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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 T⋅m2T⋅m2.
  • 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, 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.

23.4 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 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.

23.10 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).
    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=I0e−t/τ    (turning off).I=I0e−t/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=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.

23.12 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+(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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