# Faraday s Law of Induction

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Chapter 10 Faraday s Law of Induction 10.1 Faraday s Law of Induction Magnetic Flux Lenz s Law Motional EMF Induced Electric Field Generators Eddy Currents Summary Appendix: Induced Emf and Reference Frames Problem-Solving Tips: Faraday s Law and Lenz s Law Solved Problems Rectangular Loop Near a Wire Loop Changing Area Sliding Rod Moving ar Time-Varying Magnetic Field Moving Loop Conceptual Questions Additional Problems Sliding ar Sliding ar on Wedges RC Circuit in a Magnetic Field Sliding ar Rotating ar Rectangular Loop Moving Through Magnetic Field Magnet Moving Through a Coil of Wire Alternating-Current Generator EMF Due to a Time-Varying Magnetic Field Square Loop Moving Through Magnetic Field Falling Loop

2 Faraday s Law of Induction 10.1 Faraday s Law of Induction The electric fields and magnetic fields considered up to now have been produced by stationary charges and moving charges (currents), respectively. Imposing an electric field on a conductor gives rise to a current which in turn generates a magnetic field. One could then inquire whether or not an electric field could be produced by a magnetic field. In 1831, Michael Faraday discovered that, by varying magnetic field with time, an electric field could be generated. The phenomenon is known as electromagnetic induction. Figure illustrates one of Faraday s experiments. Figure Electromagnetic induction Faraday showed that no current is registered in the galvanometer when bar magnet is stationary with respect to the loop. However, a current is induced in the loop when a relative motion exists between the bar magnet and the loop. In particular, the galvanometer deflects in one direction as the magnet approaches the loop, and the opposite direction as it moves away. Faraday s experiment demonstrates that an electric current is induced in the loop by changing the magnetic field. The coil behaves as if it were connected to an emf source. Experimentally it is found that the induced emf depends on the rate of change of magnetic flux through the coil. 10-

3 Magnetic Flux Consider a uniform magnetic field passing through a surface S, as shown in Figure below: Figure Magnetic flux through a surface Let the area vector be A = Anˆ, where A is the area of the surface and ˆn its unit normal. The magnetic flux through the surface is given by where θ is the angle between and Φ = A = Acosθ (10.1.1) ˆn. If the field is non-uniform, Φ then becomes Φ = d A (10.1.) S The SI unit of magnetic flux is the weber (Wb): 1 Wb = 1 T m Faraday s law of induction may be stated as follows: The induced emf ε in a coil is proportional to the negative of the rate of change of magnetic flux: ε dφ dt = (10.1.3) For a coil that consists of N loops, the total induced emf would be N times as large: ε dφ dt = N (10.1.4) 10-3

4 Combining Eqs. (10.1.3) and (10.1.1), we obtain, for a spatially uniform field, d d da dθ ε = ( Acos θ) = Acosθ cosθ + Asinθ dt dt dt dt (10.1.5) Thus, we see that an emf may be induced in the following ways: (i) by varying the magnitude of with time (illustrated in Figure ) Figure Inducing emf by varying the magnetic field strength (ii) by varying the magnitude of A, i.e., the area enclosed by the loop with time (illustrated in Figure ) Figure Inducing emf by changing the area of the loop (iii) varying the angle between and the area vector A with time (illustrated in Figure ) 10-4

5 Figure Inducing emf by varying the angle between and A Lenz s Law The direction of the induced current is determined by Lenz s law: The induced current produces magnetic fields which tend to oppose the change in magnetic flux that induces such currents. To illustrate how Lenz s law works, let s consider a conducting loop placed in a magnetic field. We follow the procedure below: 1. Define a positive direction for the area vector A.. Assuming that is uniform, take the dot product of and A. This allows for the determination of the sign of the magnetic flux Φ. 3. Obtain the rate of flux change dφ / dt by differentiation. There are three possibilities: > 0 induced emf ε < 0 dφ : < 0 induced emf ε > 0 dt = 0 induced emf ε = 0 4. Determine the direction of the induced current using the right-hand rule. With your thumb pointing in the direction of A, curl the fingers around the closed loop. The induced current flows in the same direction as the way your fingers curl if ε > 0, and the opposite direction if ε < 0, as shown in Figure Figure Determination of the direction of induced current by the right-hand rule In Figure we illustrate the four possible scenarios of time-varying magnetic flux and show how Lenz s law is used to determine the direction of the induced current I. 10-5

6 (a) (b) (c) Figure Direction of the induced current using Lenz s law (d) The above situations can be summarized with the following sign convention: Φ + dφ / dt ε I The positive and negative signs of I correspond to a counterclockwise and clockwise currents, respectively. As an example to illustrate how Lenz s law may be applied, consider the situation where a bar magnet is moving toward a conducting loop with its north pole down, as shown in Figure (a). With the magnetic field pointing downward and the area vector A pointing upward, the magnetic flux is negative, i.e., Φ = A < 0, where A is the area of the loop. As the magnet moves closer to the loop, the magnetic field at a point on the loop increases ( d / dt > 0), producing more flux through the plane of the loop. Therefore, dφ / dt = A( d / dt) < 0, implying a positive induced emf, ε > 0, and the induced current flows in the counterclockwise direction. The current then sets up an induced magnetic field and produces a positive flux to counteract the change. The situation described here corresponds to that illustrated in Figure (c). Alternatively, the direction of the induced current can also be determined from the point of view of magnetic force. Lenz s law states that the induced emf must be in the direction that opposes the change. Therefore, as the bar magnet approaches the loop, it experiences 10-6

7 a repulsive force due to the induced emf. Since like poles repel, the loop must behave as if it were a bar magnet with its north pole pointing up. Using the right-hand rule, the direction of the induced current is counterclockwise, as view from above. Figure (b) illustrates how this alternative approach is used. Figure (a) A bar magnet moving toward a current loop. (b) Determination of the direction of induced current by considering the magnetic force between the bar magnet and the loop 10. Motional EMF Consider a conducting bar of length l moving through a uniform magnetic field which points into the page, as shown in Figure Particles with charge q > 0 inside experience a magnetic force F = qv which tends to push them upward, leaving negative charges on the lower end. Figure A conducting bar moving through a uniform magnetic field The separation of charge gives rise to an electric field E inside the bar, which in turn produces a downward electric force F e = qe. At equilibrium where the two forces cancel, 10-7

8 we have qv = qe, or E= v. etween the two ends of the conductor, there exists a potential difference given by V = V V = ε = El= lv (10..1) ab a b Sinceε arises from the motion of the conductor, this potential difference is called the motional emf. In general, motional emf around a closed conducting loop can be written as where d s is a differential length element. ε = ( v ) d s (10..) Now suppose the conducting bar moves through a region of uniform magnetic field = k ˆ (pointing into the page) by sliding along two frictionless conducting rails that are at a distance l apart and connected together by a resistor with resistance R, as shown in Figure Figure 10.. A conducting bar sliding along two conducting rails Let an external force F ˆ ext be applied so that the conductor moves to the right with a constant velocity v = vi. The magnetic flux through the closed loop formed by the bar and the rails is given by Thus, according to Faraday s law, the induced emf is Φ = A= lx (10..3) dφ d dx ε = = ( lx) = l = lv (10..4) dt dt dt where dx / dt = v is simply the speed of the bar. The corresponding induced current is I ε lv = = (10..5) R R 10-8

