College Physics for AP® Courses

# Section Summary

### 23.1Induced Emf and Magnetic Flux

• The crucial quantity in induction is magnetic flux $ΦΦ size 12{Φ} {}$, defined to be $Φ=BAcosθΦ=BAcosθ size 12{Φ= ital "BA""cos"θ} {}$, where $BB size 12{B} {}$ is the magnetic field strength over an area $AA size 12{A} {}$ at an angle $θθ size 12{θ} {}$ with the perpendicular to the area.
• Units of magnetic flux $ΦΦ size 12{Φ} {}$ are $T⋅m2T⋅m2 size 12{T cdot m rSup { size 8{2} } } {}$.
• Any change in magnetic flux $ΦΦ size 12{Φ} {}$ 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
$emf = − N Δ Φ Δt emf = − N Δ Φ Δt size 12{"emf"= - N { {ΔΦ} over {Δt} } } {}$

when flux changes by $ΔΦΔΦ size 12{ΔΦ} {}$ in a time $ΔtΔt size 12{Δt} {}$.

• If emf is induced in a coil, $N N$ is its number of turns.
• The minus sign means that the emf creates a current $II size 12{I} {}$ and magnetic field $BB size 12{B} {}$ that oppose the change in flux $ΔΦΔΦ size 12{ΔΦ} {}$ —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
$emf=Bℓv(B, ℓ, andv perpendicular),emf=Bℓv(B, ℓ, andv perpendicular), size 12{"emf"=Bℓv} {}$
where $ℓℓ size 12{ℓ} {}$ is the length of the object moving at speed $vv size 12{v} {}$ 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
$emf=NABωsinωt,emf=NABωsinωt, size 12{"emf"= ital "NAB"ω"sin"ωt} {}$
where $AA size 12{A} {}$ is the area of an $NN size 12{N} {}$-turn coil rotated at a constant angular velocity $ωω size 12{ω} {}$ in a uniform magnetic field $BB size 12{B} {}$.
• The peak emf $emf0emf0 size 12{"emf" rSub { size 8{0} } } {}$ of a generator is
$emf0=NABω.emf0=NABω. size 12{"emf" rSub { size 8{0} } = ital "NAB"ω} {}$

### 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
$VsVp=NsNp,VsVp=NsNp, size 12{ { {V rSub { size 8{s} } } over {V rSub { size 8{p} } } } = { {N rSub { size 8{s} } } over {N rSub { size 8{p} } } } } {}$
where $VpVp size 12{V rSub { size 8{p} } } {}$ and $VsVs size 12{V rSub { size 8{s} } } {}$ are the voltages across primary and secondary coils having $NpNp size 12{N rSub { size 8{p} } } {}$ and $NsNs size 12{N rSub { size 8{s} } } {}$ turns.
• The currents $IpIp size 12{I rSub { size 8{p} } } {}$ and $IsIs size 12{I rSub { size 8{s} } } {}$ in the primary and secondary coils are related by $IsIp=NpNsIsIp=NpNs size 12{ { {I rSub { size 8{s} } } over {I rSub { size 8{p} } } } = { {N rSub { size 8{p} } } over {N rSub { size 8{s} } } } } {}$.
• 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 $ΔI1/ΔtΔI1/Δt size 12{ΔI rSub { size 8{1} } /Δt} {}$ in one induces an emf $emf2emf2 size 12{"emf" rSub { size 8{2} } } {}$ in the second:
$emf2=−MΔI1Δt,emf2=−MΔI1Δt, size 12{"emf" rSub { size 8{2} } = - M { {ΔI rSub { size 8{1} } } over {Δt} } } {}$
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 $ΔI2/ΔtΔI2/Δt size 12{ΔI rSub { size 8{2} } /Δt} {}$ through the second device induces an emf $emf1emf1 size 12{"emf" rSub { size 8{1} } } {}$ in the first:
$emf1=−MΔI2Δt,emf1=−MΔI2Δt, size 12{"emf" rSub { size 8{1} } = - M { {ΔI rSub { size 8{2} } } over {Δt} } } {}$
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
$emf=−LΔIΔt,emf=−LΔIΔt, size 12{"emf"= - L { {ΔI} over {Δt} } } {}$
where $LL size 12{L} {}$ is the self-inductance of the inductor, and $ΔI/ΔtΔI/Δt size 12{ΔI/Δt} {}$ 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 $1 H=1 Ω⋅s1 H=1 Ω⋅s size 12{1H=1 %OMEGA cdot s} {}$.
• The self-inductance $LL size 12{L} {}$ of an inductor is proportional to how much flux changes with current. For an $NN size 12{N} {}$-turn inductor,
$L=NΔΦΔI .L=NΔΦΔI . size 12{L=N { {ΔΦ} over {ΔI} } } {}$
• The self-inductance of a solenoid is
$L=μ0N2Aℓ(solenoid),L=μ0N2Aℓ(solenoid), size 12{L= { {μ rSub { size 8{0} } N rSup { size 8{2} } A} over {ℓ} } } {}$
where $NN size 12{N} {}$ is its number of turns in the solenoid, $AA size 12{A} {}$ is its cross-sectional area, $ℓℓ size 12{ℓ} {}$ is its length, and $μ0=4π×10−7T⋅m/Aμ0=4π×10−7T⋅m/A size 12{μ rSub { size 8{0} } =4π times "10" rSup { size 8{"-7"} } `T cdot "m/A"} {}$ is the permeability of free space.
• The energy stored in an inductor $EindEind size 12{E rSub { size 8{"ind"} } } {}$ is
$Eind=12LI2.Eind=12LI2. size 12{E rSub { size 8{"ind"} } = { {1} over {2} } ital "LI" rSup { size 8{2} } } {}$

