# Inductance

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## Video Clips

#### Self Inductance

RealVideo®
5:28 minutes (0:00 - 5:28)

Definition, including calculation of the self inductance L for a solenoid and mention of the unit Henry for self inductance.

Prior Knowledge: None
Instructor: Prof. Walter Lewin

#### LR Circuits

RealVideo®
10:47 minutes (5:28 - 16:15)

Using Faraday's Law to find the equation for current as a function of time in a circuit containing inductors and resistors.

Prior Knowledge: Self Inductance (Beginning of video lecture 20)
Instructor: Prof. Walter Lewin

#### LR Circuits with Battery Removed

RealVideo®
9:47 minutes (16:15 - 26:02)

Finding the equation for current in an LR circuit when the battery is removed, as well as equations for power and energy in an inductor.

Prior Knowledge: LR Circuits (5:28 of video lecture 20)
Instructor: Prof. Walter Lewin

#### Demonstration: LR Circuit

RealVideo®
4:28 minutes (26:02 - 30:30)

Showing that a self-inductor will oppose the buildup of current using a circuit of light bulbs and a large inductor.

Prior Knowledge: LR Circuits (5:28 of video lecture 20)
Instructor: Prof. Walter Lewin

#### LR Circuits with Alternating Current

RealVideo®
5:22 minutes (30:30 - 35:52)

Using Faraday's Law to derive the equation for current in an LR circuit with an AC power supply, including the phase shift.

Prior Knowledge: LR Circuits (5:28 of video lecture 20)
Instructor: Prof. Walter Lewin

#### Demonstration: Reduction of High Frequencies

RealVideo®
4:08 minutes (35:52 - 40:00)

Hear the current reduction of high frequency audio due to a self-inductor. Cannot hear phase shift.

Prior Knowledge: LR Circuits with AC power (30:30 of video lecture 20)
Instructor: Prof. Walter Lewin

#### Phase Shift in Magnetic Levitation

RealVideo®
6:32 minutes (40:00 - 46:32)

How the phase lag of the current due to self-inductance in a conductor makes magnetic levitation using an AC coil possible.

Prior Knowledge: Magnetic Levitation with Alternating Current (42:04 of video lecture 19) and LR Circuits with AC power (30:30 of video lecture 20)
Instructor: Prof. Walter Lewin

#### Demonstration: Magnetic Levitation with Alternating Current

RealVideo®
5:22 minutes (46:32 - 51:54)

Levitating a conducting ring above a coil powered by an AC power supply.

Prior Knowledge: Phase Shift in Magnetic Levitation (40:00 of video lecture 20)
Instructor: Prof. Walter Lewin

## Lecture Notes

#### Self-Inductance

PDF#
Page 5 to page 8

Induced EMF and back EMF in solenoid; self-inductance definition and calculation; energy storage in inductors.

Prior Knowledge: AC Current
Instructor: Prof. Gabriella Sciolla

#### Mutual Inductance

PDF#
Page 9 to page 11

Definition; reciprocity theorem; transformers.

Prior Knowledge: Self-inductance
Instructor: Prof. Gabriella Sciolla

#### RL Circuits

PDF
Page 2 to page 4

Intuitive RL circuit description; differential equation and solution for charging and discharging; time constant.

Prior Knowledge: Self-inductance
Instructor: Prof. Gabriella Sciolla

#### LC Circuits

PDF
Page 5 to page 6

Oscillation in LC circuits; differential equation and solution; energy conservation.

Prior Knowledge: Self-inductance
Instructor: Prof. Gabriella Sciolla

## Online Textbook Chapters

#### Mutual Inductance

PDF - 1.0 MB
Page 3 to page 5

Introduction of the phenomenon of mutual inductance, whereby the magnetic field produced by one coil induces a current in a nearby coil. Example of two concentric coplanar loops.

Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao

#### Inductors and Self Inductance

PDF - 1.0 MB
Page 5 to page 9

Introduces self inductance with examples of a solenoid and a toroid; relates mutual inductance to self inductance.

Prior Knowledge: Mutual Inductance (OT11.1)
Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao

#### Energy of the Magnetic Field

PDF - 1.0 MB
Page 9 to page 15

Derives the energy stored in an inductor, with an example of an ideal solenoid, and introduces energy density of the magnetic field.

