(a) I f the resistance of the resistor in Fig. 29-38

The magnetic flux through a coil of wire containing two loops changes at a constant rate from 58 Wb to +38 Wb in 0.42 s. What is the emf induced in the coil?

The north pole of the magnet in Fig. 29-36 is being inserted into the coil. In which direction is the induced current flowing through the resistor RI fs n P FIGURE 29- 36 Problem 2.

The rectangular loop shown in Fig. 29-37 is pushed into the magnetic field which points inward. In what direction is the induced current?

A 22.0-cm-diameter loop of wire is initially oriented perpendicular to a 1.5-T magnetic field. The loop is rotated so that its plane is parallel to the field direction in 0.20 s. What is the average induced emf in the loop?

A circular wire loop of radius r = 12 cm is immersed in a uniform magnetic field B = 0.500 T with its plane normal to the direction of the field. If the field magnitude then decreases at a constant rate of 0.010 T/s, at what rate should r increase so that the induced emf within the loop is zero?

A 10.8-cm-diameter wire coil is initially oriented so that its plane is perpendicular to a magnetic field of 0.68 T pointing up. During the course of 0.16 s, the field is changed to one of 0.25 T pointing down. What is the average induced emf in the coil?

A 16-cm-diameter circular loop of wire is placed in a 0.50-T magnetic field, (a) When the plane of the loop is perpendicular to the field lines, what is the magnetic flux through the loop? (b) The plane of the loop is rotated until it makes a 35 angle with the field lines. What is the angle 6 in F,n. 291a for this situation? fr ) What is the magnetic flux

(a) I f the resistance of the resistor in Fig. 29-38 is slowly increased, what is the direction of the current induced in the small circular loop inside the larger loop? (b) What would it be if the small loop were placed outside the larger one, to the left? FIGURE 29-38 Problem 8.

f the solenoid in Fig. 29-39 is being pulled away from the loop shown, in what direction is the induced current in

The magnetic field perpendicular to a circular wire loop 8.0 cm in diameter is changed from +0.52 T to -0.45 T in 180 ms, where + means the field points away from an observer and toward the observer, (a) Calculate the induced emf. (b) In what direction does the induced current flow?

A circular loop in the plane of the paper lies in a 0.75-T magnetic field pointing into the paper. I f the loops diameter changes from 20.0 cm to 6.0 cm in 0.50 s, (a) what is the direction of the induced current, (b ) what is the magnitude of the average induced emf, and (c) if the coil resistance is 2.5 f l, what is the average induced current?

Part of a single rectangular loop of wire with dimensions shown in Fig. 29-40 is situated inside a region of uniform magnetic field of 0.650 T. The total resistance of the loop is 0.280 fl. Calculate the force required to pull the loop from the field (to the right) at a constant velocity of 3.40 m/s. Neglect gravity.FIGURE 29-40 Problem 12.

While demonstrating Faradays law to her class, a physics professor inadvertently moves the gold ring on her finger from a location where a 0.80-T magnetic field points along her finger to a zero-field location in 45 ms. The 1.5-cm-diameter ring has a resistance and mass of 55 /Lift and 15 g, respectively, (a) Estimate the thermal energy produced in the ring due to the flow of induced current. (b) Find the temperature rise of the ring, assuming all of the thermal p.nfirov nrodur.ed ones into innreasina the rinas

A 420-turn solenoid, 25 cm long, has a diameter of 2.5 cm. A 15-turn coil is wound tightly around the center of the solenoid. I f the current in the solenoid increases uniformly from 0 to 5.0 A in 0.60 s, what w ill be the induced emf in the short coil during this time?

A 22.0-cm-diameter coil consists of 28 turns of circular copper wire 2.6 mm in diameter. A uniform magnetic field, perpendicular to the plane of the coil, changes at a rate of 8.65 X 1 0 _ 3 T / s . Determine (a) the current in the loop, and (b) the rate at which thermal energy is produced.

A power line carrying a sinusoidally varying current with frequency / = 60 Hz and peak value / 0 = 55 kA runs at a height of 7.0 m across a farmers land (Fig. 29-41). The farmer constructs a vertically oriented 2.0-m-high 10-turn rectangular wire coil below the power line. The farmer hopes to use the induced voltage in this coil to power 120-Volt electrical equipment, which requires a sinusoidally varying voltage with frequency / = 60 Hz and peak value Vq = 170 V. What should the length of the coil be? Would this be unethical?FIGURE 29-41 Problem 16.

The magnetic field perpendicular to a single 18.2-cmdiameter circular loop of copper wire decreases uniformly from 0.750 T to zero. I f the wire is 2.35 mm in diameter, how much charge moves past a point in the coil during this operation?

The magnetic flux through each loop of a 75-loop coil is given by (8.81 0.5l3) X 10_2T-m 2, where the time t is in seconds, (a) Determine the emf % as a function of time, (ft) What is ^ a t t = 1.0 s and t = 4.0 s?

A 25-cm-diameter circular loop of wire has a resistance of 150 fl. It is initia lly in a 0.40-T magnetic field, with its plane perpendicular to B, but is removed from the field in 120 ms. Calculate the electric energy dissipated in this process.

The area of an elastic circular loop decreases at a constant rate, d A /d t = 3.50 X 10_2m2/s. The loop is in a magnetic field B = 0.28 T whose direction is perpendicular to the plane of the loop. A t t = 0, the loop has area A = 0.285 m2. Determine the induced emf at t = 0, and at t = 2.00 s.

