Section Summary

23.1 Induced Emf and Magnetic Flux

• The crucial quantity in induction is magnetic flux Φ, defined to be Φ=BAcosθ, where B is the magnetic field strength over an area A at an angle θ with the perpendicular to the area.
• Units of magnetic flux Φ are T⋅m2.
• Any change in magnetic flux 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 isemf=−NΔΦΔtwhen flux changes by ΔΦ in a time Δt.
• If emf is induced in a coil, N is its number of turns.
• The minus sign means that the emf creates a current I and magnetic field B that oppose the change in flux ΔΦ —this opposition is known as Lenz’s law.

23.3 Motional Emf

• An emf induced by motion relative to a magnetic field B is called a motional emf and is given byemf=Bℓv(B,ℓ, andvperpendicular),where ℓ is the length of the object moving at speed 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 byemf=NABωsinωt,where A is the area of an N-turn coil rotated at a constant angular velocity ω in a uniform magnetic field B.
• The peak emf emf0 of a generator isemf0=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 byVsVp=NsNp,where Vp and Vs are the voltages across primary and secondary coils having Np and Ns turns.
• The currents Ip and Is in the primary and secondary coils are related by IsIp=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/Δt in one induces an emf emf2 in the second:emf2=−MΔI1Δt,where 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/Δt through the second device induces an emf emf1 in the first:emf1=−MΔI2Δt,where 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 isemf=−LΔIΔt,where L is the self-inductance of the inductor, and 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.
• The self-inductance L of an inductor is proportional to how much flux changes with current. For an N-turn inductor,L=NΔΦΔI.
• The self-inductance of a solenoid isL=μ0N2Aℓ(solenoid),where N is its number of turns in the solenoid, A is its cross-sectional area, ℓ is its length, and μ0=4π×10−7T⋅m/A is the permeability of free space.
• The energy stored in an inductor Eind isEind=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 isI=I0(1−e−t/τ)    (turning on).where I0=V/R is the final current.
• The characteristic time constant τ is τ=LR , where L is the inductance and R is the resistance.
• In the first time constant τ, the current rises from zero to 0.632I0, and 0.632 of the remainder in every subsequent time interval τ.
• When the inductor is shorted through a resistor, current decreases asI=I0e−t/τ    (turning off).Here I0 is the initial current.
• Current falls to 0.368I0 in the first time interval τ, and 0.368 of the remainder toward zero in each subsequent time 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 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 isI=VXL,where V is the rms voltage across the inductor.
• XL is defined to be the inductive reactance, given byXL=2πfL,with f the frequency of the AC voltage source in hertz.
• Inductive reactance XL 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 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 isI=VXC,where V is the rms voltage across the capacitor.
• XC is defined to be the capacitive reactance, given byXC=12πfC.
• XC has units of ohms and is greatest at low frequencies.

23.12 RLC Series AC Circuits

• The AC analogy to resistance is impedance Z, the combined effect of resistors, inductors, and capacitors, defined by the AC version of Ohm’s law:I0=V0ZorIrms=VrmsZ,where I0 is the peak current and V0 is the peak source voltage.
• Impedance has units of ohms and is given by Z=R2+(XL−XC)2−−−−−−−−−−−−−√.
• The resonant frequency f0, at which XL=XC, isf0=12πLC−−−√.
• In an AC circuit, there is a phase angle ϕ between source voltage V and the current I, which can be found fromcosϕ=RZ,
• ϕ=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 byPave=IrmsVrmscosϕ,cosϕ is called the power factor, which ranges from 0 to 1.