9 and its direction is counterclockwise, according to Lenz s law. The equivalent circuit diagram is shown in Figure 10..3: Figure Equivalent circuit diagram for the moving bar The magnetic force experienced by the bar as it moves to the right is F ( ˆ) ( ˆ ) ˆ lv = Ilj k = Ili = ˆ i (10..6) R which is in the opposite direction of v. For the bar to move at a constant velocity, the net force acting on it must be zero. This means that the external agent must supply a force F ext = =+ lv F ˆ i R (10..7) The power delivered by F ext is equal to the power dissipated in the resistor: as required by energy conservation. lv ( lv) ε P= Fext v= Fextv= v= = = R R R I R (10..8) From the analysis above, in order for the bar to move at a constant speed, an external agent must constantly supply a force F. What happens if at t = 0, the speed of the rod v 0 ext is, and the external agent stops pushing? In this case, the bar will slow down because of the magnetic force directed to the left. From Newton s second law, we have or lv dv F = ma m R = = dt (10..9) dv l dt = dt = (10..10) v mr τ 10-9

10 where τ = mr / l. Upon integration, we obtain t / vt () ve τ 0 = (10..11) Thus, we see that the speed decreases exponentially in the absence of an external agent doing work. In principle, the bar never stops moving. However, one may verify that the total distance traveled is finite Induced Electric Field In Chapter 3, we have seen that the electric potential difference between two points A and in an electric field E can be written as V = V VA = E d s A (10.3.1) When the electric field is conservative, as is the case of electrostatics, the line integral of E d sis path-independent, which implies E d s = 0. Faraday s law shows that as magnetic flux changes with time, an induced current begins to flow. What causes the charges to move? It is the induced emf which is the work done per unit charge. However, since magnetic field can do not work, as we have shown in Chapter 8, the work done on the mobile charges must be electric, and the electric field in this situation cannot be conservative because the line integral of a conservative field must vanish. Therefore, we conclude that there is a non-conservative electric field E nc associated with an induced emf: ε = E Combining with Faraday s law then yields E nc d s (10.3.) dφ dt nc d s = (10.3.3) The above expression implies that a changing magnetic flux will induce a nonconservative electric field which can vary with time. It is important to distinguish between the induced, non-conservative electric field and the conservative electric field which arises from electric charges. As an example, let s consider a uniform magnetic field which points into the page and is confined to a circular region with radius R, as shown in Figure Suppose the magnitude of increases with time, i.e., d / dt > 0. Let s find the induced electric field everywhere due to the changing magnetic field

11 Since the magnetic field is confined to a circular region, from symmetry arguments we choose the integration path to be a circle of radius r. The magnitude of the induced field E nc at all points on a circle is the same. According to Lenz s law, the direction of E nc must be such that it would drive the induced current to produce a magnetic field opposing the change in magnetic flux. With the area vector A pointing out of the page, the magnetic flux is negative or inward. With d / dt > 0, the inward magnetic flux is increasing. Therefore, to counteract this change the induced current must flow counterclockwise to produce more outward flux. The direction of E nc is shown in Figure Figure Induced electric field due to changing magnetic flux Let s proceed to find the magnitude of E nc. In the region r < R, the rate of change of magnetic flux is Using Eq. (10.3.3), we have which implies d Φ d d d = ( A ) = ( A) = π r (10.3.4) dt dt dt dt E dφ d dt dt ( π ) π nc d s = Enc r = = r (10.3.5) E nc rd = (10.3.6) dt Similarly, for r > R, the induced electric field may be obtained as or E dφ d = = dt dt ( π r) π R nc (10.3.7) 10-11

12 E nc R d = (10.3.8) r dt A plot of E nc as a function of r is shown in Figure Figure Induced electric field as a function of r 10.4 Generators One of the most important applications of Faraday s law of induction is to generators and motors. A generator converts mechanical energy into electric energy, while a motor converts electrical energy into mechanical energy. Figure (a) A simple generator. (b) The rotating loop as seen from above. Figure (a) is a simple illustration of a generator. It consists of an N-turn loop rotating in a magnetic field which is assumed to be uniform. The magnetic flux varies with time, thereby inducing an emf. From Figure (b), we see that the magnetic flux through the loop may be written as The rate of change of magnetic flux is Φ = A = Acosθ = Acosωt (10.4.1) d Φ = Aω sinωt (10.4.) dt 10-1

13 Since there are N turns in the loop, the total induced emf across the two ends of the loop is dφ ε = N = NAωsinωt (10.4.3) dt If we connect the generator to a circuit which has a resistance R, then the current generated in the circuit is given by I ε NAω = = sinωt (10.4.4) R R The current is an alternating current which oscillates in sign and has an amplitude I = NAω R. The power delivered to this circuit is 0 / ( NAω ) = ε = ω (10.4.5) R P I sin t On the other hand, the torque exerted on the loop is τ = µ sinθ = µ sinωt (10.4.6) Thus, the mechanical power supplied to rotate the loop is Since the dipole moment for the N-turn current loop is P = τω = µ ωsinωt (10.4.7) m N A ω µ = NIA = sinωt (10.4.8) R the above expression becomes N A ω ( NAω) Pm = t t = R R sinω ωsinω sin ωt (10.4.9) As expected, the mechanical power put in is equal to the electrical power output Eddy Currents We have seen that when a conducting loop moves through a magnetic field, current is induced as the result of changing magnetic flux. If a solid conductor were used instead of a loop, as shown in Figure , current can also be induced. The induced current appears to be circulating and is called an eddy current

14 Figure Appearance of an eddy current when a solid conductor moves through a magnetic field. The induced eddy currents also generate a magnetic force that opposes the motion, making it more difficult to move the conductor across the magnetic field (Figure 10.5.). Figure Magnetic force arising from the eddy current that opposes the motion of the conducting slab. Since the conductor has non-vanishing resistance R, Joule heating causes a loss of power by an amount P= ε / R. Therefore, by increasing the value of R, power loss can be reduced. One way to increase R is to laminate the conducting slab, or construct the slab by using gluing together thin strips that are insulated from one another (see Figure a). Another way is to make cuts in the slab, thereby disrupting the conducting path (Figure b). Figure Eddy currents can be reduced by (a) laminating the slab, or (b) making cuts on the slab. There are important applications of eddy currents. For example, the currents can be used to suppress unwanted mechanical oscillations. Another application is the magnetic braking systems in high-speed transit cars

15 10.6 Summary The magnetic flux through a surface S is given by Φ = d A S Faraday s law of induction states that the induced emf ε in a coil is proportional to the negative of the rate of change of magnetic flux: dφ ε = N dt The direction of the induced current is determined by Lenz s law which states that the induced current produces magnetic fields which tend to oppose the changes in magnetic flux that induces such currents. A motional emf ε is induced if a conductor moves in a magnetic field. The general expression forε is ε = ( v ) d s In the case of a conducting bar of length l moving with constant velocity v through a magnetic field which points in the direction perpendicular to the bar and v, the induced emf is ε = vl. An induced emf in a stationary conductor is associated with a non-conservative electric field : E nc dφ ε = E nc d s = dt 10.7 Appendix: Induced Emf and Reference Frames In Section 10., we have stated that the general equation of motional emf is given by ε = ( v ) d s where v is the velocity of the length element d s of the moving conductor. In addition, we have also shown in Section 10.4 that induced emf associated with a stationary conductor may be written as the line integral of the non-conservative electric field: 10-15