Prior Knowledge: Inductance (OT11.2)
Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao

#### RL Circuits

PDF - 1.0 MB
Page 15 to page 21

Modifies Kirchhoff's Voltage Rule to include the induced EMF due to an inductor, and uses this rule to analyze rising and decaying currents in an RL circuit.

Prior Knowledge: Kirchhoff's Loop Rule (OT7.4), Inductors (OT11.2)
Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao

#### LC Circuits

PDF - 1.0 MB
Page 21 to page 24

Description of current oscillations in circuits with inductors and capacitors; includes analogy to harmonic oscillations in a mass-spring system.

Prior Knowledge: Storing Energy in Electric (OT5.4) and Magnetic (OT11.3) Fields
Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao

#### Problem Solving Strategy: Self Inductance

PDF - 1.0 MB
Page 38 to page 39

Enumerated procedure for calculating the self inductance of a loop.

Prior Knowledge: Self-inductance (OT11.2)
Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao

#### Problem Solving Strategy: Circuits with Inductors

PDF - 1.0 MB
Page 39

Summary of Kirchoff's Voltage Law for circuits that contain inductors.

Prior Knowledge: Inductors (OT11.2)
Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao

## Practice Problems

#### Charge in an LC Circuit

PDF
Problem on page 40 to page 43

Write down equation for charge in an LC circuit, and find the time and stored energies when energy stored in the capacitor reaches its first maximum. Solution is included after problem.

Prior Knowledge: LC Circuits
Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao

#### Inductance of a Double-Layer Solenoid

PDF
Problem on page 33 to page 35

Find the inductance of a wire coiled into the shape of two concentric solenoids. Solution is included after problem.

Prior Knowledge: Self Inductance, Ampere's Law, Magnetic Energy Density
Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao

#### Magnetic Energy Stored in a Toroid

PDF
Problem on Page 39 to page 40

Find the total energy stored in the magnetic field in a toroid. Solution is included after problem.

Prior Knowledge: Energy of the Magnetic Field (OT11.3)
Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao

#### Magnetic Energy in a Wire

PDF
Problem on page 40 to page 41

Find the total energy stored in the magnetic field in a current-carrying wire. Solution is included after problem.

Prior Knowledge: Ampere's Law (OT9.3), Energy of the Magnetic Field (OT11.3)
Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao

#### Mutual Inductance: Loop and Wire

PDF
Problem on page 41 to page 42

Find the mutual inductance of a system consisting of a rectangular conducting loop next to a long conducting wire. Solution is included after problem.

Prior Knowledge: Ampere's Law (OT9.3), Mutual Inductance (OT11.1)
Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao

PDF
Problem on page 42 to page 43

Find the initial and final currents through each resistor in a two-loop RL circuit. Solution is included after problem.

Prior Knowledge: RL Circuits (OT11.4)
Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao

#### RL Circuit Dynamics

PDF
Problem on page 43 to page 45

Find how the current through an inductor evolves in time, for a two-loop RL circuit. Solution is included after problem.

Prior Knowledge: RL Circuits (OT11.4)
Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao

#### LC Circuit Analysis

PDF
Problem on page 45 to page 47

Characterize the behavior of an LC oscillator circuit by finding the oscillation frequency, maximum charge, maximum current and total energy. Solution is included after problem.

Prior Knowledge: LC Circuits (OT11.5)
Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao

#### Conceptual Questions on Inductance

PDF
Problem on page 47

Questions about inductance (1-3), circuits involving inductors (4,6,7).

Prior Knowledge: Inductors, LC Circuits, RL Circuits, Magnetic Pressure
Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao

#### Inductance of a Solenoid

PDF
Problem on page 48

Find the inductance of a solenoid and the induced EMF when a changing current is applied.

Prior Knowledge: Self-inductance (OT11.2)
Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao

#### Inductance of a Homemade Solenoid

PDF
Problem on page 48

Find the inductance of a cylindrical coil made by tightly wrapping a length of wire.

Prior Knowledge: Self-inductance (OT11.2)
Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao

#### Inductors in Series and in Parallel

PDF
Problem on page 48 to page 49

Show that inductors combine like resistors in series and in parallel; consider the effect of their mutual inductance.

Prior Knowledge: Self-inductance, Mutual Inductance
Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao

#### RL Circuit with Added Resistance

PDF
Problem on page 49

An additional large resistance is suddenly added in series to a steady-state RL circuit: solve for the current and induced EMF of the inductor as a function of time.

