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Section Summary

6.1 Induced Emf and Magnetic Flux

  • The crucial quantity in induction is magnetic flux Φ,Φ, size 12{Φ} {} defined to be Φ=BAcosθ,Φ=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⋅m2.T⋅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.

6.2 Faraday’s Law of Induction: Lenz's Law

  • Faraday’s law of induction states that the emf induced 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 Δt.Δt. 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.

6.3 Motional 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.

6.4 Eddy Currents and Magnetic Damping

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

6.5 Electric Generators

  • An electric generator rotates a coil in a magnetic field, inducing an emf given 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 B.B. 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} {}

6.6 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.

6.7 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
    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=NpNs.IsIp=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.

6.8 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.

6.9 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 /Δt Δ I 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 Δ I 2 /Δt Δ I 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/Δt 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⋅s.1 H=1 Ω⋅s. size 12{1`H=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 N 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} } } {}

6.10 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), I= I 0 (1− e −t/τ )    (turning on),
    I 0 =V/R I 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 τ,τ,size 12{τ} {} the current rises from zero to 0.632I0,0.632I0, size 12{0 "." "632"I rSub { size 8{0} } } {} and 0.632 of the remainder in every subsequent time interval τ.τ. size 12{τ} {}
  • When the inductor is shorted through a resistor, current decreases as
    I=I0e−t/τ    (turning off).I=I0e−t/τ    (turning off). size 12{I=I rSub { size 8{0} } e rSup { size 8{ - t/τ} } } {}
    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 τ,τ, size 12{τ} {} and 0.368 of the remainder toward zero in each subsequent time τ.τ. size 12{τ} {}

6.11 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 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.

6.12 RLC 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)2.Z=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 f0,f0, size 12{f rSub { size 8{0} } } {} at which XL=XC,XL=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 I,I, 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.