Chapter 21

1. Chapter 21 Magnetic Induction

2. Magnetic Induction • Electric and magnetic forces both act only on particles carrying an electric charge • Moving electric charges create a magnetic field • A changing magnetic field creates an electric field • This effect is called magnetic induction • This links electricity and magnetism in a fundamental way • Magnetic induction is also the key to many practical applications Introduction

3. Electromagnetism • Electric and magnetic phenomena were connected by Ørsted in 1820 • He discovered an electric current in a wire can exert a force on a compass needle • Indicated a electric field can lead to a force on a magnet • He concluded an electric field can produce a magnetic field • Did a magnetic field produce an electric field? • Experiments were done by Michael Faraday Section 21.1

4. Faraday’s Experiment • Faraday attempted to observe an induced electric field • He used an ammeter instead of a light bulb • If the bar magnet was in motion, a current was observed • If the magnet is stationary, the current and the electric field are both zero Section 21.1

5. Another Faraday Experiment • A solenoid is positioned near a loop of wire with the light bulb • He passed a current through the solenoid by connecting it to a battery • When the current through the solenoid is constant, there is no current in the wire • When the switch is opened or closed, the bulb does light up Section 21.1

6. Conclusions from Experiments • An electric current is produced during those instances when the current through the solenoid is changing • Faraday’s experiments show that an electric current is produced in the wire loop only when the magnetic field at the loop is changing • A changing magnetic field produces an electric field • An electric field produced in this way is called an induced electric field • The phenomena is called electromagnetic induction Section 21.1

7. Magnetic Flux • Faraday developed a quantitative theory of induction now called Faraday’s Law • The law shows how to calculate the induced electric field in different situations • Faraday’s Law uses the concept of magnetic flux • Magnetic flux is similar to the concept of electric flux • Let A be an area of a surface with a magnetic field passing through it • The flux is ΦB = B A cos θ Section 21.2

8. Magnetic Flux, cont. • If the field is perpendicular to the surface, ΦB = B A • If the field makes an angle θ with the normal to the surface, ΦB = B A cos θ • If the field is parallel to the surface, ΦB = 0 Section 21.2

9. Magnetic Flux, final • The magnetic flux can be defined for any surface • A complicated surface can be broken into small regions and the definition of flux applied • The total flux is the sum of the fluxes through all the individual pieces of the surface • The surfaces of interest are open surfaces • With electric flux, closed surfaces were used • The SI unit of magnetic flux is the Weber (Wb) • 1 Wb = 1 T . m2 Section 21.2

10. Faraday’s Law • Faraday’s Law indicates how to calculate the potential difference that produces the induced current • Written in terms of the electromotive force induced in the wire loop • The magnitude of the induced emf equals the rate of change of the magnetic flux • The negative sign is Lenz’s Law Section 21.2

11. Applying Faraday’s Law • The ε is the induced emf in the wire loop • Its value will be indicated on the voltmeter • It is related to the electric field directly along and inside the wire loop • The induced potential difference produces the current

12. Applying Faraday’s Law, cont. • The emf is produced by changes in the magnetic flux through the circuit • A constant flux does not produce an induced voltage • The flux can change due to • Changes in the magnetic field • Changes in the area • Changes in the angle • The voltmeter will indicate the direction of the induced emf and induced current and electric field Section 21.2

13. Faraday’s Law, Summary • Only changes in the magnetic flux matter • Rapid changes in the flux produce larger values of emf than do slow changes • This dependency on frequency means the induced emf plays an important role in AC circuits • The magnitude of the emf is proportional to the rate of change of the flux • If the rate is constant, then the emf is constant • In most cases, this isn’t possible and AC currents result • The induced emf is present even if there is no current in the path enclosing an area of changing magnetic flux Section 21.2

14. Flux Though a Changing Area • A magnetic field is constant and in a direction perpendicular to the plane of the rails and the bar • Assume the bar moves at a constant speed • The magnitude of the induced emf is ε = B L v • The current leads to power dissipation in the circuit Section 21.2