16 ε = E nc d However, whether an object is moving or stationary actually depends on the reference frame. As an example, let s examine the situation where a bar magnet is approaching a conducting loop. An observer O in the rest frame of the loop sees the bar magnet moving toward the loop. An electric field E nc is induced to drive the current around the loop, and a charge on the loop experiences an electric force Fe = qenc. Since the charge is at rest according to observer O, no magnetic force is present. On the other hand, an observer O ' in the rest frame of the bar magnet sees the loop moving toward the magnet. Since the conducting loop is moving with a velocity v, a motional emf is induced. In this frame, O ' sees the charge q moving with a velocity v, and concludes that the charge experiences a magnetic force F = qv. s Figure Induction observed in different reference frames. In (a) the bar magnet is moving, while in (b) the conducting loop is moving. Since the event seen by the two observer is the same except the choice of reference frames, the force acting on the charge must be the same, F F, which implies e = E = v nc (10.7.1) In general, as a consequence of relativity, an electric phenomenon observed in a reference frame O may appear to be a magnetic phenomenon in a frame O ' that moves at a speed v relative too Problem-Solving Tips: Faraday s Law and Lenz s Law In this chapter we have seen that a changing magnetic flux induces an emf: dφ ε = N dt 10-16

17 according to Faraday s law of induction. For a conductor which forms a closed loop, the emf sets up an induced current I = ε / R, where R is the resistance of the loop. To compute the induced current and its direction, we follow the procedure below: 1. For the closed loop of area A on a plane, define an area vector A and let it point in the direction of your thumb, for the convenience of applying the right-hand rule later. Compute the magnetic flux through the loop using Determine the sign of Φ. A ( is uniform) Φ = da ( is non-uniform). Evaluate the rate of change of magnetic flux dφ / dt. Keep in mind that the change could be caused by (i) changing the magnetic field d / dt 0, (ii) changing the loop area if the conductor is moving ( da/ dt 0), or (iii) changing the orientation of the loop with respect to the magnetic field ( dθ / dt 0). Determine the sign of dφ / dt. 3. The sign of the induced emf is the opposite of that of dφ / dt. The direction of the induced current can be found by using Lenz s law discussed in Section Solved Problems Rectangular Loop Near a Wire An infinite straight wire carries a current I is placed to the left of a rectangular loop of wire with width w and length l, as shown in the Figure Figure Rectangular loop near a wire 10-17

18 (a) Determine the magnetic flux through the rectangular loop due to the current I. (b) Suppose that the current is a function of time with I ( t) = a+ bt, where a and b are positive constants. What is the induced emf in the loop and the direction of the induced current? Solutions: (a) Using Ampere s law: s = d µ 0Ienc (10.9.1) the magnetic field due to a current-carrying wire at a distance r away is µ I π r 0 = (10.9.) The total magnetic flux Φ through the loop can be obtained by summing over contributions from all differential area elements da =l dr: µ 0Il s+ wdr µ 0Il s + w Φ = dφ = da= = ln π s r π s (10.9.3) Note that we have chosen the area vector to point into the page, so that Φ > 0. (b) According to Faraday s law, the induced emf is dφ d µ 0Il s+ w µ 0l s+ w di µ 0bl s+ w ε = = ln ln ln dt dt π s = = (10.9.4) π s dt π s where we have used di / dt = b. The straight wire carrying a current I produces a magnetic flux into the page through the rectangular loop. y Lenz s law, the induced current in the loop must be flowing counterclockwise in order to produce a magnetic field out of the page to counteract the increase in inward flux Loop Changing Area A square loop with length l on each side is placed in a uniform magnetic field pointing into the page. During a time interval t, the loop is pulled from its two edges and turned 10-18

19 into a rhombus, as shown in the Figure Assuming that the total resistance of the loop is R, find the average induced current in the loop and its direction. Solution: Using Faraday s law, we have Figure Conducting loop changing area ε Φ t A t = = (10.9.5) Since the initial and the final areas of the loop are Ai = l and (recall that the area of a parallelogram defined by two vectors A= l l = ll sinθ ), the average rate of change of area is 1 1 A = l sin θ, respectively f l 1 and l is A A f Ai l (1 sin θ ) = = < 0 (10.9.6) t t t which gives l (1 sin θ ) ε = > 0 t (10.9.7) Thus, the average induced current is I ε = = R l (1 sin θ ) tr (10.9.8) Since ( A/ t) < 0, the magnetic flux into the page decreases. Hence, the current flows in the clockwise direction to compensate the loss of flux Sliding Rod A conducting rod of length l is free to slide on two parallel conducting bars as in Figure

20 Figure Sliding rod In addition, two resistors R 1 and R are connected across the ends of the bars. There is a uniform magnetic field pointing into the page. Suppose an external agent pulls the bar to the left at a constant speed v. Evaluate the following quantities: (a) The currents through both resistors; (b) The total power delivered to the resistors; (c) The applied force needed for the rod to maintain a constant velocity. Solutions: (a) The emf induced between the ends of the moving rod is The currents through the resistors are I ε dφ dt = = lv (10.9.9) ε ε =, I = ( ) 1 R1 R Since the flux into the page for the left loop is decreasing, I 1 flows clockwise to produce a magnetic field pointing into the page. On the other hand, the flux into the page for the right loop is increasing. To compensate the change, according to Lenz s law, I must flow counterclockwise to produce a magnetic field pointing out of the page. (b) The total power dissipated in the two resistors is PR = I1 ε + I ε = ( I1+ I) ε = ε + = l v + R R R R 1 1 ( ) (c) The total current flowing through the rod is I = I1+ I. Thus, the magnetic force acting on the rod is = = ε + = + R (10.9.1) F Il l l v R1 R R1 10-0

21 and the direction is to the right. Thus, an external agent must apply an equal but opposite force F = F to the left in order to maintain a constant speed. ext Alternatively, we note that since the power dissipated in the resistors must be equal to P ext, the mechanical power supplied by the external agent. The same result is obtained since P = F v= F ext ext ext v ( ) Moving ar A conducting rod of length l moves with a constant velocity v perpendicular to an infinitely long, straight wire carrying a current I, as shown in the Figure What is the emf generated between the ends of the rod? Solution: Figure A bar moving away from a current-carrying wire From Faraday s law, the motional emf is ε = lv ( ) where v is the speed of the rod. However, the magnetic field due to the straight currentcarrying wire at a distance r away is, using Ampere s law: µ I π r Thus, the emf between the ends of the rod is given by 0 = ( ) ε µ I lv π r 0 = ( ) 10-1

22 Time-Varying Magnetic Field A circular loop of wire of radius a is placed in a uniform magnetic field, with the plane of the loop perpendicular to the direction of the field, as shown in Figure Figure Circular loop in a time-varying magnetic field The magnetic field varies with time according to ( t) = 0 + bt, where 0 and b are positive constants. (a) Calculate the magnetic flux through the loop at t = 0. (b) Calculate the induced emf in the loop. (c) What is the induced current and its direction of flow if the overall resistance of the loop is R? (d) Find the power dissipated due to the resistance of the loop. Solution: (a) The magnetic flux at time t is given by ( )( π ) π ( ) Φ = A = 0 + bt a = 0 + bt a ( ) where we have chosen the area vector to point into the page, so that we have Φ > 0. At t = 0, π Φ = 0a ( ) (b) Using Faraday s Law, the induced emf is ( + ) dφ ε = = A = ( πa ) = π ba ( ) dt dt dt d d 0 bt 10-

