Chapter 21

1. Chapter 21 Magnetic Induction February 18th, 2013

2. REVISED review sessions for exam 1 All review sessions are 2 hours Saturday, Feb 16th, 10:00 AM, Chamberlin 3320, Pankhuri Saturday, Feb 16th, 5:00 PM, Chamberlin 3320, Hiren Patel Monday, Feb 18th, 10:00 AM, Chamberlin 3328, James Osborne Monday, Feb 18th, 4:35 PM, Chamberlin 3320, JunghaKim Tuesday, Feb 19th, 7:45 AM, Chamberlin 3320, Laura Dodd Tuesday, Feb 19th, 12:05 PM, Chamberlin 3328, Subin Lee Tuesday, Feb 19th, 2:25 PM, Chamberlin 3328, Nitin Ramesh Tuesday, Feb 19th, 4:35 PM, Chamberlin 3328, Nate Woods Wednesday, Feb 20th, 12:00 PM, Chamberlin 3320, Greg Lau Wednesday, Feb 20th, 4:35 PM, Chamberlin 3328, AbhishekAggarwal

3. Magnetic Induction • A changing magnetic field creates an electric field • This effect is called magnetic induction • This links electricity and magnetism • Magnetic induction is also the key to many practical applications

4. Faraday’s Experiment

5. Another Faraday Experiment Isolenoid = constant Isolenoid changes for an instant when the switch is flipped

6. Magnetic Flux • From these experiments, Faraday developed a quantitative theory now called Faraday’s Law • Faraday’s Law uses the concept of magnetic flux • Let A be an area of a surface with a magnetic field passing through it • The flux is ΦB = B A cosθ • The unit of magnetic flux is the Weber (Wb): 1 Wb = 1 T . m2

7. Faraday’s Law • Faraday’s Law tells us how to calculate the potential difference that produces the induced current • Written in terms of the voltage (emf, or e) induced in the wire loop • The magnitude of the voltage equals the rate of change of the magnetic flux • The negative sign is Lenz’s Law Section 21.2

8. demo: bar magnet trough a solenoid induces a current in the solenoid, measure by the galvanometer as the bar magnet is moved, not when it is stationary Faraday’s Law magnetic flux ΦB = B A cosθ

9. demo: jumping ring. When an AC current flows through the solenoid, B1 is constantly changing, thus a current I2is induced in the Al ring, which in turn generates a B2, opposite to B1. Inserting an Fe core makes B1 larger. Cooling the Al ring makes I2 larger (7x larger at LN2 temperature 77°K, or -196°C!) Faraday’s Law magnetic flux ΦB = B A cosθ

10. Applying Faraday’s Law Faraday’s Law magnetic flux ΦB = B A cosθ • All 3 parameters in the magnetic flux may change: B, A, or θ • If they do, there will be an induced voltage and an induced current • The current will generate a magnetic field, which will opposethe initial change in B, A, or θ • A voltmeter indicates the direction of the induced voltage, and therefore the induced current, and therefore the induced magnetic field

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

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

13. Lenz’s Law, Example 1 • TThemagnetic 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, seen from the top.

14. 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 downward and the induced current will be counterclockwise

15. Problem 21.13 A square loop of wire (edge length 45 cm) is found to have an induced emf = 1.2 V at t = 0. (a) Assuming B is perpendicular to the loop, and B varies linearly with time, what is the change in the magnetic field from t = 0 to t = 0.1 s? (b) why is it only possible to calculate the change in B and not its absolute magnitude?

16. Inductance of a Solenoid • Faraday’s Law can be used to find the inductance of a solenoid • L is the symbol for inductance • The unit of inductance is the Henry: 1 H = 1 V . s / A • The voltage across the solenoid can be expressed in terms of the inductance • This applies to all coils or loops of wire

17. Problem 21.35 An MRI magnet has an inductance of L = 5.0 H. What is the total flux through the magnet’s coils when the current is I = 100 A?

18. 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

19. 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

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

21. Up to here

22. 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

23. RL Circuit Example Section 21.5

24. 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

25. 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

26. 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

27. 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 expressed as • In terms of the magnetic field, Section 21.6

28. 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

29. 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

30. 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 • 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 Section 21.7

31. Electric Guitars • An electric guitar uses Faraday’s Law to sense the motion of the strings • The 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

32. 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 captures energy normally lost to heat and stores it in batteries • Still not a perfect conversion Section 21.7

33. Induction from a Distance • Assume a very long solenoid is inserted at the center of a single loop of wire • This will produce mutual inductance • The field from the solenoid at the outer loop is essentially zero Section 21.8

34. 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