Prior Knowledge: RL Circuits
Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao

#### Energy in an RL Circuit

PDF
Problem on page 50

Find the total energy stored and dissipated in an RL circuit when a battery is connected and then, much later, disconnected; estimate the time scale.

Prior Knowledge: RL Circuits, Energy Stored in an Inductor
Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao

#### Inductance of a Solenoid with an Iron Core

PDF
Problem on page 50 to page 51

Derive the inductance of a solenoid and then calculate the inductance of the same solenoid when filled with an iron core.

Prior Knowledge: Self-inductance, Solenoids, Magnetic Permeability
Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao

#### RL and LC Circuit

PDF
Problem on page 51 to page 52

A circuit with two switches is first configured as an RL circuit and then, much later, switched to be an LC oscillator; find the current and energy in the RL circuit and the maximum charge on the capacitor.

Prior Knowledge: RL Circuits, LC Circuits
Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao

#### Electromagnetic Energy in a Spinning Cylindrical Capacitor

PDF
Problem on page 52

Find the magnetic field energy in a spinning charged cylindrical capacitor; also find the electric field energy when the charged cylindrical capacitor is at rest.

Prior Knowledge: Gauss's Law, Ampere's Law, Energy Density of the Electric Field, Energy Density of the Magnetic Field
Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao

#### Inductance of a Spinning Loop in an External Field

PDF
Problem on page 52 to page 53

Guided questions to estimate the inductance of a conducting loop spinning like a generator in an external magnetic field.

Prior Knowledge: Faraday's Law, Lenz's Law, Self-inductance
Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao

#### RL Circuit Questions

PDF
Problem on page 1 to page 3

Identify the behavior of current as switches close in one- and two-loop RL circuits.

Prior Knowledge: RL Circuits
Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao

#### Current in a Driven Motor

PDF
Problem on page 5

How does the current in a driven electric motor change when the rotating loop is held still?

Prior Knowledge: Faraday's Law, Electric Motor
Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao

#### LC Circuit Questions

PDF
Problem on page 1 to page 2

Identify how the charge and current in an LC circuit are behaving at specific moments in the oscillation.

Prior Knowledge: LC Circuits
Instructors: Prof. John Belcher, Dr. Peter Dourmashkin, Prof. Robert Redwine, Prof. Bruce Knuteson, Prof. Gunther Roland, Prof. Bolek Wyslouch, Dr. Brian Wecht, Prof. Eric Katsavounidis, Prof. Robert Simcoe, Prof. Joseph Formaggio, Andy Neely, Matthew Strafuss, Prof. Eric Hudson, Dr. Sen-Ben Liao

#### Self-inductance and Field Energy

PDF
Problem 4

Calculating magnetic field energy and self-inductance of a current-carrying wire.

Prior Knowledge: None
Instructor: Prof. Walter Lewin

#### RL Circuit

PDF
Problem 5

3-part RL circuit problem; energy delivered and built up in circuit; time to current build-up.

Prior Knowledge: None
Instructor: Prof. Walter Lewin

#### Circuit-Breaking

PDF
Problem 2

Explaining the spark from disconnecting circuit; energy stored in L.

Prior Knowledge: None
Instructors: Dr. Peter Dourmashkin, Prof. Gunther Roland

## Exam Questions

#### LR Circuit

PDF
Problem 1

Finding current across an inductor and a resistor as switch is closed and long after switch is closed.

Prior Knowledge: None
Instructor: Prof. Walter Lewin

PDF
Problem 11

4-part problem; maximal currents across circuit elements for various driving frequencies.

Prior Knowledge: None
Instructor: Prof. Walter Lewin

#### Mutual Inductance of Solenoids

PDF#
Problem 1

Finding B-fields and mutual inductance of two concentric solenoids.

Prior Knowledge: None
Instructors: Dr. Peter Dourmashkin, Prof. Gunther Roland

#### LC Circuit

PDF
Problem 3

In ideal LC circuit, calculating Q(t) and volume of inductor given maximum field.

Prior Knowledge: None
Instructors: Dr. Peter Dourmashkin, Prof. Gunther Roland

#### LR Circuit

PDF
Problem 4

Finding current in LR circuit just after opening; graphing current change.

Prior Knowledge: None
Instructors: Dr. Peter Dourmashkin, Prof. Gunther Roland