College Physics 2e

# Section Summary

College Physics 2eSection Summary

### 23.1Induced Emf and Magnetic Flux

• The crucial quantity in induction is magnetic flux $ΦΦ$, defined to be $Φ=BAcosθΦ=BAcosθ$, where $BB$ is the magnetic field strength over an area $AA$ at an angle $θθ$ with the perpendicular to the area.
• Units of magnetic flux $ΦΦ$ are $T⋅m2T⋅m2$.
• Any change in magnetic flux $ΦΦ$ induces an emf—the process is defined to be electromagnetic induction.

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

• Faraday’s law of induction states that the emfinduced by a change in magnetic flux is
$emf = − N Δ Φ Δt emf = − N Δ Φ Δt$

when flux changes by $ΔΦΔΦ$ in a time $ΔtΔt$.

• If emf is induced in a coil, $N N$ is its number of turns.
• The minus sign means that the emf creates a current $II$ and magnetic field $BB$ that oppose the change in flux $ΔΦΔΦ$ —this opposition is known as Lenz’s law.

### 23.3Motional Emf

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

### 23.4Eddy Currents and Magnetic Damping

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

### 23.5Electric Generators

• An electric generator rotates a coil in a magnetic field, inducing an emfgiven as a function of time by
$emf=NABωsinωt,emf=NABωsinωt,$
where $AA$ is the area of an $NN$-turn coil rotated at a constant angular velocity $ωω$ in a uniform magnetic field $BB$.
• The peak emf $emf0emf0$ of a generator is
$emf0=NABω.emf0=NABω.$

### 23.6Back 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.

### 23.7Transformers

• 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
$VsVp=NsNp,VsVp=NsNp,$
where $VpVp$ and $VsVs$ are the voltages across primary and secondary coils having $NpNp$ and $NsNs$ turns.
• The currents $IpIp$ and $IsIs$ in the primary and secondary coils are related by $IsIp=NpNsIsIp=NpNs$.
• A step-up transformer increases voltage and decreases current, whereas a step-down transformer decreases voltage and increases current.

### 23.8Electrical 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.

### 23.9Inductance

• 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 $ΔI1/ΔtΔI1/Δt$ in one induces an emf $emf2emf2$ in the second:
$emf2=−MΔI1Δt,emf2=−MΔI1Δt,$
where $M M$ 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 $ΔI2/ΔtΔI2/Δt$ through the second device induces an emf $emf1emf1$ in the first:
$emf1=−MΔI2Δt,emf1=−MΔI2Δt,$
where $M M$ 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
$emf=−LΔIΔt,emf=−LΔIΔt,$
where $LL$ is the self-inductance of the inductor, and $ΔI/ΔtΔI/Δt$ 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 $1 H=1 Ω⋅s1 H=1 Ω⋅s$.
• The self-inductance $LL$ of an inductor is proportional to how much flux changes with current. For an $NN$-turn inductor,
$L=NΔΦΔI .L=NΔΦΔI .$
• The self-inductance of a solenoid is
$L=μ0N2Aℓ(solenoid),L=μ0N2Aℓ(solenoid),$
where $NN$ is its number of turns in the solenoid, $AA$ is its cross-sectional area, $ℓℓ$ is its length, and $μ0=4π×10−7T⋅m/Aμ0=4π×10−7T⋅m/A$ is the permeability of free space.
• The energy stored in an inductor $EindEind$ is
$Eind=12LI2.Eind=12LI2.$

### 23.10RL 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
where $I0=V/RI0=V/R$ is the final current.
• The characteristic time constant $ττ$ is $τ=LRτ=LR$ , where $L L$ is the inductance and $R R$ is the resistance.
• In the first time constant $ττ$, the current rises from zero to $0.632I00.632I0$, and 0.632 of the remainder in every subsequent time interval $ττ$.
• When the inductor is shorted through a resistor, current decreases as
Here $I0I0$ is the initial current.
• Current falls to $0.368I00.368I0$ in the first time interval $ττ$, and 0.368 of the remainder toward zero in each subsequent time $ττ$.

### 23.11Reactance, 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 $90º 90º$ 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
$I=VXL,I=VXL,$
where $VV$ is the rms voltage across the inductor.
• $XLXL$ is defined to be the inductive reactance, given by
$XL=2πfL,XL=2πfL,$
with $ff$ the frequency of the AC voltage source in hertz.
• Inductive reactance $XLXL$ 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 $90º 90º$ 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
$I=VXC,I=VXC,$
where $VV$ is the rms voltage across the capacitor.
• $XCXC$ is defined to be the capacitive reactance, given by
$XC=12πfC.XC=12πfC.$
• $XCXC$ has units of ohms and is greatest at low frequencies.

### 23.12RLC Series AC Circuits

• The AC analogy to resistance is impedance $Z Z$, the combined effect of resistors, inductors, and capacitors, defined by the AC version of Ohm’s law:
$I 0 = V 0 Z or I rms = V rms Z , I 0 = V 0 Z or I rms = V rms Z ,$
where $I0I0$ is the peak current and $V0V0$ is the peak source voltage.
• Impedance has units of ohms and is given by $Z=R2+(XL−XC)2Z=R2+(XL−XC)2$.
• The resonant frequency $f0f0$, at which $XL=XCXL=XC$, is
$f0=12πLC.f0=12πLC.$
• In an AC circuit, there is a phase angle $ϕϕ$ between source voltage $VV$ and the current $II$, which can be found from
$cosϕ=RZ,cosϕ=RZ,$
• $ϕ=0ºϕ=0º$ 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
$Pave=IrmsVrmscosϕ,Pave=IrmsVrmscosϕ,$
$cosϕcosϕ$ is called the power factor, which ranges from 0 to 1.
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