Chapter 29 Electromagnetic Induction In this chapter we investigate how changing the magnetic flux in a circuit induces an emf and a current. We learned in Chapter 25 that an electromotive force (E) is required for a current to flow in a circuit. Up until now, we used a battery as our emf source. However, most of our power comes from electric generating stations where the source of power is gravitational potential energy, chemical energy, and nuclear energy. How are these forms of energy converted into electrical energy? This is accomplished by electromagnetic induction. In all power-generating stations, magnets move relative to coils of wire to produce a changing magnetic flux in the coils and hence an emf. In this chapter we will study Faraday s law. This relates the induced emf to changing magnetic flux in any loop. We also discuss Lenz s law, which helps predict the direction of the induced emfs and currents. These principles are at the heart of electrical energy conversion devices such as motors, generators, and transformers. Finally, electromagnetic induction tells us that a time-varying magnetic field can act as a source of electric field. We will also see how a time-varying electric field can act as a source of magnetic field. These are the principle concepts embodied in what has been called Maxwell s equations. Just as Newton s laws describe classical kinematics and dynamics, Maxwell s equations provide the fundamental framework for electrodynamics. 1 Induction Experiments Magnetically induced emf was first of observed in the 1830 s by Michael Faraday (England), and by Joseph Henry (United States). The changing flux of magnetic field passing through a closed loop (or circuit) was observed to produce an induced current and the corresponding emf required to cause this current is call an induced emf (E). 1
Figure 1: This figure shows the measurements observed when a changing magnetic flux occurs in a coil of wire connected to a galvanometer. The following induction experiments were performed using the apparatus shown in Fig. 2 below. 2
Figure 2: This figure is used to describe the observations made below. The figure shows a coil placed inside an electromagnet and connected to a galvanometer. The strength of the magnetic field shown with the N and S pole faces can be adjusted by varying the current into the device. What was observed? 1. When there is no current in the electromagnet, in other words B = 0, the galvanometer show no current. 2. When the electromagnet is turned on, there is momentary current through the meter as B increases. 3. When B levels off at a steady value, the current drops to zero. 4. With the coil in a horizontal plane, we squeeze it so as to decrease the crosssectional area of the coil. The meter detects current only during the deformation, not before or after. When we increase the area to return the coil to its original shape, there is current in the opposite direction, but only while the area of the coil is changing. 5. If we rotate the coil a few degrees about a horizontal axis, the meter detects current during the rotation, in the same direction as when we decreased the area. When we rotate the coil back, there is a current in the opposite direction during this rotation. 6. If we decrease the number of turns in the coil by unwinding one or more turns, 3
there is a current during the unwinding, in the same direction as when we decreased the area. If we wind more turns onto the coil, there is a current in the opposite direction during the winding. 7. When the magnet is turned off, there is a momentary current in the direction opposite to the current when it was turned on. 8. The faster we carry out any of these changes, the greater the current. 9. If all these experiments are repeated with a coil that has the same shape but different material and different resistance, the current in each case is inversely proportional to the total circuit resistance. This shows that the induced emfs that are causing the current do not depend on the material of the coil but only on its shape and the magnetic field. 4
2 Faraday s Law The common feature in all induction effects is changing magnetic flux through a circuit. For an infinitesimal-area elements d A in a magnetic field B, the magnetic flux dφ B through the area is: dφ B = B d A = B da = B da cos φ Figure 3: This figure shows how the magnetic flux is calculated as it passes through a surface. Figure 4: Calculating the flux of a uniform magnetic field through a flat area. Different orientations of the area result in different amounts of flux Φ B. Φ B = B A = BA cos φ 5
