Gregg Wolfe; Erika Gasper; John Stoke; Julie Kretchman; David Anderson; Nathan Czuba; Sudhi Oberoi; Liza Pujji; Irina Lyublinskaya; Douglas Ingram

23.1 Induced Emf and Magnetic Flux

The crucial quantity in induction is magnetic flux $Φ$ , defined to be $Φ = BA cos θ$ , where $B$ is the magnetic field strength over an area $A$ at an angle $θ$ with the perpendicular to the area.
Units of magnetic flux $Φ$ are $T \cdot m^{2}$ .
Any change in magnetic flux $Φ$ induces an emf—the process is defined to be electromagnetic induction.

23.2 Faraday’s Law of Induction: Lenz’s Law

Faraday’s law of induction states that the emfinduced by a change in magnetic flux is
$emf = - N \frac{Δ Φ}{Δ t}$
when flux changes by $Δ Φ$ in a time $Δ t$ .
If emf is induced in a coil, $N$ is its number of turns.
The minus sign means that the emf creates a current $I$ and magnetic field $B$ that oppose the change in flux $Δ Φ$ —this opposition is known as Lenz’s law.

23.3 Motional Emf

An emf induced by motion relative to a magnetic field $B$ is called a motional emf and is given by
$emf = Bℓv (B, ℓ, and v perpendicular),$
where $ℓ$ is the length of the object moving at speed $v$ relative to the field.

23.4 Eddy Currents and Magnetic Damping

Current loops induced in moving conductors are called eddy currents.
They can create significant drag, called magnetic damping.

23.5 Electric Generators

An electric generator rotates a coil in a magnetic field, inducing an emfgiven as a function of time by
$emf = NAB ω sin ωt,$
where $A$ is the area of an $N$ -turn coil rotated at a constant angular velocity $ω$ in a uniform magnetic field $B$ .
The peak emf ${emf}_{0}$ of a generator is
${emf}_{0} = NAB ω .$

23.6 Back Emf

Any rotating coil will have an induced emf—in motors, this is called back emf, since it opposes the emf input to the motor.

23.7 Transformers

Transformers use induction to transform voltages from one value to another.
For a transformer, the voltages across the primary and secondary coils are related by
$\frac{V_{s}}{V_{p}} = \frac{N_{s}}{N_{p}},$
where $V_{p}$ and $V_{s}$ are the voltages across primary and secondary coils having $N_{p}$ and $N_{s}$ turns.
The currents $I_{p}$ and $I_{s}$ in the primary and secondary coils are related by $\frac{I_{s}}{I_{p}} = \frac{N_{p}}{N_{s}}$ .
A step-up transformer increases voltage and decreases current, whereas a step-down transformer decreases voltage and increases current.

23.8 Electrical Safety: Systems and Devices

Electrical safety systems and devices are employed to prevent thermal and shock hazards.
Circuit breakers and fuses interrupt excessive currents to prevent thermal hazards.
The three-wire system guards against thermal and shock hazards, utilizing live/hot, neutral, and earth/ground wires, and grounding the neutral wire and case of the appliance.
A ground fault interrupter (GFI) prevents shock by detecting the loss of current to unintentional paths.
An isolation transformer insulates the device being powered from the original source, also to prevent shock.
Many of these devices use induction to perform their basic function.

23.9 Inductance

Inductance is the property of a device that tells how effectively it induces an emf in another device.
Mutual inductance is the effect of two devices in inducing emfs in each other.
A change in current $Δ I_{1} / Δ t$ in one induces an emf ${emf}_{2}$ in the second:
${emf}_{2} = - M \frac{Δ I_{1}}{Δ t},$
where $M$ is defined to be the mutual inductance between the two devices, and the minus sign is due to Lenz’s law.
Symmetrically, a change in current $Δ I_{2} / Δ t$ through the second device induces an emf ${emf}_{1}$ in the first:
${emf}_{1} = - M \frac{Δ I_{2}}{Δ t},$
where $M$ is the same mutual inductance as in the reverse process.
Current changes in a device induce an emf in the device itself.
Self-inductance is the effect of the device inducing emf in itself.
The device is called an inductor, and the emf induced in it by a change in current through it is
$emf = - L \frac{Δ I}{Δ t},$
where $L$ is the self-inductance of the inductor, and $Δ I / Δ t$ 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- and mutual inductance is the henry (H), where $1 H = 1 Ω \cdot s$ .
The self-inductance $L$ of an inductor is proportional to how much flux changes with current. For an $N$ -turn inductor,
$L = N \frac{Δ Φ}{Δ I} .$
The self-inductance of a solenoid is
$L = \frac{μ_{0} N^{2} A}{ℓ} (solenoid),$
where $N$ is its number of turns in the solenoid, $A$ is its cross-sectional area, $ℓ$ is its length, and $μ_{0} = 4π \times 10^{−7} T \cdot m/A$ is the permeability of free space.
The energy stored in an inductor $E_{ind}$ is
$E_{ind} = \frac{1}{2} {LI}^{2} .$

