Chapter 23

1. Chapter 23 Faraday’s Law and Induction

2. Michael Faraday • 1791 – 1867 • Great experimental physicist • Contributions to early electricity include • Invention of motor, generator, and transformer • Electromagnetic induction • Laws of electrolysis

3. Induction • An induced current is produced by a changing magnetic field • There is an induced emf associated with the induced current • A current can be produced without a battery present in the circuit • Faraday’s Law of Induction describes the induced emf

4. EMF Produced by a Changing Magnetic Field, 1 • A loop of wire is connected to a sensitive ammeter • When a magnet is moved toward the loop, the ammeter deflects • The deflection indicates a current induced in the wire

5. EMF Produced by a Changing Magnetic Field, 2 • When the magnet is held stationary, there is no deflection of the ammeter • Therefore, there is no induced current • Even though the magnet is inside the loop

6. EMF Produced by a Changing Magnetic Field, 3 • The magnet is moved away from the loop • The ammeter deflects in the opposite direction

7. EMF Produced by a Changing Magnetic Field, Summary • The ammeter deflects when the magnet is moving toward or away from the loop • The ammeter also deflects when the loop is moved toward or away from the magnet • An electric current is set up in the coil as long as relative motion occurs between the magnet and the coil • This is the induced current that is produced by an induced emf

8. Faraday’s Experiment – Set Up • A primary coil is connected to a switch and a battery • The wire is wrapped around an iron ring • A secondary coil is also wrapped around the iron ring • There is no battery present in the secondary coil • The secondary coil is not electrically connected to the primary coil

9. Faraday’s Experiment – Findings • At the instant the switch is closed, the galvanometer (ammeter) needle deflects in one direction and then returns to zero • When the switch is opened, the galvanometer needle deflects in the opposite direction and then returns to zero • The galvanometer reads zero when there is a steady current or when there is no current in the primary circuit

10. Faraday’s Experiment – Conclusions • An electric current can be produced by a time-varying magnetic field • This would be the current in the secondary circuit of this experimental set-up • The induced current exists only for a short time while the magnetic field is changing • This is generally expressed as: an induced emf is produced in the secondary circuit by the changing magnetic field • The actual existence of the magnetic field is not sufficient to produce the induced emf, the field must be changing

11. Magnetic Flux • To express Faraday’s finding mathematically, the magnetic flux is used • The flux depends on the magnetic field and the area:

12. Flux and Induced emf • An emf is induced in a circuit when the magnetic flux through the surface bounded by the circuit changes with time • This summarizes Faraday’s experimental results

13. Faraday’s Law – Statements • Faraday’s Law of Induction states that the emf induced in a circuit is equal to the time rate of change of the magnetic flux through the circuit • Mathematically,

14. Faraday’s Law – Statements, cont • If the circuit consists of N identical and concentric loops, and if the field lines pass through all loops, the induced emf becomes • The loops are in series, so the emfs in the individual loops add to give the total emf

15. Faraday’s Law – Example • Assume a loop enclosing an area A lies in a uniform magnetic field • The magnetic flux through the loop is FB = B A cos q • The induced emf is

16. Ways of Inducing an emf • The magnitude of the field can change with time • The area enclosed by the loop can change with time • The angle q between the field and the normal to the loop can change with time • Any combination of the above can occur

17. Applications of Faraday’s Law – Pickup Coil • The pickup coil of an electric guitar uses Faraday’s Law • The coil is placed near the vibrating string and causes a portion of the string to become magnetized • When the string vibrates at the some frequency, the magnetized segment produces a changing flux through the coil • The induced emf is fed to an amplifier

18. Motional emf • A motional emf is one induced in a conductor moving through a magnetic field • The electrons in the conductor experience a force that is directed along l

19. Motional emf, cont • Under the influence of the force, the electrons move to the lower end of the conductor and accumulate there • As a result of the charge separation, an electric field is produced inside the conductor • The charges accumulate at both ends of the conductor until they are in equilibrium with regard to the electric and magnetic forces

20. Motional emf, final • For equilibrium, q E = q v B or E = v B • A potential difference is maintained between the ends of the conductor as long as the conductor continues to move through the uniform magnetic field • If the direction of the motion is reversed, the polarity of the potential difference is also reversed

21. Sliding Conducting Bar • A bar moving through a uniform field and the equivalent circuit diagram • Assume the bar has zero resistance • The work done by the applied force appears as internal energy in the resistor R

22. Sliding Conducting Bar, cont • The induced emf is • Since the resistance in the circuit is R, the current is

23. Sliding Conducting Bar, Energy Considerations • The applied force does work on the conducting bar • This moves the charges through a magnetic field • The change in energy of the system during some time interval must be equal to the transfer of energy into the system by work • The power input is equal to the rate at which energy is delivered to the resistor

24. AC Generators • Electric generators take in energy by work and transfer it out by electrical transmission • The AC generator consists of a loop of wire rotated by some external means in a magnetic field