### 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 $I0=V/RI0=V/R size 12{I rSub { size 8{0} } =V/R} {}$ is the final current.
• The characteristic time constant $ττ size 12{τ} {}$ is $τ=LRτ=LR size 12{τ= { {L} over {R} } } {}$ , where $L L$ is the inductance and $R R$ is the resistance.
• In the first time constant $ττ size 12{τ} {}$, the current rises from zero to $0.632I00.632I0 size 12{0 "." "632"I rSub { size 8{0} } } {}$, and 0.632 of the remainder in every subsequent time interval $ττ size 12{τ} {}$.
• When the inductor is shorted through a resistor, current decreases as
Here $I0I0 size 12{I rSub { size 8{0} } } {}$ is the initial current.
• Current falls to $0.368I00.368I0 size 12{0 "." "368"I rSub { size 8{0} } } {}$ in the first time interval $ττ size 12{τ} {}$, and 0.368 of the remainder toward zero in each subsequent time $ττ size 12{τ} {}$.

### 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
$I=VXL,I=VXL, size 12{I= { {V} over {X rSub { size 8{L} } } } } {}$
where $VV size 12{V} {}$ is the rms voltage across the inductor.
• $XLXL size 12{X rSub { size 8{L} } } {}$ is defined to be the inductive reactance, given by
$XL=2πfL,XL=2πfL, size 12{X rSub { size 8{L} } =2π ital "fL"} {}$
with $ff size 12{f} {}$ the frequency of the AC voltage source in hertz.
• Inductive reactance $XLXL size 12{X rSub { size 8{L} } } {}$ 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 $90º 90º$ 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
$I=VXC,I=VXC, size 12{I= { {V} over {X rSub { size 8{C} } } } } {}$
where $VV size 12{V} {}$ is the rms voltage across the capacitor.
• $XCXC size 12{X rSub { size 8{C} } } {}$ is defined to be the capacitive reactance, given by
$XC=12πfC.XC=12πfC. size 12{X rSub { size 8{C} } = { {1} over {2π ital "fC"} } } {}$
• $XCXC size 12{X rSub { size 8{C} } } {}$ 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:
$I 0 = V 0 Z or I rms = V rms Z , I 0 = V 0 Z or I rms = V rms Z , size 12{I rSub { size 8{0} } = { {V rSub { size 8{0} } } over {Z} } " or "I rSub { size 8{ ital "rms"} } = { {V rSub { size 8{ ital "rms"} } } over {Z} } ,} {}$
where $I0I0 size 12{I rSub { size 8{0} } } {}$ is the peak current and $V0V0 size 12{V rSub { size 8{0} } } {}$ is the peak source voltage.
• Impedance has units of ohms and is given by $Z=R2+(XL−XC)2Z=R2+(XL−XC)2 size 12{Z= sqrt {R rSup { size 8{2} } + $$X rSub { size 8{L} } - X rSub { size 8{C} }$$ rSup { size 8{2} } } } {}$.
• The resonant frequency $f0f0 size 12{f rSub { size 8{0} } } {}$, at which $XL=XCXL=XC size 12{X rSub { size 8{L} } =X rSub { size 8{C} } } {}$, is
$f0=12πLC.f0=12πLC. size 12{f rSub { size 8{0} } = { {1} over {2π sqrt { ital "LC"} } } } {}$
• In an AC circuit, there is a phase angle $ϕϕ size 12{ϕ} {}$ between source voltage $VV size 12{V} {}$ and the current $II size 12{I} {}$, which can be found from
$cosϕ=RZ,cosϕ=RZ, size 12{"cos"ϕ= { {R} over {Z} } } {}$
• $ϕ=0ºϕ=0º size 12{ϕ=0 rSup { size 8{ circ } } } {}$ 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
$Pave=IrmsVrmscosϕ,Pave=IrmsVrmscosϕ, size 12{P rSub { size 8{"ave"} } =I rSub { size 8{"rms"} } V rSub { size 8{"rms"} } "cos"ϕ} {}$
$cosϕcosϕ size 12{"cos"ϕ} {}$ is called the power factor, which ranges from 0 to 1.
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