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# Chapter 20

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1. Chapter 20 Induced Voltages and Inductance

2. 20.1 Induced emf • A current can be produced by a changing magnetic field [B=f (t)], i.e., B varies over time • First shown in an experiment by Michael Faraday • A primary coil is connected to a battery • A secondary coil is connected to an ammeter

3. Michael Faraday • Faraday is often regarded as the greatest experimental scientist of the 1800s. His contributions to the study of electricity include the invention of the electric motor, generator, and transformer.

4. Faraday’s Experiment • The purpose of the secondary circuit is to detect current that might be produced by the magnetic field • When the switch is closed, the ammeter deflects in one direction and then returns to zero • When the switch is opened, the ammeter deflects in the opposite direction and then returns to zero • When there is a steady current in the primary circuit, the ammeter reads zero

5. Faraday’s Conclusions • An electrical current is produced by a changing magnetic field • It is customary to say that an induced emf is produced in the secondary circuit by the changing magnetic field

6. Magnetic Flux • The emf is actually induced by a change in the quantity called the magnetic flux rather than simply by a change in the magnetic field • Magnetic flux is defined in a manner similar to that of electrical flux • Magnetic flux is proportional to both the strength of the magnetic field passing through the plane of a wire loop wire and the area of the loop

7. Magnetic Flux, 2 • You are given a loop of wire • The wire is in an uniform magnetic field B • The loop has an area A • The flux is defined as • ΦB = BA = B A cos θ • θ is the angle between B and the normal to the plane

8. Magnetic Flux, 3 • (a) When the field is perpendicular to the plane of the loop, θ = 0 and ΦB =ΦB, max = BA • (b) When the field is parallel to the plane of the loop, θ = 90° and ΦB = 0 • The flux can be negative, for example if θ = 180° • SI units of flux are T m² = Wb (Weber)

9. Magnetic Flux, final • The flux can be visualized with respect to magnetic field lines • The value of the magnetic flux is proportional to the total number of lines passing through the loop • When the area is perpendicular to the lines, the maximum number of lines pass through the area and the flux is a maximum • When the area is parallel to the lines, no lines pass through the area and the flux is 0

10. 20.2 Electromagnetic Induction • When a magnet moves toward a loop of wire, the ammeter shows the presence of a current (a) • When the magnet is held stationary, there is no current (b) • When the magnet moves away from the loop, the ammeter shows a current in the opposite direction (c) • If the loop is moved instead of the magnet, a current is also detected

11. Electromagnetic Induction – Results of the Experiment • A current is set up in the circuit as long as there is relative motion between the magnet and the loop • The same experimental results are found whether the loop moves or the magnet moves • The current is called an induced current because it is produced by an induced emf

12. Faraday’s Law and Electromagnetic Induction • The instantaneous emf induced in a circuit equals the time rate of change of magnetic flux through the circuit • If a circuit contains N tightly wound loops and the flux through each loop changes by ΔΦ during an interval Δt, the average emf induced is given by Faraday’s Law:

13. Faraday’s Law and Lenz’ Law • The minus sign is included because of the polarity of the emf. The induced emf in the coil gives rise to a current whose magnetic field OPPOSES (Lenz’s law) the change in magnetic flux that produced it

14. There are three possibilities to produce an emf 1) Time-varying magnetic field e=-N[(A cos)(DB/Dt)+ 2) Time-varying loop area +(B cos )(DA/Dt)+  3) Turning of the loop (generator) +BA(D[cos]/Dt)]

15. Applications of Faraday’s Law – Ground Fault Interrupters • The ground fault interrupter (GFI) is a safety device that protects against electrical shock • Wire 1 leads from the wall outlet to the appliance • Wire 2 leads from the appliance back to the wall outlet • The iron ring confines the magnetic field, which is generally 0 • If a leakage occurs, the field is no longer 0 and the induced voltage triggers a circuit breaker shutting off the current