23 (c) The induced current is I ε πba = = (10.9.0) R R and its direction is counterclockwise by Lenz s law. (d) The power dissipated due to the resistance R is πba ( πba ) P= I R= R= R R (10.9.1) Moving Loop A rectangular loop of dimensions l and w moves with a constant velocity v away from an infinitely long straight wire carrying a current I in the plane of the loop, as shown in Figure Let the total resistance of the loop be R. What is the current in the loop at the instant the near side is a distance r from the wire? Solution: Figure A rectangular loop moving away from a current-carrying wire The magnetic field at a distance s from the straight wire is, using Ampere s law: µ I π s 0 = (10.9.) The magnetic flux through a differential area element da= lds of the loop is µ 0I (10.9.3) dφ = da = π s lds where we have chosen the area vector to point into the page, so that Φ > 0. Integrating over the entire area of the loop, the total flux is 10-3

24 µ 0Il r+ wds µ 0Il r + w Φ = ln π = r s π r (10.9.4) Differentiating with respect to t, we obtain the induced emf as µ Il 0 µ Il µ Il 0 dφ d r+ w dr wv ε = = ln = = dt π dt r π r + w r dt π r( r + w) (10.9.5) where v= dr/ dt. Notice that the induced emf can also be obtained by using Eq. (10..): µ 0I µ 0I ε = ( v ) d s = vl[ ( r) ( r+ w) ] = vl πr π( r w) + µ 0Il vw = π rr ( + w) (10.9.6) The induced current is I ε µ 0Il vw = = R π Rr r w ( + ) (10.9.7) Conceptual Questions 1. A bar magnet falls through a circular loop, as shown in Figure Figure (a) Describe qualitatively the change in magnetic flux through the loop when the bar magnet is above and below the loop. (b) Make a qualitative sketch of the graph of the induced current in the loop as a function of time, choosing I to be positive when its direction is counterclockwise as viewed from above.. Two circular loops A and have their planes parallel to each other, as shown in Figure

25 Figure Loop A has a current moving in the counterclockwise direction, viewed from above. (a) If the current in loop A decreases with time, what is the direction of the induced current in loop? Will the two loops attract or repel each other? (b) If the current in loop A increases with time, what is the direction of the induced current in loop? Will the two loops attract or repel each other? 3. A spherical conducting shell is placed in a time-varying magnetic field. Is there an induced current along the equator? 4. A rectangular loop moves across a uniform magnetic field but the induced current is zero. How is this possible? Additional Problems Sliding ar A conducting bar of mass m and resistance R slides on two frictionless parallel rails that are separated by a distance and connected by a battery which maintains a constant emf ε, as shown in Figure Figure Sliding bar A uniform magnetic field is directed out of the page. The bar is initially at rest. Show that at a later time t, the speed of the bar is where τ = mr /. ε v= t / τ (1 e ) 10-5

26 Sliding ar on Wedges A conducting bar of mass m and resistance R slides down two frictionless conducting rails which make an angle θ with the horizontal, and are separated by a distance, as shown in Figure In addition, a uniform magnetic field is applied vertically downward. The bar is released from rest and slides down. Figure Sliding bar on wedges (a) Find the induced current in the bar. Which way does the current flow, from a to b or b to a? (b) Find the terminal speed v t of the bar. After the terminal speed has been reached, (c) What is the induced current in the bar? (d) What is the rate at which electrical energy has been dissipated through the resistor? (e) What is the rate of work done by gravity on the bar? RC Circuit in a Magnetic Field Consider a circular loop of wire of radius r lying in the xy plane, as shown in Figure The loop contains a resistor R and a capacitor C, and is placed in a uniform magnetic field which points into the page and decreases at a rate d / dt = α, with α > 0. Figure RC circuit in a magnetic field 10-6

27 (a) Find the maximum amount of charge on the capacitor. (b) Which plate, a or b, has a higher potential? What causes charges to separate? Sliding ar A conducting bar of mass m and resistance R is pulled in the horizontal direction across two frictionless parallel rails a distance apart by a massless string which passes over a frictionless pulley and is connected to a block of mass M, as shown in Figure A uniform magnetic field is applied vertically upward. The bar is released from rest. Figure Sliding bar (a) Let the speed of the bar at some instant be v. Find an expression for the induced current. Which direction does it flow, from a to b or b to a? You may ignore the friction between the bar and the rails. (b) Solve the differential equation and find the speed of the bar as a function of time Rotating ar A conducting bar of length l with one end fixed rotates at a constant angular speed ω, in a plane perpendicular to a uniform magnetic field, as shown in Figure Figure Rotating bar (a) A small element carrying charge q is located at a distance r away from the pivot point O. Show that the magnetic force on the element is F = qrω. 10-7

28 1 (b) Show that the potential difference between the two ends of the bar is V = ωl Rectangular Loop Moving Through Magnetic Field A small rectangular loop of length l = 10 cm and width w = 8.0 cm with resistance R =.0 Ω is pulled at a constant speed v =.0 cm/s through a region of uniform magnetic field =.0 T, pointing into the page, as shown in Figure At Figure t = 0, the front of the rectangular loop enters the region of the magnetic field. (a) Find the magnetic flux and plot it as a function of time (from t = 0 till the loop leaves the region of magnetic field.) (b) Find the emf and plot it as a function of time. (c) Which way does the induced current flow? Magnet Moving Through a Coil of Wire Suppose a bar magnet is pulled through a stationary conducting loop of wire at constant speed, as shown in Figure Figure Assume that the north pole of the magnet enters the loop of wire first, and that the center of the magnet is at the center of the loop at time t = 0. (a) Sketch qualitatively a graph of the magnetic flux of time. Φ through the loop as a function 10-8

29 (b) Sketch qualitatively a graph of the current I in the loop as a function of time. Take the direction of positive current to be clockwise in the loop as viewed from the left. (c) What is the direction of the force on the permanent magnet due to the current in the coil of wire just before the magnet enters the loop? (d) What is the direction of the force on the magnet just after it has exited the loop? (e) Do your answers in (c) and (d) agree with Lenz's law? (f) Where does the energy come from that is dissipated in ohmic heating in the wire? Alternating-Current Generator An N-turn rectangular loop of length a and width b is rotated at a frequency f in a uniform magnetic field which points into the page, as shown in Figure At time t = 0, the loop is vertical as shown in the sketch, and it rotates counterclockwise when viewed along the axis of rotation from the left. Figure (a) Make a sketch depicting this generator as viewed from the left along the axis of rotation at a time t shortly after t = 0, when it has rotated an angle θ from the vertical. Show clearly the vector, the plane of the loop, and the direction of the induced current. (b) Write an expression for the magnetic flux of time for the given parameters. Φ passing through the loop as a function (c) Show that an induced emf ε appears in the loop, given by ε = π fnbasin( π ft) = ε sin( π ft) 0 (d) Design a loop that will produce an emf with ε 0 = 10 V when rotated at 60 revolutions/sec in a magnetic field of 0.40 T. 10-9

