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General Physics (PHY 2140)

Lecture 18Lecture 18Electricity and Magnetism

Induced voltages and inductionGenerators and motorsSelf-induction

Chapter 20

http://www.physics.wayne.edu/~apetrov/PHY2140/

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Lightning ReviewLightning Review

0

2IBr

µπ

=Last lecture:

1.1. Induced voltages and inductionInduced voltages and inductionInduced EMFInduced EMFFaraday’s lawFaraday’s lawMotional EMF

cosBA θΦ =

Nt

∆Φ= −

∆E

Motional EMF Blv=E

Review Problem: Two very long, fixed wires cross each other perpendicularly. They do not touch but are close to each other, as shown. Equal currents flow in the wires, in the directions shown. Indicate the locus of points where the net magnetic field is zero.

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Next week: Prof. Claude Next week: Prof. Claude PruneauPruneau

Lectures on Monday and WednesdayExam on Friday

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20.5 Generators20.5 Generators

Generators and motors are two of the most important Generators and motors are two of the most important applications of induced applications of induced emfemf (magnetic inductance). (magnetic inductance). A generator is something that converts mechanical A generator is something that converts mechanical energy to electrical energy. energy to electrical energy.

Alternating Current (AC) generatorAlternating Current (AC) generatorDirect Current (DC) generatorDirect Current (DC) generator

A motor does the opposite, it converts electrical energy A motor does the opposite, it converts electrical energy to mechanical energy. to mechanical energy.

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AC GeneratorsAC Generators

Basic operation of the generatorBasic operation of the generator

As the As the loop rotatesloop rotates, the , the magnetic magnetic flux through it changesflux through it changes with timewith timeThis This induces an emfinduces an emf and a and a current in the external circuitcurrent in the external circuitThe ends of the loop are The ends of the loop are connected to slip rings that rotate connected to slip rings that rotate with the loopwith the loopConnections to the external Connections to the external circuit are made by stationary circuit are made by stationary brushed in contact with the slip brushed in contact with the slip ringsrings

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AC generatorAC generator

Compute EMFCompute EMFIt is only generated in BC It is only generated in BC and DA wiresand DA wiresEMF generated in BC and EMF generated in BC and DA would beDA would be

Thus, total EMF isThus, total EMF is

If the loop is rotating with If the loop is rotating with ω

AB

CD

ω

vv sin θBlv⊥= =BC DAE E

B

2 2 sinBlv Blv θ⊥= =E

A

2 sin 2 sin2aBlv t Bl tω ω ω = =

E as v=rω=aω/2

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AC generator (cont)AC generator (cont)

Generalize the result to N loopsGeneralize the result to N loops

where we also noticed that where we also noticed that A=laA=la

Note: Note: is is reached when reached when ωωt=90t=90˚ or 270or 270˚ (loop (loop parallel to the magnetic field)parallel to the magnetic field)

sinNBA tω ω=E

NBAω=maxE

EMF generated by the AC generator

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DC generatorDC generator

By a clever change to the rings and brushes By a clever change to the rings and brushes of the ac generator, we can create a dc of the ac generator, we can create a dc generator, that is, a generator where the generator, that is, a generator where the polarity of the polarity of the emfemf is always positive. is always positive. The basic idea is to use The basic idea is to use a single split ring a single split ring instead of two complete ringsinstead of two complete rings. The split ring . The split ring is arranged so that, just as the is arranged so that, just as the emfemf is about is about to change sign from positive to negative, the to change sign from positive to negative, the brushes cross the gap, and the polarity of brushes cross the gap, and the polarity of the contacts is switched. the contacts is switched. The polarity of the contacts changes in The polarity of the contacts changes in phase with the polarity of the phase with the polarity of the emfemf ---- the two the two changes essentially cancel each other out, changes essentially cancel each other out, and the and the emfemf remains always positive. remains always positive. The The emfemf still varies still varies sinusoidallysinusoidally during each during each half cycle, but every half cycle is a positive half cycle, but every half cycle is a positive emfemf. .

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MotorsMotors

A motor is basically a generator running in reverse. A A motor is basically a generator running in reverse. A current is passed through the coil, producing a torque current is passed through the coil, producing a torque and causing the coil to rotate in the magnetic field. Once and causing the coil to rotate in the magnetic field. Once turning, the coil of the motor generates a turning, the coil of the motor generates a back back emfemf, just , just as does the coil of a generator. The back as does the coil of a generator. The back emfemf cancels cancels some of the applied some of the applied emfemf, and limits the current through , and limits the current through the coil. the coil.