15. Conservation of Energy • The mechanical power put into the bar by the external agent is equal to the electrical power delivered to the resistor • Energy is converted from mechanical to electrical, but the total energy remains the same • Conservation of energy is obeyed by electromagnetic phenomena Section 21.2

16. Electrical Generator • Need to make the rate of change of the flux large enough to give a useful emf • Use rotational motion instead of linear motion • A permanent magnet produces a constant magnetic field in the region between its poles Section 21.2

17. Generator, cont. • A wire loop is located in the region of the field • The loop has a fixed area, but is mounted on a rotating shaft • The angle between the field and the plane of the loop changes as the loop rotates • If the shaft rotates with a constant angular velocity, the flux varies sinusoidally with time • This basic design could generate about 70 V so it is a practical design Section 21.2

18. Changing a Magnetic Flux, Summary • A change in magnetic flux and therefore an induced current can be produced in four ways • If the magnitude of the magnetic field changes with time • If the area changes with time • If the loop rotates so that the angle changes with time • If the loop moves from one region to another and the magnitude of the field is different in the two regions Section 21.2

19. Changing a Magnetic Flux, Summary, cont. Section 21.2

20. Lenz’s Law • Lenz’s Law gives an easy way to determine the sign of the induced emf • Lenz’s Law states the magnetic field produced by an induced current always opposes any changes in the magnetic flux Section 21.3

21. Lenz’s Law, Example 1 • Assume a metal loop in which the magnetic field passes upward through it • Assume the magnetic flux increases with time • The magnetic field produced by the induced emf must oppose the change in flux • Therefore, the induced magnetic field must be downward and the induced current will be clockwise Section 21.3

22. Lenz’s Law, Example 2 • Assume a metal loop in which the magnetic field passes upward through it • Assume the magnetic flux decreases with time • The magnetic field produced by the induced emf must oppose the change in flux • Therefore, the induced magnetic field must be upward and the induced current will be counterclockwise Section 21.3

23. Problem Solving Strategy • Recognize the principle • The induced emf always opposes changes in flux through the Lenz’s Law loop or path • Sketch the problem • Show the closed path that runs along the perimeter of a surface crossed by the magnetic field lines • Identify • Is the magnetic flux increasing or decreasing with time? Section 21.3

24. Problem Solving Strategy, cont. • Solve • Treat the perimeter of the surface as a wire loop • Suppose there is a current in the loop • Determine the direction of the resulting magnetic field • Find the current direction for which this induced magnetic field opposes the change in the magnetic flux • This current direction gives the sign (direction) of the induced emf • Check • Consider what your answer means • Check that your answer makes sense Section 21.3

25. Lenz’s Law and Conservation of Energy • Mathematically, Lenz’s Law is just the negative sign in Faraday’s Law • It is actually a consequence of conservation of energy • Therefore, conservation of energy is contained in Faraday’s Law • Nowhere in the laws of electricity and magnetism is there any explicit mention of energy or conservation of energy • Physicists believe all laws of physics must satisfy the principle of conservation of energy Section 21.3

26. Inductance • In some cases, you must include the induced flux • When the switch is closed, a sudden change in current occurs in the coil • This current produces a magnetic field • An emf and current are induced in the coil Section 21.4

27. Inductor • A coil is type of circuit element called an inductor • Many inductors are constructed as small solenoids • Almost any coil or loop will act as an inductor • Whenever the current through an inductor changes, a voltage is induced in the inductor that opposes this change • This phenomenon is called self-inductance • The current changing through a coil induces a current in the same coil • The induced current opposes the original applied current, from Lenz’s Law Section 21.4

28. Inductance of a Solenoid • Faraday’s Law can be used to find the inductance of a solenoid • L is the symbol for inductance • The voltage across the solenoid can be expressed in terms of the inductance Section 21.4