30 EMF Due to a Time-Varying Magnetic Field A uniform magnetic field is perpendicular to a one-turn circular loop of wire of negligible resistance, as shown in Figure The field changes with time as shown (the z direction is out of the page). The loop is of radius r = 50 cm and is connected in series with a resistor of resistance R = 0 Ω. The "+" direction around the circuit is indicated in the figure. Figure (a) What is the expression for EMF in this circuit in terms of z ( t) for this arrangement? (b) Plot the EMF in the circuit as a function of time. Label the axes quantitatively (numbers and units). Watch the signs. Note that we have labeled the positive direction of the emf in the left sketch consistent with the assumption that positive is out of the paper. [Partial Ans: values of EMF are 1.96 V, 0.0 V, 0.98 V]. (c) Plot the current I through the resistor R. Label the axes quantitatively (numbers and units). Indicate with arrows on the sketch the direction of the current through R during each time interval. [Partial Ans: values of current are 98 ma, 0.0 ma, 49 ma] (d) Plot the rate of thermal energy production in the resistor. [Partial Ans: values are 19 mw, 0.0 mw, 48 mw] Square Loop Moving Through Magnetic Field An external force is applied to move a square loop of dimension l l and resistance R at a constant speed across a region of uniform magnetic field. The sides of the square loop make an angle θ = 45 with the boundary of the field region, as shown in Figure At t = 0, the loop is completely inside the field region, with its right edge at the boundary. Calculate the power delivered by the external force as a function of time

31 Figure Falling Loop A rectangular loop of wire with mass m, width w, vertical length l, and resistance R falls out of a magnetic field under the influence of gravity, as shown in Figure The magnetic field is uniform and out of the paper ( = ˆi) within the area shown and zero outside of that area. At the time shown in the sketch, the loop is exiting the magnetic field at speed v= vkˆ. Figure (a) What is the direction of the current flowing in the circuit at the time shown, clockwise or counterclockwise? Why did you pick this direction? (b) Using Faraday's law, find an expression for the magnitude of the emf in this circuit in terms of the quantities given. What is the magnitude of the current flowing in the circuit at the time shown? (c) esides gravity, what other force acts on the loop in the magnitude and direction in terms of the quantities given. ˆ ±k direction? Give its (d) Assume that the loop has reached a terminal velocity and is no longer accelerating. What is the magnitude of that terminal velocity in terms of given quantities? (e) Show that at terminal velocity, the rate at which gravity is doing work on the loop is equal to the rate at which energy is being dissipated in the loop through Joule heating

### Chapter 10. Faraday s Law of Induction

10 10 10-0 Chapter 10 Faraday s Law of Induction 10.1 Faraday s Law of Induction... 10-3 10.1.1 Magnetic Flux... 10-5 10.2 Motional EMF... 10-5 10.3 Faraday s Law (see also Faraday s Law Simulation in

### Eðlisfræði 2, vor 2007

[ Assignment View ] [ Pri Eðlisfræði 2, vor 2007 29a. Electromagnetic Induction Assignment is due at 2:00am on Wednesday, March 7, 2007 Credit for problems submitted late will decrease to 0% after the

### Physics 1653 Exam 3 - Review Questions

Physics 1653 Exam 3 - Review Questions 3.0 Two uncharged conducting spheres, A and B, are suspended from insulating threads so that they touch each other. While a negatively charged rod is held near, but

### Electromagnetism Laws and Equations

Electromagnetism Laws and Equations Andrew McHutchon Michaelmas 203 Contents Electrostatics. Electric E- and D-fields............................................. Electrostatic Force............................................2

### AP Physics C Chapter 23 Notes Yockers Faraday s Law, Inductance, and Maxwell s Equations

AP Physics C Chapter 3 Notes Yockers Faraday s aw, Inductance, and Maxwell s Equations Faraday s aw of Induction - induced current a metal wire moved in a uniform magnetic field - the charges (electrons)

### Faraday s Law; Inductance

This test covers Faraday s Law of induction, motional emf, Lenz s law, induced emf and electric fields, eddy currents, self-inductance, inductance, RL circuits, and energy in a magnetic field, with some

### Faraday s Law Challenge Problem Solutions

Faraday s Law Challenge Problem Solutions Problem 1: A coil of wire is above a magnet whose north pole is pointing up. For current, counterclockwise when viewed from above is positive. For flux, upwards

### Direction of Induced Current

Direction of Induced Current Bar magnet moves through coil Current induced in coil A S N v Reverse pole Induced current changes sign B N S v v Coil moves past fixed bar magnet Current induced in coil as

### 1 of 7 4/13/2010 8:05 PM

Chapter 33 Homework Due: 8:00am on Wednesday, April 7, 2010 Note: To understand how points are awarded, read your instructor's Grading Policy [Return to Standard Assignment View] Canceling a Magnetic Field

### MASSACHUSETTS INSTINUTE OF TECHNOLOGY ESG Physics. Problem Set 9 Solution

MASSACHUSETTS INSTINUTE OF TECHNOLOGY ESG Physics 8. with Kai Spring 3 Problem 1: 3-7 and 8 Problem Set 9 Solution A conductor consists of a circular loop of radius R =.1 m and two straight, long sections,

### Chapter 14 Magnets and

Chapter 14 Magnets and Electromagnetism How do magnets work? What is the Earth s magnetic field? Is the magnetic force similar to the electrostatic force? Magnets and the Magnetic Force! We are generally

### Physics 2212 GH Quiz #4 Solutions Spring 2015

Physics 1 GH Quiz #4 Solutions Spring 15 Fundamental Charge e = 1.6 1 19 C Mass of an Electron m e = 9.19 1 31 kg Coulomb constant K = 8.988 1 9 N m /C Vacuum Permittivity ϵ = 8.854 1 1 C /N m Earth s

### Chapter 14: Magnets and Electromagnetism

Chapter 14: Magnets and Electromagnetism 1. Electrons flow around a circular wire loop in a horizontal plane, in a direction that is clockwise when viewed from above. This causes a magnetic field. Inside

### The purposes of this experiment are to test Faraday's Law qualitatively and to test Lenz's Law.

260 17-1 I. THEORY EXPERIMENT 17 QUALITATIVE STUDY OF INDUCED EMF Along the extended central axis of a bar magnet, the magnetic field vector B r, on the side nearer the North pole, points away from this

### Force on Moving Charges in a Magnetic Field

[ Assignment View ] [ Eðlisfræði 2, vor 2007 27. Magnetic Field and Magnetic Forces Assignment is due at 2:00am on Wednesday, February 28, 2007 Credit for problems submitted late will decrease to 0% after

### Solution Derivations for Capa #10

Solution Derivations for Capa #10 1) A 1000-turn loop (radius = 0.038 m) of wire is connected to a (25 Ω) resistor as shown in the figure. A magnetic field is directed perpendicular to the plane of the

### Chapter 29 Electromagnetic Induction

Chapter 29 Electromagnetic Induction - Induction Experiments - Faraday s Law - Lenz s Law - Motional Electromotive Force - Induced Electric Fields - Eddy Currents - Displacement Current and Maxwell s Equations

### 11. Rotation Translational Motion: Rotational Motion:

11. Rotation Translational Motion: Motion of the center of mass of an object from one position to another. All the motion discussed so far belongs to this category, except uniform circular motion. Rotational

### Physics 126 Practice Exam #3 Professor Siegel

Physics 126 Practice Exam #3 Professor Siegel Name: Lab Day: 1. Which one of the following statements concerning the magnetic force on a charged particle in a magnetic field is true? A) The magnetic force

