### 23.1 Induced Emf and Magnetic Flux

- The crucial quantity in induction is magnetic flux
*$\Phi $*, defined to be $\Phi =\text{BA}\phantom{\rule{0.25em}{0ex}}\text{cos}\phantom{\rule{0.25em}{0ex}}\theta $, where*$B$*is the magnetic field strength over an area*$A$*at an angle $\theta $ with the perpendicular to the area. - Units of magnetic flux
*$\Phi $*are $\text{T}\cdot {\text{m}}^{2}$. - Any change in magnetic flux
*$\Phi $*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
$$\text{emf}=-N\frac{\mathrm{\Delta}\Phi}{\mathrm{\Delta}t}$$
when flux changes by $\mathrm{\Delta}\Phi $ in a time $\mathrm{\Delta}t$.

- 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*$\mathrm{\Delta}\Phi $ —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$$\text{emf}=\mathrm{B\ell v}\phantom{\rule{3.00em}{0ex}}\text{(}B\text{,}\phantom{\rule{0.25em}{0ex}}\ell \text{, and}\phantom{\rule{0.25em}{0ex}}v\phantom{\rule{0.25em}{0ex}}\text{perpendicular),}$$where $\ell $ 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
$$\text{emf}=\text{NAB}\omega \phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\mathrm{\omega t}\text{,}$$where $A$ is the area of an $N$-turn coil rotated at a constant angular velocity $\omega $ in a uniform magnetic field $B$.
- The peak emf ${\text{emf}}_{0}$ of a generator is
$${\text{emf}}_{0}=\text{NAB}\omega \text{.}$$

### 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
$$\frac{{V}_{\text{s}}}{{V}_{\text{p}}}=\frac{{N}_{\text{s}}}{{N}_{\text{p}}}\text{,}$$where ${V}_{\text{p}}$ and ${V}_{\text{s}}$ are the voltages across primary and secondary coils having ${N}_{\text{p}}$ and ${N}_{\text{s}}$ turns.
- The currents ${I}_{\text{p}}$ and ${I}_{\text{s}}$ in the primary and secondary coils are related by $\frac{{I}_{\text{s}}}{{I}_{\text{p}}}=\frac{{N}_{\text{p}}}{{N}_{\text{s}}}$.
- 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 $\mathrm{\Delta}{I}_{1}/\mathrm{\Delta}t$ in one induces an emf ${\text{emf}}_{2}$ in the second:
$${\text{emf}}_{2}=-M\frac{\mathrm{\Delta}{I}_{1}}{\mathrm{\Delta}t}\text{,}$$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 $\mathrm{\Delta}{I}_{2}/\mathrm{\Delta}t$ through the second device induces an emf ${\text{emf}}_{1}$ in the first:
$${\text{emf}}_{1}=-M\frac{\mathrm{\Delta}{I}_{2}}{\mathrm{\Delta}t}\text{,}$$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
$$\text{emf}=-L\frac{\mathrm{\Delta}I}{\mathrm{\Delta}t}\text{,}$$where $L$ is the self-inductance of the inductor, and $\mathrm{\Delta}I/\mathrm{\Delta}t$ 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 $\mathrm{1\; H}=1\; \Omega \cdot \text{s}$.
- The self-inductance $L$ of an inductor is proportional to how much flux changes with current. For an $N$-turn inductor,
$$L=N\frac{\mathrm{\Delta}\Phi}{\mathrm{\Delta}I}\text{.}$$
- The self-inductance of a solenoid is
$$L=\frac{{\mu}_{0}{N}^{2}A}{\ell}\text{(solenoid),}$$where $N$ is its number of turns in the solenoid, $A$ is its cross-sectional area, $\ell $ is its length, and ${\text{\mu}}_{0}=\mathrm{4\pi}\times {\text{10}}^{\text{\u22127}}\phantom{\rule{0.25em}{0ex}}\text{T}\cdot \text{m/A}\phantom{\rule{0.10em}{0ex}}$ is the permeability of free space.
- The energy stored in an inductor ${E}_{\text{ind}}$ is
$${E}_{\text{ind}}=\frac{1}{2}{\text{LI}}^{2}\text{.}$$

### 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$$I={I}_{0}(1-{e}^{-t/\tau})\text{(turning on).}$$where ${I}_{0}=V/R$ is the final current. - The characteristic time constant $\tau $ is $\tau =\frac{L}{R}$ , where $L$ is the inductance and $R$ is the resistance.
- In the first time constant $\tau $, the current rises from zero to $0\text{.}\text{632}{I}_{0}$, and 0.632 of the remainder in every subsequent time interval $\tau $.
- When the inductor is shorted through a resistor, current decreases as
$$I={I}_{0}{e}^{-t/\tau}\text{(turning off).}$$Here ${I}_{0}$ is the initial current.
- Current falls to $0\text{.}\text{368}{I}_{0}$ in the first time interval $\tau $, and 0.368 of the remainder toward zero in each subsequent time $\tau $.

### 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 $\text{90\xba}$ 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
$$I=\frac{V}{{X}_{L}}\text{,}$$where $V$ is the rms voltage across the inductor.
- ${X}_{L}$ is defined to be the inductive reactance, given by
$${X}_{L}=\mathrm{2\pi}\text{fL}\text{,}$$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 $\text{90\xba}$ 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
$$I=\frac{V}{{X}_{C}}\text{,}$$where $V$ is the rms voltage across the capacitor.
- ${X}_{C}$ is defined to be the capacitive reactance, given by
$${X}_{C}=\frac{1}{\mathrm{2\pi}\text{fC}}\text{.}$$
- ${X}_{C}$ 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:
$${I}_{0}=\frac{{V}_{0}}{Z}\phantom{\rule{0.25em}{0ex}}\text{or}\phantom{\rule{0.25em}{0ex}}{I}_{\text{rms}}=\frac{{V}_{\text{rms}}}{Z},$$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
$${f}_{0}=\frac{1}{\mathrm{2\pi}\sqrt{\text{LC}}}\text{.}$$
- In an AC circuit, there is a phase angle
*$\varphi $*between source voltage $V$ and the current $I$, which can be found from$$\text{cos}\phantom{\rule{0.25em}{0ex}}\varphi =\frac{R}{Z}\text{,}$$ - $\varphi =\mathrm{0\xba}$ 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$${P}_{\text{ave}}={I}_{\text{rms}}{V}_{\text{rms}}\phantom{\rule{0.25em}{0ex}}\text{cos}\phantom{\rule{0.25em}{0ex}}\varphi \text{,}$$$\text{cos}\phantom{\rule{0.25em}{0ex}}\varphi $ is called the power factor, which ranges from 0 to 1.