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Motors and Back emfMotors and Back emf

The phrase The phrase back emfback emf is used is used for an emf that tends to reduce for an emf that tends to reduce the applied currentthe applied currentWhen a motor is turned on, When a motor is turned on, there is no back emf initiallythere is no back emf initiallyThe current is very large The current is very large because it is limited only by because it is limited only by the resistance of the coilthe resistance of the coil

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Example: coil in magnetic fieldExample: coil in magnetic field

A coil of area 0.10 m² is rotating at 60 rev/s with its axis of A coil of area 0.10 m² is rotating at 60 rev/s with its axis of rotation rotation perpendicular to a 0.20T magnetic field. (a) If there are 1000 tperpendicular to a 0.20T magnetic field. (a) If there are 1000 turns on urns on the coil, what is the maximum voltage induced in the coil? (b) Wthe coil, what is the maximum voltage induced in the coil? (b) When the hen the maximum induced voltage occurs, what is the orientation of the cmaximum induced voltage occurs, what is the orientation of the coil oil with respect to the magnetic field? with respect to the magnetic field?

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20.6 Eddy currents (application)20.6 Eddy currents (application)Magnetic Levitation (Maglev) TrainsMagnetic Levitation (Maglev) Trains

Induced surface (Induced surface (“eddy”“eddy”) currents produce field in opposite direction) currents produce field in opposite directionRepels magnetRepels magnetLevitates trainLevitates train

Maglev trains today can travel up to 310 mphMaglev trains today can travel up to 310 mphTwice the speed of AmtrakTwice the speed of Amtrak’’s fastest conventional train!s fastest conventional train!

May eventually use superconducting loops to produce BMay eventually use superconducting loops to produce B--fieldfieldNo power dissipation in resistance of wires!No power dissipation in resistance of wires!

NS

rails“eddy” current

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20.7 Self20.7 Self--inductanceinductance

When a current flows through a loop, the magnetic field created by that current has a magnetic flux through the area of the loop.If the current changes, the magnetic field changes, and so the flux changes giving rise to an induced emf. This phenomenon is called self-induction because it si the loop's own current, and not an external one, that gives rise to the induced emf.Faraday’s law states

Nt

∆Φ= −

∆E

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The magnetic flux is proportional to the magnetic field, The magnetic flux is proportional to the magnetic field, which is proportional to the current in the circuitwhich is proportional to the current in the circuitThus, the selfThus, the self--induced EMF must be proportional to the induced EMF must be proportional to the time rate of change of the currenttime rate of change of the current

where where LL is called the is called the inductance inductance of the deviceof the device

Units: SI: Units: SI: henryhenry (H)(H)

If flux is initially zero,

ILt

∆= −

∆E

1 1V sH A⋅=

If flux is initially zero, NL NI I

∆Φ Φ= =

∆

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Example: solenoidExample: solenoid

A solenoid of radius 2.5cm has 400 turns and a length of 20 cm. A solenoid of radius 2.5cm has 400 turns and a length of 20 cm. Find Find (a) its inductance and (b) the rate at which current must change(a) its inductance and (b) the rate at which current must changethrough it to produce an through it to produce an emfemf of 75mV. of 75mV.

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20.9 Energy stored in a magnetic field20.9 Energy stored in a magnetic fieldThe battery in any circuit that contains a coil has to do The battery in any circuit that contains a coil has to do work to produce a currentwork to produce a currentSimilar to the capacitor, any coil (or inductor) would store Similar to the capacitor, any coil (or inductor) would store potential energypotential energy

212LPE LI=

PEL = L I² / 2PEC = C V² / 20energy stored

00P = I V = I² R = V² / Rpower dissipated

emf = -L (∆I / ∆t)Q = C VV = I Rrelation

LCRsymbol

henry, H = V s / Afarad, F = C / Vohm, Ω = V / Aunits

InductorCapacitorResistorSummary of the properties of circuit elements.

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Example: stored energyExample: stored energy

A 24V battery is connected in series with a resistor and an induA 24V battery is connected in series with a resistor and an inductor, ctor, where R = 8.0W and L = 4.0H. Find the energy stored in the inducwhere R = 8.0W and L = 4.0H. Find the energy stored in the inductor tor (a) when the current reaches its maximum value and (b) one time (a) when the current reaches its maximum value and (b) one time constant after the switch is closed. constant after the switch is closed.