### Faraday s Law of Induction Motional emf

Faraday s Law of Induction Motional emf Motors and Generators Lenz s Law Eddy Currents Te needle deflects momentarily wen te switc is closed A current flows troug te loop wen a magnet is moved near it,

### Objectives for the standardized exam

III. ELECTRICITY AND MAGNETISM A. Electrostatics 1. Charge and Coulomb s Law a) Students should understand the concept of electric charge, so they can: (1) Describe the types of charge and the attraction

### ElectroMagnetic Induction. AP Physics B

ElectroMagnetic Induction AP Physics B What is E/M Induction? Electromagnetic Induction is the process of using magnetic fields to produce voltage, and in a complete circuit, a current. Michael Faraday

### Magnetic Field of a Circular Coil Lab 12

HB 11-26-07 Magnetic Field of a Circular Coil Lab 12 1 Magnetic Field of a Circular Coil Lab 12 Equipment- coil apparatus, BK Precision 2120B oscilloscope, Fluke multimeter, Wavetek FG3C function generator,

### Electromagnetic Induction

. Electromagnetic Induction Concepts and Principles Creating Electrical Energy When electric charges move, their electric fields vary. In the previous two chapters we considered moving electric charges

### Chapter 22: Electric Flux and Gauss s Law

22.1 ntroduction We have seen in chapter 21 that determining the electric field of a continuous charge distribution can become very complicated for some charge distributions. t would be desirable if we

### AP2 Magnetism. (c) Explain why the magnetic field does no work on the particle as it moves in its circular path.

A charged particle is projected from point P with velocity v at a right angle to a uniform magnetic field directed out of the plane of the page as shown. The particle moves along a circle of radius R.

### Physics 12 Study Guide: Electromagnetism Magnetic Forces & Induction. Text References. 5 th Ed. Giancolli Pg

Objectives: Text References 5 th Ed. Giancolli Pg. 588-96 ELECTROMAGNETISM MAGNETIC FORCE AND FIELDS state the rules of magnetic interaction determine the direction of magnetic field lines use the right

### Physics 9 Fall 2009 Homework 8 - Solutions

1. Chapter 34 - Exercise 9. Physics 9 Fall 2009 Homework 8 - s The current in the solenoid in the figure is increasing. The solenoid is surrounded by a conducting loop. Is there a current in the loop?

### University of California, Berkeley Physics H7B Spring 1999 (Strovink) SOLUTION TO PROBLEM SET 10 Solutions by P. Pebler

University of California, Berkeley Physics H7B Spring 1999 (Strovink) SOLUTION TO PROBLEM SET 10 Solutions by P Pebler 1 Purcell 66 A round wire of radius r o carries a current I distributed uniformly

### Induced voltages and Inductance Faraday s Law

Induced voltages and Inductance Faraday s Law concept #1, 4, 5, 8, 13 Problem # 1, 3, 4, 5, 6, 9, 10, 13, 15, 24, 23, 25, 31, 32a, 34, 37, 41, 43, 51, 61 Last chapter we saw that a current produces a magnetic

### Chapter 14 Magnets and Electromagnetism

Chapter 14 Magnets and Electromagnetism Magnets and Electromagnetism In the 19 th century experiments were done that showed that magnetic and electric effects were just different aspect of one fundamental

### Your Comments. I got so confused with Faraday's Law and Lenz's Law... AREN'T THESE TWO LAWS SUPPOSED TO MEAN THE SAME THING???

Your Comments There were some parts of this prelecture I grasped well while other parts like the generator and the loops I still have trouble with. Can you please clarify how Faraday's and Lenz's law are

### PHYSICS 212 INDUCED VOLTAGES AND INDUCTANCE WORKBOOK ANSWERS

PHYSICS 212 CHAPTER 20 INDUCED VOLTAGES AND INDUCTANCE WORKOOK ANSWERS STUDENT S FULL NAME (y placing your name above and submitting this for credit you are affirming this to be predominantly your own

### Chapter 23 Magnetic Flux and Faraday s Law of Induction

Chapter 23 Magnetic Flux and Faraday s Law of Induction 23.1 Induced EMF 23.2 Magnetic Flux 23.3 Faraday s Law of Induction 23.4 Lenz s Law 23.5 Mechanical Work and Electrical Energy 23.6 Generators and

### Linear DC Motors. 15.1 Magnetic Flux. 15.1.1 Permanent Bar Magnets

Linear DC Motors The purpose of this supplement is to present the basic material needed to understand the operation of simple DC motors. This is intended to be used as the reference material for the linear

### Electromagnetic Induction. Physics 231 Lecture 9-1

Electromagnetic Induction Physics 231 Lecture 9-1 Induced Current Past experiments with magnetism have shown the following When a magnet is moved towards or away from a circuit, there is an induced current

### Eðlisfræði 2, vor 2007

[ Assignment View ] [ Pri Eðlisfræði 2, vor 2007 28. Sources of Magnetic Field Assignment is due at 2:00am on Wednesday, March 7, 2007 Credit for problems submitted late will decrease to 0% after the deadline

### Physics 2220 Module 09 Homework

Physics 2220 Module 09 Homework 01. A potential difference of 0.050 V is developed across the 10-cm-long wire of the figure as it moves though a magnetic field perpendicular to the page. What are the strength

### Problem 1 (25 points)

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02 Spring 2012 Exam Three Solutions Problem 1 (25 points) Question 1 (5 points) Consider two circular rings of radius R, each perpendicular

### Chapter 22 Magnetism

22.6 Electric Current, Magnetic Fields, and Ampere s Law Chapter 22 Magnetism 22.1 The Magnetic Field 22.2 The Magnetic Force on Moving Charges 22.3 The Motion of Charged particles in a Magnetic Field

### 2. In Newton s universal law of gravitation the masses are assumed to be. B. masses of planets. D. spherical masses.

Practice Test: 41 marks (55 minutes) Additional Problem: 19 marks (7 minutes) 1. A spherical planet of uniform density has three times the mass of the Earth and twice the average radius. The magnitude

### 1. The diagram below represents magnetic lines of force within a region of space.

1. The diagram below represents magnetic lines of force within a region of space. 4. In which diagram below is the magnetic flux density at point P greatest? (1) (3) (2) (4) The magnetic field is strongest

### Lecture 10 Induction and Inductance Ch. 30

Lecture 10 Induction and Inductance Ch. 30 Cartoon - Faraday Induction Opening Demo - Thrust bar magnet through coil and measure the current Warm-up problem Topics Faraday s Law Lenz s Law Motional Emf

### 6. ELECTROMAGNETIC INDUCTION

6. ELECTROMAGNETIC INDUCTION Questions with answers 1. Name the phenomena in which a current induced in coil due to change in magnetic flux linked with it. Answer: Electromagnetic Induction 2. Define electromagnetic

### Phys222 Winter 2012 Quiz 4 Chapters 29-31. Name

Name If you think that no correct answer is provided, give your answer, state your reasoning briefly; append additional sheet of paper if necessary. 1. A particle (q = 5.0 nc, m = 3.0 µg) moves in a region

### 1 of 7 10/1/2012 3:17 PM

Assignment Previewer http://www.webassign.net/v4cgijfederici@njit/control.pl 1 of 7 10/1/2012 3:17 PM HW11Faraday (2861550) Question 1 2 3 4 5 6 7 8 9 10 1. Question Details SerPSE8 31.P.011.WI. [1742725]

