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The crucial quantity in induction is magnetic fluxΦΦ, defined to beΦ=BAcosθΦ=BAcosθ, whereBBis the magnetic field strength over an areaAAat an angleθθwith the perpendicular to the area.
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Units of magnetic fluxΦΦareTâ‹m2Tâ‹m2.
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Any change in magnetic fluxΦΦinduces an emf—the process is defined to be electromagnetic induction.
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Faraday’s law of induction states that the emfinduced by a change in magnetic flux isemf=−NΔΦΔtemf=−NΔΦΔtwhen flux changes byΔΦΔΦin a timeΔtΔt.
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when flux changes byΔΦΔΦin a timeΔtΔt.
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If emf is induced in a coil,NNis its number of turns.
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The minus sign means that the emf creates a currentIIand magnetic fieldBBthatoppose the change in fluxΔΦΔΦ—this opposition is known as Lenz’s law.
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An emf induced by motion relative to a magnetic fieldBBis called amotional emfand is given byemf=Bℓv(B,ℓ, andvperpendicular),emf=Bℓv(B,ℓ, andvperpendicular),whereℓℓis the length of the object moving at speedvvrelative to the field.
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Current loops induced in moving conductors are called eddy currents.
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They can create significant drag, called magnetic damping.
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An electric generator rotates a coil in a magnetic field, inducing an emfgiven as a function of time byemf=NABωsinωt,emf=NABωsinωt,whereAAis the area of anNN-turn coil rotated at a constant angular velocityωωin a uniform magnetic fieldBB.
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The peak emfemf0emf0of a generator isemf0=NABω.emf0=NABω.
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Any rotating coil will have an induced emf—in motors, this is called back emf, since it opposes the emf input to the motor.
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Transformers use induction to transform voltages from one value to another.
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For a transformer, the voltages across the primary and secondary coils are related byVsVp=NsNp,VsVp=NsNp,whereVpVpandVsVsare the voltages across primary and secondary coils havingNpNpandNsNsturns.
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The currentsIpIpandIsIsin the primary and secondary coils are related byIsIp=NpNsIsIp=NpNs.
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A step-up transformer increases voltage and decreases current, whereas a step-down transformer decreases voltage and increases current.
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Electrical safety systems and devices are employed to prevent thermal and shock hazards.
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Circuit breakers and fuses interrupt excessive currents to prevent thermal hazards.
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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.
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A ground fault interrupter (GFI) prevents shock by detecting the loss of current to unintentional paths.
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An isolation transformer insulates the device being powered from the original source, also to prevent shock.
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Many of these devices use induction to perform their basic function.
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Inductance is the property of a device that tells how effectively it induces an emf in another device.
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Mutual inductance is the effect of two devices in inducing emfs in each other.
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A change in currentΔI1/ΔtΔI1/Δtin one induces an emfemf2emf2in the second:emf2=−MΔI1Δt,emf2=−MΔI1Δt,whereMMis defined to be the mutual inductance between the two devices, and the minus sign is due to Lenz’s law.
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Symmetrically, a change in currentΔI2/ΔtΔI2/Δtthrough the second device induces an emfemf1emf1in the first:emf1=−MΔI2Δt,emf1=−MΔI2Δt,whereMMis the same mutual inductance as in the reverse process.
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Current changes in a device induce an emf in the device itself.
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Self-inductance is the effect of the device inducing emf in itself.
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The device is called an inductor, and the emf induced in it by a change in current through it isemf=−LΔIΔt,emf=−LΔIΔt,whereLLis the self-inductance of the inductor, andΔI/ΔtΔI/Δtis the rate of change of current through it. The minus sign indicates that emf opposes the change in current, as required by Lenzâ...
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The unit of self- and mutual inductance is the henry (H), where1 H=1 Ωâ‹s1 H=1 Ωâ‹s.
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The self-inductanceLLof an inductor is proportional to how much flux changes with current. For anNN-turn inductor,L=NΔΦΔI.L=NΔΦΔI.
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The self-inductance of a solenoid isL=μ0N2Aℓ(solenoid),L=μ0N2Aℓ(solenoid),whereNNis its number of turns in the solenoid,AAis its cross-sectional area,ℓℓis its length, andμ0=4π×10−7Tâ‹m/Aμ0=4π×10−7Tâ‹m/Ais the permeability of free space.