Faraday s Law of Induction states: E = E = dφ B ( db da A cos φ + B (Faraday s Law) (1) cos φ BA sin φ ( )) dφ There are 3 ways to induce an emf E: 1. Change the strength of the magnetic field B, 2. Change the area of the loop or circuit, and 3. Change the angle the area makes with respect to the B field. 2.1 Direction of Induced emf Figure 5: The magnetic flux Φ B is increasing in (a) and (d) thus inducing an negative emf. In figures (b) and (c) the magnetic flux Φ B is decreasing, thereby inducing a positive emf. 6
2.2 Magnitude and Direction of an Induced EMF Figure 6: In this figure the uniform magnetic field is decreased at a rate of 0.200 T/s. What is the induced emf in this 500-loop wire coil of radius 4.00 cm? 2.3 A Simple Alternator Figure 7: A schematic diagram of an alternator is shown (a). The conducting loop rotates in a magnetic field, producing an emf shown in (b). Connections from each end of the loop to the external circuit are made by means of that end s slip ring. One cycle of the generated emf is shown in the figure. E = dφ = BA d cos φ = BA d (cos ωt) = BAω sin(ωt) This is the starting point for producing AC power. 7
A DC Generator Figure 8: This figure is used to describe the observations made on the following page. The time-averaged back-emf for a coil having N loops is: 2.4 The Slidewire Generator E av = 2NωBA π Figure 9: This figure is used to describe the observations made on the following page. In this case the magnetic flux changes due to the changing area da/. E = dφ = B da = B (Lv ) = BLv 8
The power required to move the bar is equal to the power dissipated by the I 2 R loss in the current loop. 3 Lenz s Law P applied = B2 L 2 v 2 The direction of any magnetic induction effect is such as to oppose the cause of the effect Example: Lenz s Law and the Slidewire Generator R 3.1 Lenz s Law and the direction of Induced Current Figure 10: This figure shows how to visualize the induced emf, and current, when the strength of the external B-field is increased. Lenz s law says that there will be an opposing B-field, ( B induced ) is produced in response to the increasing external field. 9
Figure 11: This figure shows four possible ways of increasing or decreasing the flux with two magnetic poles. In all cases, the induced B-field is created in a manner that opposes the change to the electric field. The induced B-field is produced as a result of the induced emf and its associated current. 10
4 Motional Electromotive Force Figure 12: This figure shows a conducting rod moving in a uniform magnetic field. The velocity of the rod and the field are mutually perpendicular. In part (a) an electric field is induced in the bar as it moves through the B field, and positive charges migrate upward. In part (b) of the figure shows the direction of the induced current in the circuit when the bar is connected to a conducting loop. The magnitude of the potential difference V ab V a V b is equal to the electric-field magnitude E multipled by the length L of the rod. Since E = vb, we have: V ab = EL = vbl The motional emf generated in part (b) of the figure is: E = vbl 11
4.1 Motional emf: General Form Figure 13: This figure shows how the motional emf arises for a current loop moving in a static magnetic field. The velocity v can be different for different elements if the loop is rotating or changing shape. The magnetic field B can also have different values at different points around the loop. The motional emf produced for a segment of the loop is: de = ( v B) d l For any closed conducting loop, the total emf is: E = E nc d l 0 ( v B) d l (2) (non-conservative E field) 12
5 Induced Electric Fields Figure 14: This figure shows the winding on a solenoid carrying current I that is increasing at a rate di/. The magnetic flux in the solenoid is increasing at a rate dφ/, and this changing flux passes through a wire loop. An emf E = dφ/ is induced in the loop, inducing a current I that is measured by the galvanometer G. The magnetic flux created by the solenoid is: Φ = BA = µ o nia When the solenoid current changes, so does the flux Φ. E = dφ = µ o na di 13
There is an induced electric field in the conductor caused by the changing magnetic flux. 14
5.1 Nonelectrostatic Electric Fields Figure 15: Here are some applications of induced electric fields. (a) This hybrid automobile has both a gasoline engine and an electric motor. As the car comes to a halt, the spinning wheels run the motor backward so that it acts as a generator. The induced current is used to recharge the car s batteries. (b) The rotating crankshaft of a piston-engine airplane spins a magnet (i.e., a magneto) inducing an emf in an adjacent coil and generating the spark that ignites fuel in the engine cylinders. This process continues even if the airplane s other electrical systems fail. 15