23.10 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
$I = I_{0} (1 - e^{- t / τ}) (turning on).$
where $I_{0} = V / R$ is the final current.
The characteristic time constant $τ$ is $τ = \frac{L}{R}$ , where $L$ is the inductance and $R$ is the resistance.
In the first time constant $τ$ , the current rises from zero to $0.632 I_{0}$ , and 0.632 of the remainder in every subsequent time interval $τ$ .
When the inductor is shorted through a resistor, current decreases as
$I = I_{0} e^{- t / τ} (turning off).$
Here $I_{0}$ is the initial current.
Current falls to $0.368 I_{0}$ in the first time interval $τ$ , and 0.368 of the remainder toward zero in each subsequent time $τ$ .

23.11 Reactance, Inductive and Capacitive

For inductors in AC circuits, we find that when a sinusoidal voltage is applied to an inductor, the voltage leads the current by one-fourth of a cycle, or by a $90º$ phase angle.
The opposition of an inductor to a change in current is expressed as a type of AC resistance.
Ohm’s law for an inductor is
$I = \frac{V}{X_{L}},$
where $V$ is the rms voltage across the inductor.
$X_{L}$ is defined to be the inductive reactance, given by
$X_{L} = 2π fL,$
with $f$ the frequency of the AC voltage source in hertz.
Inductive reactance $X_{L}$ has units of ohms and is greatest at high frequencies.
For capacitors, we find that when a sinusoidal voltage is applied to a capacitor, the voltage follows the current by one-fourth of a cycle, or by a $90º$ phase angle.
Since a capacitor can stop current when fully charged, it limits current and offers another form of AC resistance; Ohm’s law for a capacitor is
$I = \frac{V}{X_{C}},$
where $V$ is the rms voltage across the capacitor.
$X_{C}$ is defined to be the capacitive reactance, given by
$X_{C} = \frac{1}{2π fC} .$
$X_{C}$ has units of ohms and is greatest at low frequencies.

23.12 RLC Series AC Circuits

The AC analogy to resistance is impedance $Z$ , the combined effect of resistors, inductors, and capacitors, defined by the AC version of Ohm’s law:
$I_{0} = \frac{V_{0}}{Z} or I_{rms} = \frac{V_{rms}}{Z},$
where $I_{0}$ is the peak current and $V_{0}$ is the peak source voltage.
Impedance has units of ohms and is given by $Z = \sqrt{R^{2} + (X_{L} - X_{C})^{2}}$ .
The resonant frequency $f_{0}$ , at which $X_{L} = X_{C}$ , is
$f_{0} = \frac{1}{2π \sqrt{LC}} .$
In an AC circuit, there is a phase angle $ϕ$ between source voltage $V$ and the current $I$ , which can be found from
$cos ϕ = \frac{R}{Z},$
$ϕ = 0º$ for a purely resistive circuit or an RLC circuit at resonance.
The average power delivered to an RLC circuit is affected by the phase angle and is given by
$P_{ave} = I_{rms} V_{rms} cos ϕ,$
$cos ϕ$ is called the power factor, which ranges from 0 to 1.

Section Summary

23.1 Induced Emf and Magnetic Flux

23.2 Faraday’s Law of Induction: Lenz’s Law

23.3 Motional Emf

23.4 Eddy Currents and Magnetic Damping

23.5 Electric Generators

23.6 Back Emf

23.7 Transformers

23.8 Electrical Safety: Systems and Devices

23.9 Inductance

23.10 RL Circuits

23.11 Reactance, Inductive and Capacitive

23.12 RLC Series AC Circuits