14.8: Inductance (Summary)

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

14.2 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.

14.3 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.

14.4 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

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

14.5 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 \(\displaystyle τ\).

14.6 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

\(\displaystyle q(t)=q_0cos(ωt+ϕ)\),

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

14.7 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

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