6 Eddy Currents In previous examples we observed the induction effects and the induced currents they produced. However, many pieces of electrical equipment contain masses of metal moving in magnetic fields or located in changing magnetic fields (e.g., transformers). In these situations we have induced currents that circulate throughout the volume of the material. Because their flow patterns resemble swirling eddies in a river, we call them eddy currents. Figure 16: These figures (a and b) shows the eddy currents induced in a rotating metal disk as it rotates through a magnetic field. In figure (b) you can see the braking force F that occurs from the I L B opposing the rotation of the disk. 16
Figure 17: Eddy currents can be used in (a) metal detectors at airport security checkpoints. When the transmitting coil generates an alternating magnetic field B o, this induces eddy currents in a conducting object carried through the detector. The eddy currents in turn produce an alternating magnetic field B, which induces a current in the detector s receiver coil. The same principle is used for portable metal detectors shown in fig. (b). 7 Displacement Current and Maxwell s Equations In the previous sections we have seen how varying magnetic fields give rise to an induced electric field. One of the more remarkable symmetries of nature is that a varying electric field gives rise to a magnetic field. This symmetric phenomenon explains the existence of radio waves, gamma rays, and visible light, as well as all other forms of electromagnetic waves. In order to illustrate this symmetric relationship between varying electric fields and magnetic fields, we return to Ampère s law: B d l = µ o I encl However, when we apply this law to the charging of a capacitor (see the figure below), we see that this equation is incomplete. No conduction current i C is observed between the capacitor plates; however, there is current i C coming into the positiely charged plate and current i C leaving the negatively charged plate. While the 17
right-hand side of Ampère s law is equal to µ o i C for the planar surface, it s equal to zero if we use the bulging surface. However, it cannot be both, so, something is missing. However, something else is happening at the bulging surface. While the charge is building up on the capacitor plates, the E-field is changing between the plates. In fact, the flux of the E-field Φ E is increasing. Let s try to find a relationship between the conducting current i C and the changing electric field de/. The instantaneous charge on the capacitor q is related to the instantaneous voltage v by the following equation: q = vc where C is the capacitance ɛ/ad. Figure 18: This figure shows a current charging a parallel-plate capacitor. The conduction current through the plane surface is i C, but there is no conduction current through the surface bulging out between the plates. The two surfaces have a common boundary, so this difference in i encl leads to an apparent contradiction in applying Ampère s law. q = Cv = ɛa d (Ed) = ɛea = ɛ Φ E (3) By taking a time derivative of both sides of this equation, we can relate the current coming in to the capacitor to the changing E-field building up between the plates where i C = dq = ɛ dφ E 18
is called the displacement current. i D = ɛ dφ E We include this fictitious current, along with the real conduction current i C, to rewrite Ampère s law as: B d l = µ o (i C + i D ) encl (4) Figure 19: This figure shows a capacitor being charged by a current i C with a displacement current equal to i C between the plates, with displacement current density j D = ɛ de/. This can be regarded as the source of magnetic field between the plates. The displacement current, along with the magnetic field it produces, is more than an artifice; it is real and a fundamental property of nature. 19
8 Maxwells Equations of Electromagnetism We can now summarize all the relationships between electric and magnetic fields into a package of four equations, called Maxwell s equations. Two of Maxwell s equations involve an integral of E or B over a closed surface. The third and fourth equations involve a line integral of E or B around a closed path. Faraday s law states that a changing magnetic flux acts as a source of electric field. 1. E d A = Q encl ɛ o 2. B d A = 0 3. E d l = dφ B (Faraday s Law) 4. ) B d dφ E l = µ o (i C + ɛ o The electric field in Maxwell s equations includes both conservative and nonconservative components of the E field: E = E c + E nc N.B. These four equations are highly symmetric in empty space. The third and fourth equations can be rewritten in empty space (i C = 0 and Q encl = 0) as follow: E d l = d B d A (5) B d l = µ o ɛ o d E d A (6) 20