### Magnetic Fields; Sources of Magnetic Field

This test covers magnetic fields, magnetic forces on charged particles and current-carrying wires, the Hall effect, the Biot-Savart Law, Ampère s Law, and the magnetic fields of current-carrying loops

### Lesson 11 Faraday s Law of Induction

Lesson 11 Faraday s Law of Induction Lawrence B. Rees 006. You may make a single copy of this document for personal use without written permission. 11.0 Introduction In the last lesson, we introduced Faraday

### My lecture slides are posted at Information for Physics 112 midterm, Wednesday, May 2

My lecture slides are posted at http://www.physics.ohio-state.edu/~humanic/ Information for Physics 112 midterm, Wednesday, May 2 1) Format: 10 multiple choice questions (each worth 5 points) and two show-work

### ) 0.7 =1.58 10 2 N m.

Exam 2 Solutions Prof. Paul Avery Prof. Andrey Korytov Oct. 29, 2014 1. A loop of wire carrying a current of 2.0 A is in the shape of a right triangle with two equal sides, each with length L = 15 cm as

### Chapter 33. The Magnetic Field

Chapter 33. The Magnetic Field Digital information is stored on a hard disk as microscopic patches of magnetism. Just what is magnetism? How are magnetic fields created? What are their properties? These

### Faraday's Law da B B r r Φ B B A d dφ ε B = dt

Faraday's Law da Φ ε = r da r dφ dt Applications of Magnetic Induction AC Generator Water turns wheel rotates magnet changes flux induces emf drives current Dynamic Microphones (E.g., some telephones)

### Conceptual: 1, 3, 5, 6, 8, 16, 18, 19. Problems: 4, 6, 8, 11, 16, 20, 23, 27, 34, 41, 45, 56, 60, 65. Conceptual Questions

Conceptual: 1, 3, 5, 6, 8, 16, 18, 19 Problems: 4, 6, 8, 11, 16, 20, 23, 27, 34, 41, 45, 56, 60, 65 Conceptual Questions 1. The magnetic field cannot be described as the magnetic force per unit charge

### Chapter 27 Magnetic Field and Magnetic Forces

Chapter 27 Magnetic Field and Magnetic Forces - Magnetism - Magnetic Field - Magnetic Field Lines and Magnetic Flux - Motion of Charged Particles in a Magnetic Field - Applications of Motion of Charged

### Review Questions PHYS 2426 Exam 2

Review Questions PHYS 2426 Exam 2 1. If 4.7 x 10 16 electrons pass a particular point in a wire every second, what is the current in the wire? A) 4.7 ma B) 7.5 A C) 2.9 A D) 7.5 ma E) 0.29 A Ans: D 2.

### 1. Units of a magnetic field might be: A. C m/s B. C s/m C. C/kg D. kg/c s E. N/C m ans: D

Chapter 28: MAGNETIC FIELDS 1 Units of a magnetic field might be: A C m/s B C s/m C C/kg D kg/c s E N/C m 2 In the formula F = q v B: A F must be perpendicular to v but not necessarily to B B F must be

### Magnetic Forces and Magnetic Fields

1 Magnets Magnets are metallic objects, mostly made out of iron, which attract other iron containing objects (nails) etc. Magnets orient themselves in roughly a north - south direction if they are allowed

### Chapter 20. Magnetic Induction Changing Magnetic Fields yield Changing Electric Fields

Chapter 20 Magnetic Induction Changing Magnetic Fields yield Changing Electric Fields Introduction The motion of a magnet can induce current in practical ways. If a credit card has a magnet strip on its

### PHY2049 Exam #2 Solutions Fall 2012

PHY2049 Exam #2 Solutions Fall 2012 1. The diagrams show three circuits consisting of concentric circular arcs (either half or quarter circles of radii r, 2r, and 3r) and radial segments. The circuits

### Ch.20 Induced voltages and Inductance Faraday s Law

Ch.20 Induced voltages and Inductance Faraday s Law Last chapter we saw that a current produces a magnetic field. In 1831 experiments by Michael Faraday and Joseph Henry showed that a changing magnetic

### Homework #11 203-1-1721 Physics 2 for Students of Mechanical Engineering

Homework #11 203-1-1721 Physics 2 for Students of Mechanical Engineering 2. A circular coil has a 10.3 cm radius and consists of 34 closely wound turns of wire. An externally produced magnetic field of

### Pearson Physics Level 30 Unit VI Forces and Fields: Chapter 12 Solutions

Concept Check (top) Pearson Physics Level 30 Unit VI Forces and Fields: Chapter 1 Solutions Student Book page 583 Concept Check (bottom) The north-seeking needle of a compass is attracted to what is called

### (b) Draw the direction of for the (b) Draw the the direction of for the

2. An electric dipole consists of 2A. A magnetic dipole consists of a positive charge +Q at one end of a bar magnet with a north pole at one an insulating rod of length d and a end and a south pole at

### VELOCITY, ACCELERATION, FORCE

VELOCITY, ACCELERATION, FORCE velocity Velocity v is a vector, with units of meters per second ( m s ). Velocity indicates the rate of change of the object s position ( r ); i.e., velocity tells you how

### PHYS 155: Final Tutorial

Final Tutorial Saskatoon Engineering Students Society eric.peach@usask.ca April 13, 2015 Overview 1 2 3 4 5 6 7 Tutorial Slides These slides have been posted: sess.usask.ca homepage.usask.ca/esp991/ Section

### Outline. Tom Browder (University of Hawaii) Faraday s Law and Magnetic Induction. AC electric generator is based on Faraday s Law

Outline Tom Browder (University of Hawaii) Faraday s Law and Magnetic Induction AC electric generator is based on Faraday s Law Headline Solar flare: Biggest in six years hits the Earth Solar flare: The

### ( )( 10!12 ( 0.01) 2 2 = 624 ( ) Exam 1 Solutions. Phy 2049 Fall 2011

Phy 49 Fall 11 Solutions 1. Three charges form an equilateral triangle of side length d = 1 cm. The top charge is q = - 4 μc, while the bottom two are q1 = q = +1 μc. What is the magnitude of the net force

### Question Bank. 1. Electromagnetism 2. Magnetic Effects of an Electric Current 3. Electromagnetic Induction

1. Electromagnetism 2. Magnetic Effects of an Electric Current 3. Electromagnetic Induction 1. Diagram below shows a freely suspended magnetic needle. A copper wire is held parallel to the axis of magnetic

### ! = "d# B. ! = "d. Chapter 31: Faraday s Law! One example of Faraday s Law of Induction. Faraday s Law of Induction. Example. ! " E da = q in.

