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Chapter 29: Induction Updated 5/12/08 After the discovery that currents create magnetic fields it was widely speculated that the reverse should also be true, and that magnetic fields should produce currents. It took Michael Faraday more than a decade to discover how it happened. The key was motion; the magnetic field had to be changing with time in order to generate a current in nearby wires. The phenomenon is now called "electromagnetic induction" and nearly all electricity in generated in this manner today. The basic principle at work is Faraday's law, which states that a region of changing magnetic flux generates an Emf as if a battery where connected (dm /dt= -E, where m = B.dA). The negative sign in the equation is known as "Lenz' law" and it establishes the direction of the induced current. It also makes Faraday's law consistent with the conservation of energy principle. Faraday's law can be stated in another way that turns out to be equivalent and is called the"motional Emf"(E= (vxB) .dl). This formulation of Faraday's law is very useful when the induction is caused by a conductor moving through a magnetic field. As our understanding of induction improved we came to realize that changing magnetic fields generate electric fields in space which in turn can drive currents in wires. The final version of Faraday's law is E .dl = -dm /dt. In this assignment you will learn to apply Faraday's law in all its different versions. 1. Answer the following questions. Always explain your reasoning. a) In Chapter 27, we pointed out that a static magnetic field cannot change the energy of a charged particle. Is this true of a changing magnetic field? Discuss. b) Show that the volt is the correct SI unit for the rate of change of magnetic flux, making Faraday's law dimensionally correct. c) Is it possible to produce an induced current that never changes? How or why not? Could you produce an induced current that was steady for some finite time? How or whv not? d) When you push a bar magnet into a conducting loop, you do work. What happens when you pull it out the other sides. e) An electric generator is being turned at constant speed. A load resistor R is connected across the generator terminals. If the electrical resistance of the load is lowered, does the generator get easier or harder to turn? f) Could you tell whether a given electric field arises from electric charge or from a changing magnetic field? How or why not? 2. The figure shows two concentric conducting loops, the outer connected to a batterv and a switch. The switch is initially open. It is then closed, left closed for a while, and then reopened. Describe the currents in the inner loop during the entire procedure.

+

3. A bar magnet is moved steadily through a conducting ring, as shown. Sketch qualitatively the current and power dissipation in the ring as functions of time. Take as positive a current flowing out of the plane of the page at the top of the ring, and indicate the position of the magnet on your time axis.

S N

x x B x

x x x

x x x

4. A square wire loop 3.0 m on a side is perpendicular to a uniform magnetic field of 2.0 T. A 6-V light bulb is in series with the loop, as shown. The magnetic field is reduced steadily to zero over a time t. (a) Find t such that the light will shine at full brightness during this time.

(b) Which way will the loop current flow? 5. A 2.0-m-long solenoid is 15 cm in diameter and consists of 2000 turns of wire. The current in the solenoid is increasing at the rate of 1.0 kA/s. (a) Find the current in a wire loop with diameter 10 cm and resistance 5.0 , lying inside the solenoid in a plane perpendicular to the loop axis. (b) Repeat for a similarly oriented loop with diameter 25 cm, lying entirely outside the solenoid. (c) Now assume that there is a +q charge at rest inside the solenoid 5.0 cm from the solenoid axis. Will there be a force on it? In which direction would this force be? 6. A solenoid 2.0 m long and 30 cm in diameter consists of 5000 turns of wire. A 5-turn coil with negligible resistance is wrapped around the solenoid and connected to a 180- resistor, as shown in the figure. The direction of the current in the solenoid is such that the solenoid's magnetic field points to the right. At time t = 0 the solenoid current begins to decay exponentially, being given by I= Ioe-t/, where Io= 85 A, = 2.5 s, and t is the time in seconds. (a) What is the direction of the current in the resistor as the solenoid current decays? (b) What is the value of the resistor current at t = 1.0 s and t = 5.0 s? (c) Sketch a rough graph of the resistor current vs time.