16. Applications of Faraday’s Law – Electric Guitar • A vibrating string induces an emf in a coil • A permanent magnet inside the coil magnetizes a portion of the string nearest the coil • As the string vibrates at some frequency, its magnetized segment produces a changing flux through the pickup coil • The changing flux produces an induced emf that is fed to an amplifier

17. Applications of Faraday’s Law – Apnea Monitor • The coil of wire attached to the chest carries an alternating current • An induced emf produced by the varying field passes through a pick up coil • When breathing stops, the pattern of induced voltages stabilizes and external monitors sound an alert

18. 20.3 Application of Faraday’s Law – Motional emf • A straight conductor of length ℓ moves perpendicularly with constant velocity through a uniform field • The electrons in the conductor experience a magnetic force • F = q v B • The electrons tend to move to the lower end of the conductor ℓ

19. Motional emf • As the negative charges accumulate at the base, a net positive charge exists at the upper end of the conductor • As a result of this charge separation, an electric field is produced in the conductor • Charges build up at the ends of the conductor until the downward magnetic force is balanced by the upward electric force • There is a potential difference between the upper and lower ends of the conductor

20. Motional emf, cont. V =Eℓ F=qvB V=Bℓv, voltage across the conductor • If the motion is reversed, the polarity of the potential difference is also reversed F=qE=q (V/ℓ )=qvB

21. Magnitude of the Motional emf

22. Motional emf in a Circuit • A conducting bar sliding with v along two conducting rails under the action of an applied force Fapp. The magnetic force Fm opposes the motion, and a counterclockwise current is induced.

23. Motional emf in a Circuit, cont. • The changing magnetic flux through the loop and the corresponding induced emf in the bar result from the change in area of the loop • The induced, motional emf, acts like a battery in the circuit

24. Example: Operating a light bulb Rod and rail have negligible resistance but the bulb has a resistance of 96 W, B=0.80 T, v=5.0 m/s and ℓ=1.6 m. Calculate (a) emf in the rod, (b) induced current (c) power delivered to the bulb and (d) the energy used by the bulb in 60 s. (a) e=vBℓ e =(5.0 m/s)(0.80 T)(1.6 m)=6.4 V (b) I=e/R I=(6.4V)/(96 W)=0.067 A (c) P=eI P=eI=(6.4 V)(0.067 A)=0.43 W (d) E=Pt E=(0.43 W)(60 s)=26 J (=26 Ws)

25. 20.4 Lenz’ Law Revisited – Moving Bar Example • As the bar moves to the right, the magnetic flux through the circuit increases with time because the area of the loop increases • The induced current must be in a direction such that it opposes the change in the external magnetic flux

26. Lenz’ Law, Bar Example, cont • The flux due to the external field is increasing into the page • The flux due to the induced current must be out of the page • Therefore the current must be counterclockwise when the bar moves to the right

27. Lenz’ Law, Bar Example, final • The bar is moving toward the left • The magnetic flux through the loop is decreasing with time • The induced current must be clockwise to to produce its own flux into the page

28. Lenz’ Law Revisited, Conservation of Energy • Assume the bar is moving to the right • Assume the induced current is clockwise • The magnetic force on the bar would be to the right • The force would cause an acceleration and the velocity would increase • This would cause the flux to increase and the current to increase and the velocity to increase… • This would violate Conservation of Energy and so therefore, the current must be counterclockwise

29. Lenz’ Law, Moving Magnet Example • (a) A bar magnet is moved to the right toward a stationary loop of wire. As the magnet moves, the magnetic flux increases with time • (b) The induced current produces a flux to the left to counteract the increasing external flux to the right

30. Lenz’ Law, Final Note • When applying Lenz’ Law, there are two magnetic fields to consider • The external changing magnetic field that induces the current in the loop • The magnetic field produced by the current in the loop

31. Application – Tape Recorder • A magnetic tape moves past a recording and playback head • The tape is a plastic ribbon coated with iron oxide or chromium oxide

32. Application – Tape Recorder, cont. • To record, the sound is converted to an electrical signal which passes to an electromagnet that magnetizes the tape in a particular pattern • To playback, the magnetized pattern is converted back into an induced current driving a speaker