Chapter 29 Electromagnetic Induction and Faraday’s Law Chapter 29 opener. One of the great laws of physics is Faraday’s law of induction, which says that a changing magnetic flux produces an induced emf. This photo shows a bar magnet moving inside a coil of wire, and the galvanometer registers an induced current. This phenomenon of electromagnetic induction is the basis for many practical devices, including generators, alternators, transformers, tape recording, and computer memory.

Faraday’s Law

29-3 EMF Induced in a Moving Conductor This image shows another way the magnetic flux can change: R Figure 29-12a. A conducting rod is moved to the right on a U-shaped conductor in a uniform magnetic field B that points out of the page. The induced current is clockwise.

29-3 EMF Induced in a Moving Conductor

ConcepTest 29.9 Motional EMF A conducting rod slides on a conducting track in a constant B field directed into the page. What is the direction of the induced current? 1) clockwise 2) counterclockwise 3) no induced current x x x x x x x x x x x v

ConcepTest 29.9 Motional EMF A conducting rod slides on a conducting track in a constant B field directed into the page. What is the direction of the induced current? 1) clockwise 2) counterclockwise 3) no induced current The B field points into the page. The flux is increasing since the area is increasing. The induced B field opposes this change and therefore points out of the page. Thus, the induced current runs counterclockwise, according to the right-hand rule. x x x x x x x x x x x v Follow-up: What direction is the magnetic force on the rod as it moves?

29-3 EMF Induced in a Moving Conductor The induced emf has magnitude This equation is valid as long as B, l, and v are mutually perpendicular (if not, it is true for their perpendicular components).

29-3 EMF Induced in a Moving Conductor The induced current is in a direction that tends to slow the moving bar _ Conductor Figure 29-12b. Upward force on an electron in the metal rod (moving to the right) due to B pointing out of the page. _

29-3 EMF Induced in a Moving Conductor

29-3 EMF Induced in a Moving Conductor Example 29-6: Does a moving airplane develop a large emf? An airplane travels 1000 km/h in a region where the Earth’s magnetic field is about 5 x 10-5 T and is nearly vertical. What is the potential difference induced between the wing tips that are 70 m apart? Solution: E = Blv = 1 V.

29-4 Electric Generators A generator is the opposite of a motor – it transforms mechanical energy into electrical energy. This is an ac generator: The axle is rotated by an external force such as falling water or steam. The brushes are in constant electrical contact with the slip rings. Figure 29-15. An ac generator.

29-4 Electric Generators If the loop is rotating with constant angular velocity ω, the induced emf is sinusoidal: For a coil of N loops, Figure 29-16. An ac generator produces an alternating current. The output emf E = E0 sin ωt, where E0 = NABω.

29-4 Electric Generators Example 29-9: An ac generator. The armature of a 60-Hz ac generator rotates in a 0.15-T magnetic field. If the area of the coil is 2.0 x 10-2 m2, how many loops must the coil contain if the peak output is to be V0 = 170 V? Solution: N = E0/BAω = 150 turns. Remember to convert 60 Hz to angular units.

29-4 Electric Generators A dc generator is similar, except that it has a split-ring commutator instead of slip rings. Figure 29-18. (a) A dc generator with one set of commutators, and (b) a dc generator with many sets of commutators and windings.

ConcepTest 29.10 Generators A generator has a coil of wire rotating in a magnetic field. If the rotation rate increases, how is the maximum output voltage of the generator affected? 1) increases 2) decreases 3) stays the same 4) varies sinusoidally

ConcepTest 29.10 Generators A generator has a coil of wire rotating in a magnetic field. If the rotation rate increases, how is the maximum output voltage of the generator affected? 1) increases 2) decreases 3) stays the same 4) varies sinusoidally The maximum voltage is the leading term that multiplies sin wt and is given by e0 = NBAw. Therefore, if w increases, then e0 must increase as well.

29-5 Back EMF and Counter Torque; Eddy Currents An electric motor turns because there is a torque on it due to the current. We would expect the motor to accelerate unless there is some sort of drag torque. That drag torque exists, and is due to the induced emf, called a back emf.

29-5 Back EMF and Counter Torque; Eddy Currents Example 29-10: Back emf in a motor. The armature windings of a dc motor have a resistance of 5.0 Ω. The motor is connected to a 120-V line, and when the motor reaches full speed against its normal load, the back emf is 108 V. Calculate (a) the current into the motor when it is just starting up, and (b) the current when the motor reaches full speed. Solution: a. At startup, I = V/R = 24 A. b. The back emf means that the total emf in the circuit is 12 V, so the current is 2.4 A.

29-5 Back EMF and Counter Torque; Eddy Currents Conceptual Example 29-11: Motor overload. When using an appliance such as a blender, electric drill, or sewing machine, if the appliance is overloaded or jammed so that the motor slows appreciably or stops while the power is still connected, the device can burn out and be ruined. Explain why this happens. The motor is designed to operate at a particular speed, which means there will be a particular back emf. If the appliance slows or stops, the back emf becomes much less, the current becomes much more than designed, and the appliance may burn out.

29-5 Back EMF and Counter Torque; Eddy Currents A similar effect occurs in a generator – if it is connected to a circuit, current will flow in it, and will produce a counter torque. This means the external applied torque must increase to keep the generator turning.

29-5 Back EMF and Counter Torque; Eddy Currents Induced currents can flow in bulk material as well as through wires. These are called eddy currents, and can dramatically slow a conductor moving into or out of a magnetic field. Figure 29-21. Production of eddy currents in a rotating wheel. The grey lines in (b) indicate induced current.

29-6 Transformers and Transmission of Power A transformer consists of two coils, either interwoven or linked by an iron core. A changing emf in one induces an emf in the other. The ratio of the emfs is equal to the ratio of the number of turns in each coil:

29-6 Transformers and Transmission of Power This is a step-up transformer – the emf in the secondary coil is larger than the emf in the primary: Figure 29-24. Step-up transformer (NP = 4, NS = 12).

29-6 Transformers and Transmission of Power Energy must be conserved; therefore, in the absence of losses, the ratio of the currents must be the inverse of the ratio of turns:

29-7 A Changing Magnetic Flux Produces an Electric Field A changing magnetic flux induces an electric field; this is a generalization of Faraday’s law. The electric field will exist regardless of whether there are any conductors around: .

29-7 A Changing Magnetic Flux Produces an Electric Field Example 29-14: E produced by changing B. A magnetic field B between the pole faces of an electromagnet is nearly uniform at any instant over a circular area of radius r0. The current in the windings of the electromagnet is increasing in time so that B changes in time at a constant rate dB/dt at each point. Beyond the circular region (r > r0), we assume B = 0 at all times. Determine the electric field E at any point P a distance r from the center of the circular area due to the changing B. Figure 29-27. (a) Side view of nearly constant B. (b) Top view, for determining the electric field E at point P. (c) Lines of E produced by increasing B (pointing outward). (d) Graph of E vs. r. Example 29–14. Solution: Because of symmetry, E will be perpendicular to B and constant at radius r. Integrate around a circle of radius r as shown. For r < r0, the enclosed flux is Bπr2, and E = r/2 dB/dt. For r > r0, the enclosed flux is Bπr02, and E = r02/2r dB/dt.

HW # 10 Chapter 29 – 32, 38, 50, 70