# 14.S: Inductance (Summary)

- Page ID
- 10309

## Key Terms

henry (H) |
unit of inductance, \(\displaystyle 1H=1Ω⋅s\); it is also expressed as a volt second per ampere |

inductance |
property of a device that tells how effectively it induces an emf in another device |

inductive time constant |
denoted by \(\displaystyle τ\), the characteristic time given by quantity \(\displaystyle L/R\) of a particular series \(\displaystyle RL\) circuit |

inductor |
part of an electrical circuit to provide self-inductance, which is symbolized by a coil of wire |

LC circuit |
circuit composed of an ac source, inductor, and capacitor |

magnetic energy density |
energy stored per volume in a magnetic field |

mutual inductance |
geometric quantity that expresses how effective two devices are at inducing emfs in one another |

RLC circuit |
circuit with an ac source, resistor, inductor, and capacitor all in series. |

self-inductance |
effect of the device inducing emf in itself |

## Key Equations

Mutual inductance by flux | \(\displaystyle M=\frac{N_2Φ_2}{1_I}=\frac{N_1Φ_{12}}{I_2}\) |

Mutual inductance in circuits | \(\displaystyle ε_1=−M\frac{dI_2}{dt}\) |

Self-inductance in terms of magnetic flux | \(\displaystyle NΦ_m=LI\) |

Self-inductance in terms of emf | \(\displaystyle ε=−L\frac{dI}{dt}\) |

Self-inductance of a solenoid | \(\displaystyle L_{solenoid}=\frac{μ_0N^2A}{l}\) |

Self-inductance of a toroid | \(\displaystyle L_{toroid}=\frac{μ_0N^2h}{2π}ln\frac{R_2}{R_1}\). |

Energy stored in an inductor | \(\displaystyle U=\frac{1}{2}LI^2\) |

Current as a function of time for a RL circuit |
\(\displaystyle I(t)=\frac{ε}{R}(1−e^{−t/τ_L})\) |

Time constant for a RL circuitτ |
\(\displaystyle τ_L=L/R\) |

Charge oscillation in LC circuits |
\(\displaystyle q(t)=q_0cos(ωt+ϕ)\) |

Angular frequency in LC circuits |
\(\displaystyle ω=\sqrt{\frac{1}{LC}}\) |

Current oscillations in LC circuits |
\(\displaystyle i(t)=−ωq_0sin(ωt+ϕ)\) |

Charge as a function of time in RLC circuit |
\(\displaystyle q(t)=q_0e^{−Rt/2L}cos(ω't+ϕ)\) |

Angular frequency in RLC circuit |
\(\displaystyle ω'=\sqrt{\frac{1}{LC}−(\frac{R}{2L})^2}\) |

## Summary

### 14.1 Mutual Inductance

- Inductance is the property of a device that expresses how effectively it induces an emf in another device.
- Mutual inductance is the effect of two devices inducing emfs in each other.
- A change in current \(\displaystyle dI_1/dt\) in one circuit induces an emf (\(\displaystyle ε_2\)) in the second:

\(\displaystyle ε_2=−M\frac{dI_1}{dt}\),

where *M* is defined to be the mutual inductance between the two circuits and the minus sign is due to Lenz’s law.

- Symmetrically, a change in current \(\displaystyle dI_2/dt\) through the second circuit induces an emf (\(\displaystyle ε_1\)) in the first:

\(\displaystyle ε_1=−M\frac{dI_2}{dt}\),

where *M* is the same mutual inductance as in the reverse process.

### 14.2 Self-Inductance and Inductors

- Current changes in a device induce an emf in the device itself, called self-inductance,

\(\displaystyle ε=−L\frac{dI}{dt}\),

where *L* is the self-inductance of the inductor and \(\displaystyle dI/dt\) is the rate of change of current through it. The minus sign indicates that emf opposes the change in current, as required by Lenz’s law. The unit of self-inductance and inductance is the henry (H), where \(\displaystyle 1H=1Ω⋅s\).

- The self-inductance of a solenoid is

\(\displaystyle L=\frac{μ_0N^2A}{l}\),

where *N* is its number of turns in the solenoid, *A* is its cross-sectional area, *l* is its length, and \(\displaystyle μ_0=4π×10^{−7}T⋅m/A\) is the permeability of free space.

- The self-inductance of a toroid is

\(\displaystyle L=\frac{μ_0N^2h}{2π}ln\frac{R_2}{R_1}\),

where *N* is its number of turns in the toroid, \(\displaystyle R_1\) and \(\displaystyle R_2\) are the inner and outer radii of the toroid, *h* is the height of the toroid, and \(\displaystyle μ_0=4π×10^{−7}T⋅m/A\) is the permeability of free space.

### 14.3 Energy in a Magnetic Field

- The energy stored in an inductor
*U*is

\(\displaystyle U=\frac{1}{2}LI^2\).

- The self-inductance per unit length of coaxial cable is

\(\displaystyle \frac{L}{l}=\frac{μ_0}{2π}ln\frac{R_2}{R_1}\).

### 14.4 RL Circuits

- When a series connection of a resistor and an inductor—an
*RL*circuit—is connected to a voltage source, the time variation of the current is

\(\displaystyle I(t)=\frac{ε}{R}(1−e^{−Rt/L})=\frac{ε}{R}(1−e^{−t/τ_L})\) (turning on),

where the initial current is \(\displaystyle I_0=ε/R\)..

- The characteristic time constant \(\displaystyle τ\) is \(\displaystyle τ_L=L/R\), where
*L*is the inductance and*R*is the resistance. - In the first time constant \(\displaystyle τ\), the current rises from zero to \(\displaystyle 0.632I_0\), and to 0.632 of the remainder in every subsequent time interval \(\displaystyle τ\).
- When the inductor is shorted through a resistor, current decreases as

\(\displaystyle I(t)=\frac{ε}{R}e^{−t/τ_L}\) (turning off).

Current falls to \(\displaystyle 0.368I_0\) in the first time interval \(\displaystyle τ\), and to 0.368 of the remainder toward zero in each subsequent time \(\displaystyle τ\).

### 14.5 Oscillations in an LC Circuit

- The energy transferred in an oscillatory manner between the capacitor and inductor in an
*LC*circuit occurs at an angular frequency \(\displaystyle ω=\sqrt{\frac{1}{LC}}\). - The charge and current in the circuit are given by

\(\displaystyle q(t)=q_0cos(ωt+ϕ)\),

\(\displaystyle i(t)=−ωq_0sin(ωt+ϕ)\).

### 14.6 RLC Series Circuits

- The underdamped solution for the capacitor charge in an
*RLC*circuit is

\(\displaystyle q(t)=q_0e^{−Rt/2L}cos(ω't+ϕ).\)

- The angular frequency given in the underdamped solution for the
*RLC*circuit is

\(\displaystyle ω′=\sqrt{\frac{1}{LC}−(\frac{R}{2L})^2}\).

## 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).