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Physics

Inductance and RL Circuits

Self-Inductance, Back-EMF, and Transient Behavior in RL Circuits — A TLDR Primer

Inductance shows up on the AP Physics C exam and in every intro college E&M course — and it consistently trips students up. The math looks manageable, but the concepts underneath (why does a coil resist changing current? where does the energy actually go?) rarely get explained clearly in a textbook chapter buried between Maxwell's equations and AC circuits.

**TLDR: Inductance and RL Circuits** covers exactly what you need and nothing you don't. Concise and comprehensive, you'll move from the definition of self-inductance through the behavior of series RL circuits — both the build-up and decay of current — and finish with a clear account of how energy is stored in a magnetic field. Every key result is derived step by step, with worked examples and the common misconceptions called out by name.

This guide is written for students preparing for the AP Physics C: Electricity & Magnetism exam, anyone in a first- or second-semester college physics course hitting inductance for the first time, and tutors or parents who want a clean reference before a study session. If you've searched for a no-fluff intro college physics electromagnetics primer that gets to the point, this is it.

It's short by design. You don't need bloated textbooks to understand RL circuits — you need the right material, to the point. Pick it up, work through it in one sitting, and walk into your exam knowing the material.

What you'll learn
  • Explain what inductance is and where it comes from physically
  • Compute self-inductance for simple geometries like a solenoid
  • Apply Kirchhoff's voltage law to RL circuits and solve the resulting differential equation
  • Predict current and voltage as a function of time when an RL circuit is energized or de-energized
  • Calculate energy stored in an inductor and energy density in a magnetic field
What's inside
  1. 1. What Is Inductance?
    Introduces self-inductance as a circuit element's resistance to changes in current, grounded in Faraday's law.
  2. 2. Calculating Inductance: Solenoids and Other Geometries
    Derives the inductance of a long solenoid from first principles and surveys toroids and coaxial cables.
  3. 3. The RL Circuit: Current Build-Up
    Analyzes a series RL circuit connected to a battery, deriving the exponential approach to steady-state current.
  4. 4. RL Circuits: Current Decay and Switching
    Treats the de-energizing case when the battery is removed and discusses what happens when switches open suddenly.
  5. 5. Energy in Inductors and Magnetic Fields
    Shows that work is required to establish a current through an inductor and locates that energy in the magnetic field itself.
  6. 6. Why It Matters: From Power Supplies to Wireless Charging
    Connects inductance to real devices students encounter and previews LC oscillations and AC circuits.
Published by Solid State Press
Inductance and RL Circuits cover
TLDR STUDY GUIDES

Inductance and RL Circuits

Self-Inductance, Back-EMF, and Transient Behavior in RL Circuits — A TLDR Primer
Solid State Press

Inductance and RL Circuits

Self-Inductance, Back-EMF, and Transient Behavior in RL Circuits — A TLDR Primer

Copyright © 2026 Solid State Press.

Published by Solid State Press.

All rights reserved. No part of this book may be reproduced or transmitted in any form without prior written permission, except for brief quotations in critical reviews and educational use.

Part of the TLDR Study Guides series.

Contents

  1. 1 What Is Inductance?
  2. 2 Calculating Inductance: Solenoids and Other Geometries
  3. 3 The RL Circuit: Current Build-Up
  4. 4 RL Circuits: Current Decay and Switching
  5. 5 Energy in Inductors and Magnetic Fields
  6. 6 Why It Matters: From Power Supplies to Wireless Charging
Chapter 1

What Is Inductance?

Every circuit element resists something. A resistor resists current itself. A capacitor resists changes in voltage. An inductor — any loop or coil of wire — resists changes in current. That resistance to change is measured by a quantity called self-inductance, usually just called inductance, given the symbol $L$.

To understand where inductance comes from, you need two ideas you have already met: magnetic flux and Faraday's law.

Magnetic flux $\Phi_B$ through a surface is the total amount of magnetic field passing through it:

$\Phi_B = \int \vec{B} \cdot d\vec{A}$

For a flat loop of area $A$ in a uniform field $B$ perpendicular to the loop, this simplifies to $\Phi_B = BA$. Think of flux as counting field lines — more field, bigger loop, more flux.

Faraday's law says that whenever the magnetic flux through a loop changes with time, an electromotive force (EMF) — a voltage — is induced in that loop:

$\mathcal{E} = -\frac{d\Phi_B}{dt}$

The minus sign is not a bookkeeping accident. It encodes Lenz's law: the induced EMF always opposes the change that caused it. If flux is increasing, the induced EMF drives a current that tries to reduce that flux. If flux is decreasing, the induced EMF drives a current that tries to maintain it. Nature pushes back.

The loop that fights itself

Here is the key insight: a current-carrying wire creates its own magnetic field. That field passes through the wire's own loop, creating flux through the very loop that made the field. When the current changes, that self-generated flux changes, and by Faraday's law, an EMF is induced in the loop itself.

This self-induced EMF is called the back-EMF (or counter-EMF). It opposes whatever change in current is happening. If you try to increase the current, the back-EMF pushes back against the increase. If you try to decrease the current, the back-EMF tries to keep it flowing. An inductor has memory — it does not let current change instantaneously.

For a coil with $N$ turns, the total flux linkage is $N\Phi_B$ (each turn contributes its own flux). The self-inductance $L$ is defined as the ratio of total flux linkage to current:

$L = \frac{N\Phi_B}{I}$

This tells you how much flux the coil generates per ampere of current. A larger $L$ means the coil is better at linking flux with current — a stronger electromagnetic "memory."

Combining this definition with Faraday's law gives the voltage across an inductor:

About This Book

If you're staring down the AP Physics C Electromagnetism exam, working through an intro college physics electromagnetics primer, or just trying to make sense of inductance and magnetic fields in high school, this book was written for you. It also works for tutors who need a fast refresh before a session and parents helping a student through a tricky unit.

This guide covers the full arc of the topic: self-inductance, solenoid inductance with practice problems, the math behind current build-up in an RL circuit, exponential current decay in an RL circuit, and a focused energy stored in an inductor quick review. Every concept is shown with worked numbers, not just formulas. A concise overview with no filler.

Read it straight through the first time — each section builds on the last. When you hit a worked example, cover the solution and try it yourself. Then use the problem set at the end to confirm you've actually got it. Think of this as RL circuits explained for beginners who intend to finish the exam with time to spare.

Keep reading

You've read the first half of Chapter 1. The complete book covers 6 chapters in roughly fifteen pages — readable in one sitting.

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