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The energy stored in an inductorEindEindisEind=12LI2.Eind=12LI2.
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When a series connection of a resistor and an inductor—anRLcircuit—is connected to a voltage source, the time variation of the current isI=I0(1−e−t/τ)(turning on).I=I0(1−e−t/τ)(turning on).whereI0=V/RI0=V/Ris the final current.
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The characteristic time constantττisτ=LRτ=LR, whereLLis the inductance andRRis the resistance.
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In the first time constantττ, the current rises from zero to0.632I00.632I0, and 0.632 of the remainder in every subsequent time intervalττ.
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When the inductor is shorted through a resistor, current decreases asI=I0e−t/τ(turning off).I=I0e−t/τ(turning off).HereI0I0is the initial current.
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Current falls to0.368I00.368I0in the first time intervalττ, and 0.368 of the remainder toward zero in each subsequent timeττ.
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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 a90º90ºphase angle.
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The opposition of an inductor to a change in current is expressed as a type of AC resistance.
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Ohm’s law for an inductor isI=VXL,I=VXL,whereVVis the rms voltage across the inductor.
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XLXLis defined to be the inductive reactance, given byXL=2πfL,XL=2πfL,withffthe frequency of the AC voltage source in hertz.
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Inductive reactanceXLXLhas units of ohms and is greatest at high frequencies.
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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 a90º90ºphase angle.
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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,I=VXC,whereVVis the rms voltage across the capacitor.
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XCXCis defined to be the capacitive reactance, given byXC=12πfC.XC=12πfC.
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XCXChas units of ohms and is greatest at low frequencies.
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The AC analogy to resistance is impedanceZZ, the combined effect of resistors, inductors, and capacitors, defined by the AC version of Ohm’s law:I0=V0ZorIrms=VrmsZ,I0=V0ZorIrms=VrmsZ,whereI0I0is the peak current andV0V0is the peak source voltage.
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Impedance has units of ohms and is given byZ=R2+(XL−XC)2Z=R2+(XL−XC)2.
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The resonant frequencyf0f0, at whichXL=XCXL=XC, isf0=12πLC.f0=12πLC.
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In an AC circuit, there is a phase angleϕϕbetween source voltageVVand the currentII, which can be found fromcosϕ=RZ,cosϕ=RZ,
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ϕ=0ºϕ=0ºfor a purely resistive circuit or anRLCcircuit at resonance.
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The average power delivered to anRLCcircuit is affected by the phase angle and is given byPave=IrmsVrmscosϕ,Pave=IrmsVrmscosϕ,cosϕcosϕis called the power factor, which ranges from 0 to 1.
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amplitude : the height, or magnitude, of an electromagnetic wave
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amplitude modulation (AM) : a method for placing information on electromagnetic waves by modulating the amplitude of a carrier wave with an audio signal, resulting in a wave with constant frequency but varying amplitude
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carrier wave : an electromagnetic wave that carries a signal by modulation of its amplitude or frequency
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electric field : a vector quantity (E); the lines of electric force per unit charge, moving radially outward from a positive charge and in toward a negative charge
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electric field lines : a pattern of imaginary lines that extend between an electric source and charged objects in the surrounding area, with arrows pointed away from positively charged objects and toward negatively charged objects. The more lines in the pattern, the stronger the electric field in that region
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electric field strength : the magnitude of the electric field, denotedE-field
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electromagnetic spectrum : the full range of wavelengths or frequencies of electromagnetic radiation
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electromagnetic waves : radiation in the form of waves of electric and magnetic energy
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electromotive force (emf) : energy produced per unit charge, drawn from a source that produces an electrical current
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extremely low frequency (ELF) : electromagnetic radiation with wavelengths usually in the range of 0 to 300 Hz, but also about 1kHz
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frequency : the number of complete wave cycles (up-down-up) passing a given point within one second (cycles/second)
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frequency modulation (FM) : a method of placing information on electromagnetic waves by modulating the frequency of a carrier wave with an audio signal, producing a wave of constant amplitude but varying frequency
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gamma ray : (γγray); extremely high frequency electromagnetic radiation emitted by the nucleus of an atom, either from natural nuclear decay or induced nuclear processes in nuclear reactors and weapons. The lower end of theγγ-ray frequency range overlaps the upper end of the X-ray range, butγγrays can have the hi...
