# 13.S: Electromagnetic Induction (Summary)

## Key Terms

back emf |
emf generated by a running motor, because it consists of a coil turning in a magnetic field; it opposes the voltage powering the motor |

eddy current |
current loop in a conductor caused by motional emf |

electric generator |
device for converting mechanical work into electric energy; it induces an emf by rotating a coil in a magnetic field |

Faraday’s law |
induced emf is created in a closed loop due to a change in magnetic flux through the loop |

induced electric field |
created based on the changing magnetic flux with time |

induced emf |
short-lived voltage generated by a conductor or coil moving in a magnetic field |

Lenz’s law |
direction of an induced emf opposes the change in magnetic flux that produced it; this is the negative sign in Faraday’s law |

magnetic damping |
drag produced by eddy currents |

magnetic flux |
measurement of the amount of magnetic field lines through a given area |

motionally induced emf |
voltage produced by the movement of a conducting wire in a magnetic field |

peak emf |
maximum emf produced by a generator |

## Key Equations

Magnetic flux | \(\displaystyle Φ_m=∫_S\vec{B}⋅\hat{n}dA\) |

Faraday’s law | \(\displaystyle ε=−N\frac{dΦ_m}{dt}\) |

Motionally induced emf | \(\displaystyle ε=Blv\) |

Motional emf around a circuit | \(\displaystyle ε=∮\vec{E}⋅d\vec{l}=−\frac{dΦ_m}{dt}\) |

Emf produced by an electric generator | \(\displaystyle ε=NBAωsin(ωt)\) |

## Summary

#### 13.1 Faraday’s Law

- The magnetic flux through an enclosed area is defined as the amount of field lines cutting through a surface area A defined by the unit area vector.
- The units for magnetic flux are webers, where \(\displaystyle 1Wb=1T⋅m^2\).
- The induced emf in a closed loop due to a change in magnetic flux through the loop is known as Faraday’s law. If there is no change in magnetic flux, no induced emf is created.

#### 13.2 Lenz's Law

- We can use Lenz’s law to determine the directions of induced magnetic fields, currents, and emfs.
- The direction of an induced emf always opposes the change in magnetic flux that causes the emf, a result known as Lenz’s law.

#### 13.3 Motional Emf

- The relationship between an induced emf εε in a wire moving at a constant speed
*v*through a magnetic field*B*is given by \(\displaystyle ε=Blv\). - An induced emf from Faraday’s law is created from a motional emf that opposes the change in flux.

#### 13.4 Induced Electric Fields

- A changing magnetic flux induces an electric field.
- Both the changing magnetic flux and the induced electric field are related to the induced emf from Faraday’s law.

#### 13.5 Eddy Currents

- Current loops induced in moving conductors are called eddy currents. They can create significant drag, called magnetic damping.
- Manipulation of eddy currents has resulted in applications such as metal detectors, braking in trains or roller coasters, and induction cooktops.

#### 13.6 Electric Generators and Back Emf

- An electric generator rotates a coil in a magnetic field, inducing an emf given as a function of time by \(\displaystyle ε=NBAωsin(ωt)\) where
*A*is the area of an*N*-turn coil rotated at a constant angular velocity \(\displaystyle ω\) in a uniform magnetic field \(\displaystyle \vec{B}\). - The peak emf of a generator is \(\displaystyle ε_0=NBAω\).
- Any rotating coil produces an induced emf. In motors, this is called back emf because it opposes the emf input to the motor.

#### 13.7 Applications of Electromagnetic Induction

- Hard drives utilize magnetic induction to read/write information.
- Other applications of magnetic induction can be found in graphics tablets, electric and hybrid vehicles, and in transcranial magnetic stimulation.

## Contributors

Paul Peter Urone (Professor Emeritus at California State University, Sacramento) and Roger Hinrichs (State University of New York, College at Oswego) with Contributing Authors: Kim Dirks (University of Auckland) and Manjula Sharma (University of Sydney). This work is licensed by OpenStax University Physics under a Creative Commons Attribution License (by 4.0).