Skip to ContentGo to accessibility pageKeyboard shortcuts menu
OpenStax Logo
College Physics 2e

Section Summary

College Physics 2eSection Summary

23.1 Induced 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 Tm2Tm2.
  • Any change in magnetic flux ΦΦ 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

    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.

23.3 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, , andv perpendicular),emf=Bℓv(B, , andv perpendicular),
    where is the length of the object moving at speed vv 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,
    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.

23.6 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,
    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.

23.8 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 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 Ωs1 H=1 Ωs.
  • 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=μ0N2A(solenoid),L=μ0N2A(solenoid),
    where NN is its number of turns in the solenoid, AA is its cross-sectional area, is its length, and μ0=×10−7Tm/Aμ0=×10−7Tm/A is the permeability of free space.
  • The energy stored in an inductor EindEind is
    Eind=12LI2.Eind=12LI2.

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(1et/τ)    (turning on).I=I0(1et/τ)    (turning on).
    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
    I=I0et/τ    (turning off).I=I0et/τ    (turning off).
    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.

23.11 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,
    where VV is the rms voltage across the inductor.
  • XLXL is defined to be the inductive reactance, given by
    XL=fL,XL=fL,
    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=1fC.XC=1fC.
  • XCXC 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 ,
    where I0I0 is the peak current and V0V0 is the peak source voltage.
  • Impedance has units of ohms and is given by Z=R2+(XLXC)2Z=R2+(XLXC)2.
  • The resonant frequency f0f0, at which XL=XCXL=XC, is
    f0=1LC.f0=1LC.
  • 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=ϕ= 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.
Order a print copy

As an Amazon Associate we earn from qualifying purchases.

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 Citation information

© Jan 19, 2024 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.