Chapter 29 Electromagnetic Induction

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1 Chapter 29 Electromagnetic Induction Lecture by Dr. Hebin Li

2 Goals of Chapter 29 To examine experimental evidence that a changing magnetic field induces an emf To learn how Faraday s law relates the induced emf to the change in flux To determine the direction of an induced emf To calculate the emf induced by a moving conductor To learn how a changing magnetic flux generates an electric field To study the four fundamental equations that describe electricity and magnetism

3 Induced current A changing magnetic flux causes an induced current. The induced emf is the corresponding emf causing the current. A time-varying magnetic field can act as a source of electric field

4 Magnetic flux through an area element Figure below shows how to calculate the magnetic flux through an element of area.

5 Magnetic flux through a finite area The flux depends on the orientation of the surface with respect to the magnetic field.

6 Faraday s law Faraday s law: The induced emf in a closed loop equals the negative of the time rate of change of magnetic flux through the loop. IF the coil has N identical turns, and if the flux varies at the same rate through each turn, then Change of flux can have various causes.

7 Example: emf and the current induced in a loop A coil of 5.00 cm in radius, containing 100 turns, is placed in a uniform magnetic field that varies with time according to B = T t + ( T s s 4)t4. The coil is connected to a 500-Ω resistor, and its plane is perpendicular to the magnetic field. You can ignore the resistance of the coil. (a) Find the magnitude of the induced emf in the coil as a function of time. (b) What is the current in the resistor at time t = 5.00 s?

8 Example: A rectangle measuring 30.0 cm by 40.0 cm is located inside a region of a spatially uniform magnetic field of 1.25 T, with the field perpendicular to the plane of the coil. The coil is pulled out at a steady rate of 2.00 cm/s traveling perpendicular to the field lines. The region of the field ends abruptly as shown. Find the emf induced in this coil when it is (a) all inside the field; (b) partly inside the field; (c) all outside the field.

9 Direction of the induced emf Define a positive direction With the defined positive direction, determine the signs of the magnetic flux Φ B and its rate of change dφ B /dt. Determine the sign of the induced emf ( dφ B /dt). Use the right hand rule to determine the direction of the induced emf and current. Note: an easier way is to use Lenz s law.

10 A simple alternator

11 DC generator and back emf in a motor

12 Lenz s law Lenz s law: The direction of any magnetic induction effect is such as to oppose the cause of the effect.

13 Examples: applications of Lenz s law There is a current in the straight wire going from right to left. What is the direction of the induced current in the round coil if the current increases?

14 Examples: applications of Lenz s law In the figure below, the bar is moving to the left. What is the direction of the induced current?

15 Slidewire generator In a slidewire generator, a U-shaped conductor is placed in a uniform magnetic field that is perpendicular to the plane. We lay a metal rod (slidewire) across the two arms of the conductor, forming a circuit, and move it to the right with constant velocity. This induces an emf and a current. Find the magnitude and direction of the resulting induced emf.

16 Motional electromotive force The motional electromotive force across the ends of a rod moving perpendicular to a magnetic field is

17 Examples: The conducting rod ab shown in the figure below makes contact with metal rails ca and db. The apparatus is in a uniform magnetic field of T, perpendicular to the plane of the figure (a) Find the magnitude of the emf induced in the rod when it is moving toward the right with a speed 7.50 m/s. (b) In what direction does the current flow in the rod? (c) If the resistance of the circuit abdc is 1.5 Ω and does not change, find the force (magnitude and direction) required to keep the rod moving to the right with a constant speed of 7.50 m/s. (d) Compare the rate at which mechanical work is done by the force (Fv) with the rate at which thermal energy is developed in the circuit (I 2 R).

18 Induced electric fields Changing magnetic flux causes an induced electric field. The induced electric field for a solenoid can be measured. The line integral of the induced electric field around a closed path is equal to the induced emf. Faraday s law can be restated in terms of the induced electric field.

19 Induced electric fields The magnetic field along the solenoid axis Neglect the small field outside the solenoid, then the magnetic flux through the loop is

20 Maxwell s equations Maxwell s equations consist of

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