Chapter 31: Faraday s Law So far, we e looked at: Electric Fields for stationary charges E = k dq " E da = q in r 2 Magnetic fields of moing charges d = µ o I # o One example of Faraday s Law of Induction

### Induction and Inductance

Induction and Inductance How we generate E by B, and the passive component inductor in a circuit. 1. A review of emf and the magnetic flux. 2. Faraday s Law of Induction 3. Lentz Law 4. Inductance and

### CET Moving Charges & Magnetism

CET 2014 Moving Charges & Magnetism 1. When a charged particle moves perpendicular to the direction of uniform magnetic field its a) energy changes. b) momentum changes. c) both energy and momentum

### Physics 112 Homework 5 (solutions) (2004 Fall) Solutions to Homework Questions 5

Solutions to Homework Questions 5 Chapt19, Problem-2: (a) Find the direction of the force on a proton (a positively charged particle) moving through the magnetic fields in Figure P19.2, as shown. (b) Repeat

### Q28.1 A positive point charge is moving to the right. The magnetic field that the point charge produces at point P (see diagram below) P

Q28.1 A positive point charge is moving to the right. The magnetic field that the point charge produces at point P (see diagram below) P r + v r A. points in the same direction as v. B. points from point

### PSS 27.2 The Electric Field of a Continuous Distribution of Charge

Chapter 27 Solutions PSS 27.2 The Electric Field of a Continuous Distribution of Charge Description: Knight Problem-Solving Strategy 27.2 The Electric Field of a Continuous Distribution of Charge is illustrated.

### Quiz: Work and Energy

Quiz: Work and Energy A charged particle enters a uniform magnetic field. What happens to the kinetic energy of the particle? (1) it increases (2) it decreases (3) it stays the same (4) it changes with

### Midterm Solutions. mvr = ω f (I wheel + I bullet ) = ω f 2 MR2 + mr 2 ) ω f = v R. 1 + M 2m

Midterm Solutions I) A bullet of mass m moving at horizontal velocity v strikes and sticks to the rim of a wheel a solid disc) of mass M, radius R, anchored at its center but free to rotate i) Which of

### Physics 202: Lecture 10, Pg 1

Physics 132: Lecture e 21 Elements of Physics II Forces on currents Agenda for Today Currents are moving charges Torque on current loop Torque on rotated loop Currents create B-fields Adding magnetic fields

### Eðlisfræði 2, vor 2007

[ Assignment View ] [ Print ] Eðlisfræði 2, vor 2007 30. Inductance Assignment is due at 2:00am on Wednesday, March 14, 2007 Credit for problems submitted late will decrease to 0% after the deadline has

### 4/16/ Bertrand

Physics B AP Review: Electricity and Magnetism Name: Charge (Q or q, unit: Coulomb) Comes in + and The proton has a charge of e. The electron has a charge of e. e = 1.602 10-19 Coulombs. Charge distribution

### Profs. A. Petkova, A. Rinzler, S. Hershfield. Exam 2 Solution

PHY2049 Fall 2009 Profs. A. Petkova, A. Rinzler, S. Hershfield Exam 2 Solution 1. Three capacitor networks labeled A, B & C are shown in the figure with the individual capacitor values labeled (all units

### General Physics (PHY 2140)

General Physics (PHY 2140) Lecture 12 Electricity and Magnetism Magnetism Magnetic fields and force Application of magnetic forces http://www.physics.wayne.edu/~apetrov/phy2140/ Chapter 19 1 Department

### March 20. Physics 272. Spring 2014 Prof. Philip von Doetinchem

Physics 272 March 20 Spring 2014 http://www.phys.hawaii.edu/~philipvd/pvd_14_spring_272_uhm.html Prof. Philip von Doetinchem philipvd@hawaii.edu Phys272 - Spring 14 - von Doetinchem - 129 Summary No magnetic

### Chap 21. Electromagnetic Induction

Chap 21. Electromagnetic Induction Sec. 1 - Magnetic field Magnetic fields are produced by electric currents: They can be macroscopic currents in wires. They can be microscopic currents ex: with electrons

### Chapter 30 - Magnetic Fields and Torque. A PowerPoint Presentation by Paul E. Tippens, Professor of Physics Southern Polytechnic State University

Chapter 30 - Magnetic Fields and Torque A PowerPoint Presentation by Paul E. Tippens, Professor of Physics Southern Polytechnic State University 2007 Objectives: After completing this module, you should

### 2015 Pearson Education, Inc. Section 24.5 Magnetic Fields Exert Forces on Moving Charges

Section 24.5 Magnetic Fields Exert Forces on Moving Charges Magnetic Fields Sources of Magnetic Fields You already know that a moving charge is the creator of a magnetic field. Effects of Magnetic Fields

### Induction. d. is biggest when the motor turns fastest.

Induction 1. A uniform 4.5-T magnetic field passes perpendicularly through the plane of a wire loop 0.10 m 2 in area. What flux passes through the loop? a. 5.0 T m 2 c. 0.25 T m 2 b. 0.45 T m 2 d. 0.135

### Electrostatics. Ans.The particles 1 and 2 are negatively charged and particle 3 is positively charged.

Electrostatics [ Two marks each] Q1.An electric dipole with dipole moment 4 10 9 C m is aligned at 30 with the direction of a uniform electric field of magnitude 5 10 4 N C 1. Calculate the net force and

### Physics Notes for Class 12 Chapter 4 Moving Charges and Magnetrism

1 P a g e Physics Notes for Class 12 Chapter 4 Moving Charges and Magnetrism Oersted s Experiment A magnetic field is produced in the surrounding of any current carrying conductor. The direction of this

### 45. The peak value of an alternating current in a 1500-W device is 5.4 A. What is the rms voltage across?

PHYS Practice Problems hapters 8- hapter 8. 45. The peak value of an alternating current in a 5-W device is 5.4 A. What is the rms voltage across? The power and current can be used to find the peak voltage,

### Physics 6C, Summer 2006 Homework 1 Solutions F 4

Physics 6C, Summer 006 Homework 1 Solutions All problems are from the nd edition of Walker. Numerical values are different for each student. Chapter Conceptual Questions 18. Consider the four wires shown

### Faraday s Law and Inductance

Historical Overview Faraday s Law and Inductance So far studied electric fields due to stationary charges and magentic fields due to moving charges. Now study electric field due to a changing magnetic

### Force on a square loop of current in a uniform B-field.

Force on a square loop of current in a uniform B-field. F top = 0 θ = 0; sinθ = 0; so F B = 0 F bottom = 0 F left = I a B (out of page) F right = I a B (into page) Assume loop is on a frictionless axis

### Test - A2 Physics. Primary focus Magnetic Fields - Secondary focus electric fields (including circular motion and SHM elements)

Test - A2 Physics Primary focus Magnetic Fields - Secondary focus electric fields (including circular motion and SHM elements) Time allocation 40 minutes These questions were ALL taken from the June 2010

### Magnetic Fields. I. Magnetic Field and Magnetic Field Lines

Magnetic Fields I. Magnetic Field and Magnetic Field Lines A. The concept of the magnetic field can be developed in a manner similar to the way we developed the electric field. The magnitude of the magnetic

### Lab 7: Faraday Effect and Lenz law Physics 208. What should I be thinking about before I start this lab?

Lab 7: Faraday Effect and Lenz law Physics 208 Name Section This sheet is the lab document your TA will use to score your lab. It is to be turned in at the end of lab. To receive full credit you must use

### Exam 1 Practice Problems Solutions

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8 Spring 13 Exam 1 Practice Problems Solutions Part I: Short Questions and Concept Questions Problem 1: Spark Plug Pictured at right is a typical

### 5. Measurement of a magnetic field

H 5. Measurement of a magnetic field 5.1 Introduction Magnetic fields play an important role in physics and engineering. In this experiment, three different methods are examined for the measurement of

### Chapter 27 Electromagnetic Induction

For us, who took in Faraday s ideas so to speak with our mother s milk, it is hard to appreciate their greatness and audacity. Albert Einstein 27.1 ntroduction Since a current in a wire produces a magnetic