Skip to ContentGo to accessibility pageKeyboard shortcuts menu
OpenStax Logo
College Physics for AP® Courses

Section Summary

College Physics for AP® CoursesSection Summary
Search for key terms or text.

23.1 Induced 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.

23.2 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 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.

23.3 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.

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=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} {}

23.6 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
    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.

23.8 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 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{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 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} } } {}

23.10 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=I0(1−e−t/τ)    (turning on).I=I0(1−e−t/τ)    (turning on). size 12{I=I rSub { size 8{0} } \( 1 - e rSup { size 8{ - t/τ} } \) } {}
    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
    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 ττ size 12{τ} {}, and 0.368 of the remainder toward zero in each subsequent time ττ size 12{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 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.

23.12 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)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.
PreviousNext
Order a print copy
Citation/Attribution

This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax.

Attribution information
  • If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:
    Access for free at https://openstax.org/books/college-physics-ap-courses/pages/1-connection-for-ap-r-courses
  • If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:
    Access for free at https://openstax.org/books/college-physics-ap-courses/pages/1-connection-for-ap-r-courses
Citation information

© Mar 3, 2022 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.