0. On the axes below, sketch the graphs of the current in the 100Ω resistor R3 versus time t for the three cases. Label the graphs R for the resistor, L for the inductor, and C for the capacitor.

Answers:

Multiple Choice Answers

(C) Quadrupled (Since $U = \frac{1}{2}LI^2$ , doubling I quadruples U)
(B) L/R
(B) 1/\sqrt{LC}

Free Response Answers

(a)

i. When a 50Ω resistor is connected, the equivalent resistance is 100Ω + 50Ω = 150Ω. The current through the circuit is $I = \frac{V}{R} = \frac{12}{150} = 0.08A$ . The voltage across R2 is $V = IR = 0.08 * 50 = 4V$ (2 points)

ii. An inductor acts like an open switch immediately after the switch is closed, so no current flows through the far right branch. Treat this like a series circuit. The equivalent resistance is 100 + 100 = 200. The current through the circuit is $I = \frac{V}{R} = \frac{12}{200} = 0.06A$ . The voltage across R2 is $V = IR = 0.06 * 100 = 6V$ (2 points)

iii. The voltage drop across the capacitor is 0 immediately after the switch is closed so we can ignore it. The equivalent resistance is 100 + 50 = 150. The current through the circuit is $I = \frac{V}{R} = \frac{12}{150} = 0.08A$ . The voltage across R2 is $V = IR = 0.08 * 50 = 4V$ (2 points)

(b)

The resistor (R) will have a constant current. (1 point)
The capacitor (C) will start with a very high current, then exponentially decrease. (1 point)
The inductor (L) will begin allowing no current through, then level off at some max current. (1 point)

Good luck, you've got this! Let me know if you need more help!