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hertz : an SI unit denoting the frequency of an electromagnetic wave, in cycles per second
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infrared radiation (IR) : a region of the electromagnetic spectrum with a frequency range that extends from just below the red region of the visible light spectrum up to the microwave region, or from0.74μm0.74μmto300μm300μm
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intensity : the power of an electric or magnetic field per unit area, for example, Watts per square meter
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magnetic field : a vector quantity (B); can be used to determine the magnetic force on a moving charged particle
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magnetic field lines : a pattern of continuous, imaginary lines that emerge from and enter into opposite magnetic poles. The density of the lines indicates the magnitude of the magnetic field
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magnetic field strength : the magnitude of the magnetic field, denotedB-field
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maximum field strength : the maximum amplitude an electromagnetic wave can reach, representing the maximum amount of electric force and/or magnetic flux that the wave can exert
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Maxwell’s equations : a set of four equations that comprise a complete, overarching theory of electromagnetism
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microwaves : electromagnetic waves with wavelengths in the range from 1 mm to 1 m; they can be produced by currents in macroscopic circuits and devices
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oscillate : to fluctuate back and forth in a steady beat
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radar : a common application of microwaves. Radar can determine the distance to objects as diverse as clouds and aircraft, as well as determine the speed of a car or the intensity of a rainstorm
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radio waves : electromagnetic waves with wavelengths in the range from 1 mm to 100 km; they are produced by currents in wires and circuits and by astronomical phenomena
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resonant : a system that displays enhanced oscillation when subjected to a periodic disturbance of the same frequency as its natural frequency
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RLCcircuit : an electric circuit that includes a resistor, capacitor and inductor
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speed of light : in a vacuum, such as space, the speed of light is a constant 3 x 108m/s
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standing wave : a wave that oscillates in place, with nodes where no motion happens
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thermal agitation : the thermal motion of atoms and molecules in any object at a temperature above absolute zero, which causes them to emit and absorb radiation
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transverse wave : a wave, such as an electromagnetic wave, which oscillates perpendicular to the axis along the line of travel
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TV : video and audio signals broadcast on electromagnetic waves
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ultra-high frequency (UHF) : TV channels in an even higher frequency range than VHF, of 470 to 1000 MHz
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ultraviolet radiation (UV) : electromagnetic radiation in the range extending upward in frequency from violet light and overlapping with the lowest X-ray frequencies, with wavelengths from 400 nm down to about 10 nm
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very high frequency (VHF) : TV channels utilizing frequencies in the two ranges of 54 to 88 MHz and 174 to 222 MHz
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visible light : the narrow segment of the electromagnetic spectrum to which the normal human eye responds
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wavelength : the distance from one peak to the next in a wave
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X-ray : invisible, penetrating form of very high frequency electromagnetic radiation, overlapping both the ultraviolet range and theγγ-ray range
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Electromagnetic waves consist of oscillating electric and magnetic fields and propagate at the speed of lightcc. They were predicted by Maxwell, who also showed thatc=1μ0ε0,c=1μ0ε0,whereμ0μ0is the permeability of free space andε0ε0is the permittivity of free space.
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whereμ0μ0is the permeability of free space andε0ε0is the permittivity of free space.
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Maxwell’s prediction of electromagnetic waves resulted from his formulation of a complete and symmetric theory of electricity and magnetism, known as Maxwell’s equations.
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These four equations are paraphrased in this text, rather than presented numerically, and encompass the major laws of electricity and magnetism. First is Gauss’s law for electricity, second is Gauss’s law for magnetism, third is Faraday’s law of induction, including Lenz’s law, and fourth is Ampere’s law in a...
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Electromagnetic waves are created by oscillating charges (which radiate whenever accelerated) and have the same frequency as the oscillation.
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Since the electric and magnetic fields in most electromagnetic waves are perpendicular to the direction in which the wave moves, it is ordinarily a transverse wave.
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The strengths of the electric and magnetic parts of the wave are related byEB=c,EB=c,which implies that the magnetic fieldBBis very weak relative to the electric fieldEE.
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which implies that the magnetic fieldBBis very weak relative to the electric fieldEE.
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