#LR Circuits: Your Night-Before-the-Exam Guide

Hey there! Let's get you prepped for the AP Physics C: E&M exam with a focused review of LR circuits. We'll break it down, keep it clear, and make sure you're feeling confident. Let's dive in!

#Introduction to LR Circuits

LR circuits are all about the interplay between resistors and inductors. They're super important because they show how energy is stored and released in a circuit over time. Think of it as a dance between resistance and inductance, where the music is the changing current. Let's explore the key concepts:

• Energy Dissipation: Resistors turn electrical energy into heat. 🔥
• Time-Dependent Behavior: Inductors resist changes in current, leading to time-dependent effects.
• Kirchhoff's Loop Rule: This rule helps us analyze how voltage is distributed in the circuit.
• Time Constant: This tells us how quickly the circuit reaches a stable state. 🕰️

#Resistor-Inductor Circuit Properties

#Energy Dissipation in Resistors

• Heat Conversion: As current changes in an LR circuit, resistors convert the energy stored in the inductor into heat. Think of it like friction slowing things down.
• Energy Loss: This energy dissipation leads to a decrease in the inductor's stored energy over time. The inductor's energy is being used to overcome the resistance.

#Kirchhoff's Loop Rule for LR Circuits

• Loop Analysis: Applying Kirchhoff's loop rule to a series LR circuit with a battery (emf $\mathcal{E}$) gives us a differential equation that describes the current in the loop.
• The Equation: The key equation is: $$\mathcal{E} = IR + L \frac{dI}{dt}$$
  • $IR$ is the voltage drop across the resistor.
  • $L \frac{dI}{dt}$ is the voltage drop across the inductor, which depends on the rate of change of current.

#Time Constant in LR Circuits

• Definition: The time constant ($\tau$) tells us how quickly an LR circuit reaches its steady state. It's like a circuit's "response time."
• Calculation: $$\tau = \frac{L}{R_{eq}}$$, where $L$ is the inductance and $R_{eq}$ is the equivalent resistance.
• Interpretation:
  • If the inductor starts with zero current, $\tau$ is the time it takes for the current to reach about 63% of its final value.
  • If the inductor starts with an initial current, $\tau$ is the time it takes for the current to decrease to about 37% of its initial value.

#Steady State Behavior of Inductors

• Dynamic to Static: Inductors have time-dependent electrical properties when the current is changing, but they reach a steady state after a long time.
• Switch Action: When you close or open a switch in an LR circuit, the induced emf in the inductor equals the applied voltage but acts in the opposite direction. This is Lenz's Law in action.
• Exponential Curves: The potential difference, current, and stored energy in an inductor follow exponential curves with time. 📈
• Long-Term Behavior: After a long time (much greater than $\tau$), an inductor acts like a simple wire with zero resistance. Current flows freely.

#Transient vs. Steady State in LR Circuits

• Transient State: This is the initial period after a change (like flipping a switch). During this time, current and voltage are changing rapidly.
  • The inductor's properties (induced emf, stored energy) are time-dependent and follow exponential functions.
• Steady State: This occurs after a long time (much greater than $\tau$). Current and voltage have stabilized and no longer change.
  • The inductor acts like a wire with no resistance. The current is determined by the battery's emf and the total resistance. 🔌

#Final Exam Focus

Okay, let's get down to the nitty-gritty. Here’s what you should focus on for the exam:

• High-Priority Topics:
  • Time Constant ($\tau$): Know how to calculate it and what it means for the circuit's behavior.
  • Kirchhoff's Loop Rule: Be able to apply it to LR circuits and set up the differential equation.
  • Transient and Steady-State Analysis: Understand how the inductor behaves in both states.
  • Energy in Inductors: Understand how energy is stored and dissipated in LR circuits.
• Common Question Types:
  • Multiple Choice: Expect questions on calculating $\tau$, interpreting graphs of current vs. time, and understanding the behavior of inductors in different states.
  • Free Response: Be prepared to analyze a circuit, derive equations, and interpret the results.
• Last-Minute Tips:
  • Time Management: Don't get bogged down on one question. Move on if you're stuck and come back later.
  • Common Pitfalls: Watch out for sign errors in Kirchhoff's loop rule and ensure you're using the correct time constant formula.
  • Strategies: Draw clear diagrams of the circuits, label all variables, and show all your work.

#

Practice Question

Practice Questions

#Multiple Choice Questions

1. An LR circuit has a time constant $\tau$. If the inductance is doubled and the resistance is halved, what is the new time constant? (A) $\frac{\tau}{4}$ (B) $\frac{\tau}{2}$ (C) $2\tau$ (D) $4\tau$

2. In an LR circuit, what happens to the current in the circuit a very long time after the switch is closed? (A) It becomes zero. (B) It reaches a maximum value determined by the battery and the resistance. (C) It oscillates indefinitely. (D) It decreases exponentially.

3. The current in an LR circuit is initially zero. At time $t = \tau$, the current has reached approximately what percentage of its maximum value? (A) 37% (B) 50% (C) 63% (D) 100%

#Free Response Question

Consider the circuit below, which contains a battery with emf $\mathcal{E}$, a resistor with resistance $R$, and an inductor with inductance $L$. The switch is initially open. At time $t = 0$, the switch is closed.

(a) Using Kirchhoff's loop rule, write a differential equation for the current $I$ in the circuit as a function of time $t$.

(b) Solve the differential equation to find an expression for $I(t)$.

(c) What is the current in the circuit a long time after the switch is closed?

(d) What is the time constant of the circuit?

(e) Sketch a graph of current $I$ as a function of time $t$.

#Scoring Breakdown

(a) (3 points) - 1 point for correctly applying Kirchhoff's loop rule. - 1 point for correctly identifying the voltage drop across the resistor as $IR$. - 1 point for correctly identifying the voltage drop across the inductor as $L \frac{dI}{dt}$. - Correct equation: $\mathcal{E} = IR + L \frac{dI}{dt}$

(b) (4 points) - 1 point for separating variables. - 1 point for correctly integrating both sides. - 1 point for applying initial conditions ($I(0) = 0$). - 1 point for the correct solution: $I(t) = \frac{\mathcal{E}}{R} (1 - e^{-t/\tau})$, where $\tau = L/R$

(c) (2 points) - 1 point for understanding that as $t \rightarrow \infty$, $e^{-t/\tau} \rightarrow 0$. - 1 point for the correct answer: $I_{max} = \frac{\mathcal{E}}{R}$

(d) (1 point) - 1 point for the correct time constant: $\tau = L/R$

(e) (2 points) - 1 point for a graph that starts at 0 and increases towards $I_{max}$. - 1 point for a graph that approaches $I_{max}$ asymptotically.

That's it! You've got this. Go ace that exam!