R

Problem 6,7,8

I

Problem7

B I time

7. Now assume that the current in the same solenoid above has the form shown in the graph. Sketch a qualitative plot of the resistor current vs. time. Take a left-to-right resistor current as positive. What is the basic mathematical relationship between the two graphs? 8. Now assume that the solenoid current is given by I = Io sin t, where Io = 85 A and = 210 rad/s (a) Find an expression for the resistor current. (b) What is the peak current in the resistor? (c) What is the resistor current when the solenoid current is a maximum?

B 9. A square conducting loop of side s = 0.50 m and resistance R = 5.0 moves x x x to the right with speed v = 0.25 m /s. At time t = 0 its rightmost edge enters a v s uniform magnetic field B = 1.0 T pointing into the page, as shown. The x x x magnetic field covers a square region of width w = 0.75 m. Plot (a) the current and (b) the power dissipation in the loop as functions of time, taking a clock-wise current as positive and covering the time until the entire loop has exited the field region. w (c) Derive a general expression for the total work done by the agent pulling the loop, then determine the value in this specific case. x x x

10. Repeat the problem above with the following changes: the sides of the square loop are s=0.75 m and the dimensions of the square region of the magnetism is w= 0.5 m. 11. In problem 6 above, describe qualitatively the changes that would occur in the answers if the conducting loop had a triangular shape such as the one illustrated, instead of a square shape.

s s

12. A generator consists of a rectangular coil 75 cm by 1.6 m, spinning in a 0.25-T magnetic field. The generator is to produce a 60-Hz alternating emf (i.e., E = Eo sin 2ft, where f = 60 Hz) with peak value 6.7 kV. (a) how many turns must it have? (b) What is the maximum flux through the coils? (c) If the resistance of the wires is 50, sketch the current and power vs. time. (d) What would change if the frequency doubled?..the resistance halved?

13. The figure below shows a pair of parallel conducting rails a distance apart in a uniform magnetic field B. A resistance R is connected across the rails, and a conducting bar of negligible resistance is being pulled along the rails with velocity v to the right. (a) What is the direction of the current in the resistor? (b) At what rate must work be done by the external agent pulling the bar? (c) The resistor is replaced by an ideal voltmeter. To which rail should the positive meter terminal be connected if the meter is to indicate a positive voltage? At what rate must work be done by the agent pulling the bar?

x

R

x x x v

x x x

x B l x

V

x x x

x x x v

x x x

x B l x

R

x x

+E

x x x v

x x x

x B l x

x x

x

Problem 13&15

Problem 13

Probem 14

14. A battery of emf E is inserted in series with the resistor in the same circuit above, with its positive terminal toward the top rail. The bar is initially at rest, and now no external agent pulls it. (a) Describe the bar's subsequent motion. (b) The bar eventually reaches a constant speed. Why? (c) What is that constant speed? Express your answers in terms of the magnetic field, the battery emf, and the rail spacing . Does the resistance R affect the final speed? If not, what role does it play? 15.The bar in Problem 13 has mass m and is initially at rest. A constant force F is applied to the bar, pulling it to the right. (a) Describe the motion of the bar as the force continues to pull on it and sketch a rough graph of the velocity of the bar vs. time. (b) Formulate Newton's second law for the bar as a differential equation involving both v and a = dv/dt. (c) Use your equation to show that the bar's acceleration becomes zero when its speed reaches the value (FR/B2 2). (c) Show by direct substitution that your equation is satisfied if v as vs. time is given by v(t) = (FR/B2 2)(1-e-t/), where , the time constant is mR/B22. 16. A pair of parallel conducting rails 10 cm apart lie at right angles to a uniform magnetic field B of magnitude 2.0 T. as shown in the figure. A 5.0- and a 10- resistor lie across the rails and are free to slide along them. (a) The 5- resistor is held fixed, and the 10- resistor is pulled to the right at 50 cm/s. What are the direction and magnitude of the x x x x induced current? B 5 10 (b) Now the 10- resistor is held fixed, and the 5- resistor is pulled x x x to the left at 50 cm/s. What are the direction and magnitude of the induced x x x x current? (c) What is the power dissipation in the 10- resistor in both cases? 17. In same circuit as above the 10- resistor is being moved to the right at a constant 50 cm/s. The 5- l resistor, initially at rest, is placed across the conducting rails. Describe qualitatively its subsequent motion, and determine its final speed. 18. A rectangular conducting loop of resistance R, mass m, and width w falls into a uniform magnetic field B. If the loop is long enough and the field region has a great enough vertical extent, the loop will reach a terminal speed. (a) Why? (b) Find an expression for the terminal speed. (c) What will be the direction of the loop current as the loop enters the field? (d) Describe the motion of the loop once it is fully within the region of magnetism. 19. A toroidal coil of square cross section has inner radius a and outer radius b. It consists of N turns of wire and carries a time-varying current I=Iosin t. A single-turn wire loop encircles the toroid, passing through its center hole as shown. Find an expression for the peak emf induced in the loop. Hint: You will first have