29. Inductance, final • The results apply to all coils or loops of wire • The value of L depends on the physical size and shape of the circuit element • The voltage drop across an inductor is • The unit of inductance is the Henry • 1 H = 1 V . s / A Section 21.4

30. Mutual Inductance • It is possible for the magnetic field of one coil to produce an induced current in a second coil • The coils are connected indirectly through the magnetic flux • The effect is called mutual inductance Section 21.4

31. RL Circuit • DC circuits may contain resistors, inductors, and capacitors • The voltage source is a battery or some other source that provides a constant voltage across its output terminals • Behavior of DC circuits with inductors • Immediately after any switch is closed or opened, the induced emfs keep the current through all inductors equal to the values they had the instant before the switch was thrown • After a switch has been closed or opened for a very long time, the induced emfs are zero Section 21.5

32. RL Circuit Example Section 21.5

33. RL Circuit Example, Analysis • The presence of resistors and an inductor make the circuit an RL circuit • The current starts at zero since the switch has been open for a very long time • At t = 0, the switch is closed, inducing a potential across the inductor • Just after t = 0, the current in the second loop is zero • After the switch has been closed for a long time, the voltage across the inductor is zero Section 21.5

34. Time Constant for RL Circuit • The current at time t is found by • τ is called the time constant of the circuit • For a single resistor in series with a single inductor, τ = L / R • The voltage is given by VL = V e-t/τ Section 21.5

35. Real Inductors • Most practical inductors are constructed by wrapping a wire coil around a magnetic material • Filling a coil with magnetic material greatly increases the magnetic flux through the coil and therefore increases the induced emf • The presence of magnetic material increases the inductance • Most inductors contain a magnetic material inside which produces a larger value of L in a smaller package Section 21.5

36. Energy in an Inductor • Energy is stored in the magnetic field of an inductor • The energy stored in an inductor is PEind = ½ L I2 • Very similar in form to the energy stored in the electric field of a capacitor • The expression for energy can also be stated as • In terms of the magnetic field, Section 21.6

37. Energy in an Inductor, cont. • Energy contained in the magnetic field actually exists anywhere there is a magnetic field, not just in a solenoid • Can exist in “empty” space • The potential energy can also be expressed in terms of the energy density in the magnetic field • This expression is similar to the energy density contained in an electric field Section 21.6

38. Bicycle Odometers • An odometer control unit is shown • A permanent magnet is attached to a wheel • A pickup coil is mounted on the axle support • When the magnet passes over the pickup coil, a pulse is generated • A computer keep tracks of the number of pulses Section 21.7

39. Ground Fault Interrupters • A ground fault interrupter (GFI) is a safety device used in many household circuits • It uses Faraday’s Law along with an electromechanical relay • The relay uses the current through a coil to exert a force on a magnetic metal bar in a switch Section 21.7

40. GFI, cont. • During normal operation, there is zero magnetic field in the relay • If the current in the return coil is smaller, a non-zero magnetic field opens the relay switch and the current turns off

41. Electric Guitars • An electric guitar uses Faraday’s Law to sense the motion of the strings • The metal string passes near a pickup coil wound around a permanent magnet • As the string vibrates, it produces a changing magnetic flux • The resulting emf is sent to an amplifier and the signal can be played through speakers Section 21.7

42. Generators, Motors and Cars • Motors and generators provide examples of conservation of energy and the conversion of energy from one type to another • A hybrid car contains two motors and a generator • The hybrid car “recaptures” some of the energy normally converted to heat when braking and stores it in batteries • A hybrid car is a practical example of the conversion between mechanical and electrical energy Section 21.7

43. Induction from a Distance • Assume a very long solenoid is inserted at the center of a single loop of wire • The field from the solenoid at the outer loop is essentially zero Section 21.8

44. Induction from a Distance, cont. • The field inside the solenoid at the center of the loop still produces a magnetic flux through the inner portion of the loop • Energy is transferred across the empty space between the two conductors • The energy is carried from the solenoid to the outer loop by an electromagnetic wave Section 21.8