25. Induced emf In an AC Generator • The induced emf in the loops is • This is sinusoidal, with emax = N A B w

26. Lenz’ Law • Faraday’s Law indicates the induced emf and the change in flux have opposite algebraic signs • This has a physical interpretation that has come to be known as Lenz’ Law • It was developed by a German physicist, Heinrich Lenz

27. Lenz’ Law, cont • Lenz’ Law states the polarity of the induced emf in a loop is such that it produces a current whose magnetic field opposes the change in magnetic flux through the loop • The induced current tends to keep the original magnetic flux through the circuit from changing

28. Lenz’ Law – Example 1 • When the magnet is moved toward the stationary loop, a current is induced as shown in a • This induced current produces its own magnetic field that is directed as shown in b to counteract the increasing external flux

29. Lenz’ Law – Example 2 • When the magnet is moved away the stationary loop, a current is induced as shown in c • This induced current produces its own magnetic field that is directed as shown in d to counteract the decreasing external flux

30. Induced emf and Electric Fields • An electric field is created in the conductor as a result of the changing magnetic flux • Even in the absence of a conducting loop, a changing magnetic field will generate an electric field in empty space • This induced electric field has different properties than a field produced by stationary charges

31. Induced emf and Electric Fields, cont • The emf for any closed path can be expressed as the line integral of over the path • Faraday’s Law can be written in a general form

32. Induced emf and Electric Fields, final • The induced electric field is a nonconservative field that is generated by a changing magnetic field • The field cannot be an electrostatic field because if the field were electrostatic, and hence conservative, the line integral would be zero and it isn’t

33. Self-Induction • When the switch is closed, the current does not immediately reach its maximum value • Faraday’s Law can be used to describe the effect

34. Self-Induction, 2 • As the current increases with time, the magnetic flux through the circuit loop due to this current also increases with time • The corresponding flux due to this current also increases • This increasing flux creates an induced emf in the circuit

35. Self-Inductance, 3 • The direction of the induced emf is such that it would cause an induced current in the loop which would establish a magnetic field opposing the change in the original magnetic field • The direction of the induced emf is opposite the direction of the emf of the battery • Sometimes called a back emf • This results in a gradual increase in the current to its final equilibrium value

36. Self-Induction, 4 • This effect is called self-inductance • Because the changing flux through the circuit and the resultant induced emf arise from the circuit itself • The emf eL is called a self-induced emf

37. Self-Inductance, Equations • An induced emf is always proportional to the time rate of change of the current • L is a constant of proportionality called the inductance of the coil • It depends on the geometry of the coil and other physical characteristics

38. Inductance of a Coil • A closely spaced coil of N turns carrying current I has an inductance of • The inductance is a measure of the opposition to a change in current • Compared to resistance which was opposition to the current

39. Inductance Units • The SI unit of inductance is a Henry (H) • Named for Joseph Henry

40. Joseph Henry • 1797 – 1878 • Improved the design of the electromagnet • Constructed one of the first motors • Discovered the phenomena of self-inductance

41. Inductance of a Solenoid • Assume a uniformly wound solenoid having N turns and length l • Assume l is much greater than the radius of the solenoid • The interior magnetic field is

42. Inductance of a Solenoid, cont • The magnetic flux through each turn is • Therefore, the inductance is • This shows that L depends on the geometry of the object

43. RL Circuit, Introduction • A circuit element that has a large self-inductance is called an inductor • The circuit symbol is • We assume the self-inductance of the rest of the circuit is negligible compared to the inductor • However, even without a coil, a circuit will have some self-inductance

44. RL Circuit, Analysis • An RL circuit contains an inductor and a resistor • When the switch is closed (at time t=0), the current begins to increase • At the same time, a back emf is induced in the inductor that opposes the original increasing current

45. RL Circuit, Analysis, cont • Applying Kirchhoff’s Loop Rule to the previous circuit gives • Looking at the current, we find

46. RL Circuit, Analysis, Final • The inductor affects the current exponentially • The current does not instantly increase to its final equilibrium value • If there is no inductor, the exponential term goes to zero and the current would instantaneously reach its maximum value as expected

47. RL Circuit, Time Constant • The expression for the current can also be expressed in terms of the time constant, t, of the circuit • where t = L / R • Physically, t is the time required for the current to reach 63.2% of its maximum value

48. RL Circuit, Current-Time Graph, 1 • The equilibrium value of the current is e/R and is reached as t approaches infinity • The current initially increases very rapidly • The current then gradually approaches the equilibrium value

49. RL Circuit, Current-Time Graph, 2 • The time rate of change of the current is a maximum at t = 0 • It falls off exponentially as t approaches infinity • In general,

50. Energy in a Magnetic Field • In a circuit with an inductor, the battery must supply more energy than in a circuit without an inductor • Part of the energy supplied by the battery appears as internal energy in the resistor • The remaining energy is stored in the magnetic field of the inductor