x x x B

x w x v x

x x x

a b

to integrate to find an expression for the flux through the square cross-section. 20. A conducting bar of mass m slides down the conducting wedges shown. The wedges are separated by a distance , connected at the top by a resistance R, and make an angle with the vertical. A uniform magnetic field B points horizontally, as shown. When released from rest the bar soon reaches a constant speed. (a) Find an expression for this speed. (b) Which end of the bar is +? Hint: Draw a free-body diagram for the forces on the bar. (c) Repeat the problem with the magnetic field directed vertically downward. 21. The illustration shows an unusual design for a generator, consisting of a conducting bar that rotates about a central axis while making contact with a conducting ring of radius R. A uniform magnetic field is perpendicular to the ring. Wires from the axis and ring carry power to a load. Find an expression for the emf induced in this generator when the bar rotates with angular speed . 22. A copper disk 90 cm in diameter is spinning at 3600 rpm about a conducting axle through its center, as shown in the figure. A uniform 1.5-T magnetic field is perpendicular to the disk, as shown. A stationary conducting brush maintains contact with the disk's rim, and a voltmeter is placed across the contact. Determine the voltmeter reading.

R v B l x x x x B x x x x x x R x x x

B V

23. A circular wire loop of resistance R and radius a lies with its plane perpendicular to a uniform magnetic field. The field strength changes from an initial value B1 to a final value B2. Show, by integrating the loop current over time, that the total charge that moves around the ring is q =a2(B2 ­ B1)/R. Note that this result is independent of how the field changes with time. 24. A flip coil consists of a small coil used to measure magnetic fields. The flip coil is placed in a magnetic field with its plane perpendicular to the field, and then rotated abruptly through 180' about an axis in the plane of the coil. The coil is connected to instrumentation to measure the total charge Q that flows during this process. If the coil has N turns of area A and if its rotation axis is perpendicular to the magnetic field, show that the field strength is given by B = QR/2NA, where R is the coil resistance. 25. A uniform magnetic field points into the page in the figure. In the same region an electric field points straight up, but increases with position at the rate of 10 V/m2 as you move to the right. Apply Faraday's law to a rectangular loop to show that the magnetic field must be changing with time, and calculate the rate of change. Problem 25 Problem 26 Problem 27

x x x x B x x x x x 1.2m x x v x x x x x E x x x x x E x x x x 0.50m B + E B x x x h

Problem 28 26. The figure shows a top view of a tokamak. The magnetic field in the center is confined to a circular area of radius 50 cm, and during a pulse it increases at the rate of 5.1 T/ms. (a) What is the magnitude of the induced electric field in the tokamak, 1.2 m from the center of the B field region? (b) What is the field direction? (c) If a proton circles the tokamak once at this radius, going with the electric field, how much energy does it gain?

27. The figure shows a magnetic field pointing into the page; the field is confined to a layer of thickness h in the vertical direction, but extends infinitely to the left and right. The field strength is increasing with time: B = bt, where b is a constant. Find an expression for the electric field at all points outside the field region. Hint: Apply Faraday's law in integral form to an appropriate path above and below the layer. 28. Use Faraday's law to show that the electric field produced by charges on the plates of a parallel-plate capacitor cannot end abruptly at the edges of the plates. Hint: Apply Faraday's law to the loop shown in the figure.

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