Suppose the radius of the elastic loop in Problem 20 increases at a constant rate, d r /d t = 4.30 cm/s. Determine the emf induced in the loon at t = 0 and at

A single circular loop of wire is placed inside a long solenoid with its plane perpendicular to the axis of the solenoid. The area of the loop is A x and that of the solenoid, which has n turns per unit length, is A 2. A current I = Iqcoscot flows in the solenoid turns. What is the induced emf in the small loop?

We are looking down on an elastic conducting loop with resistance R = 2.0 H, immersed in a magnetic field. The fields magnitude is uniform spatially, but varies with time t according to B (t) = at, where a = 0.60 T/s. The area A of the loop also increases at a constant rate, according to A ( t) = A 0 + p t, where A q = 0.50 m2 and /3 = 0.70 m2/s. Find the magnitude and direction (clockwise or counterclockwise, when viewed from above the page) of the induced current within the loop at time t = 2.0 s if the magnetic field (a) is parallel to the plane of the loop to the right; (b) is perpendicular to the plane of the loop, down.

Inductive battery chargers, which allow transfer of electrical power without the need fo r exposed electrical contacts, are commonly used in appliances that need to be safely immersed in water, such as electric toothbrushes. Consider the following simple model for the power transfer in an inductive charger (Fig. 29-42). Within the chargers plastic base, a primary coil o f diameter d with nF turns per unit length is connected to a homes ac wall outlet so that a current I = Iq sin^Tr/Y) flows within it. When the toothbrush is seated on the base, an iV-turn secondary coil inside the toothbrush has a diameter only slightly greater than d and is centered on the primary. Find an expression fo r the emf induced in the secondary coil. [This y induced emf recharges the battery.]FIGURE 29-42 Problem 24.

(a) Determine the magnetic flux through a square loop of side a (Fig. 29-43) if one side is parallel to, and a distance b from, a straight wire that carries a current I. (b) I f the loop is pulled away from the wire at speed v, what emf is induced in it? (c) Does the induced current flow clockwise or counterclockwise? (d) Determine the force F required to pull the loop away. FIGURE 29-43 Problems 25 and 26.

Determine the emf induced in the square loop in Fig. 29-43 if the loop stays at rest and the current in the straight wire is given by I { t ) = (15.0 A ) sin(2500?) where t is in seconds. The distance a is 12.0 cm, and b is 15.0 cm.

The moving rod in Fig. 29-12b is 13.2 cm long and generates an emf of 120 mV while moving in a 0.90-T magnetic field. What is its speed?

The moving rod in Fig. 29-12b is 12.0 cm long and is pulled at a speed of 15.0 cm/s. I f the magnetic field is 0.800 T, calculate the emf developed.

In Fig. 29-12a, the rod moves to the right with a speed of 1.3 m/s and has a resistance of 2.5 fl. The rail separation is = 25.0 cm. The magnetic field is 0.35 T, and the resistance of the U-shaped conductor is 25.0 XI at a given instant. Calculate (a) the induced emf, (b) the current in the U-shaped conductor, and (c) the external force needed to keep the rods velocity constant at that instant.

If the U-shaped conductor in Fig. 29-12a has resistivity p, whereas that of the moving rod is negligible, derive a formula for the current I as a function of time. Assume the rod starts at the bottom of the U at t = 0, and moves with uniform speed v in the magnetic field B. The cross-sectional area of the rod and all parts of the U is A.

Suppose that the U-shaped conductor and connecting rod in Fig. 2 9-12a are oriented vertically (but still in contact) so that the rod is falling due to the gravitational force. Find the terminal speed of the rod if it has mass m = 3.6 grams, length i = 18 cm, and resistance R = 0.001312. It is falling in a uniform horizontal field B = 0.060 T. Neglect the resistance of the U-shaped conductor.

When a car drives through the Earths magnetic field, an emf is induced in its vertical 75.0-cm-long radio antenna. If the Earths field (5.0 X 10-5 T) points north with a dip angle of 45, what is the maximum emf induced in the antenna and which direction(s) w ill the car be moving to produce this maximum value? The cars speed is 30.0 m/s on a horizontal road.

A conducting rod rests on two long frictionless parallel rails in a magnetic field B (_L to the rails and rod) as in Fig. 29-44. (a) I f the rails are horizontal and the rod is given an in itia l push, w ill the rod travel at constant speed even though a magnetic field is present? (b ) Suppose at t = 0, when the rod has speed v = v0 , the two rails are connected electrically by a wire from point a to point b. Assuming the rod has resistance R and the rails have negligible resistance, determine the speed of the rod as a function of time. Discuss your answer.FIGURE 29-44 Problems 33 and 34.

Suppose a conducting rod (mass ra, resistance R) rests on two frictionless and resistanceless parallel rails a distance I apart in a uniform magnetic field B (_L to the rails and to the rod) as in Fig. 29-44. A t t = 0, the rod is at rest and a source of emf is connected to the points a and b. Determine the speed of the rod as a function of time if (a) the source puts out a constant current I, (b) the source puts out a constant emf %. (c) Does the rod reach a terminal speed in either case? I f so, what is it?

A short section of wire, of length a, is moving with velocity v, parallel to a very long wire carrying a current I as shown in Fig. 29-45. The near end of the wire section is a distance b from the long wire. Assuming the vertical wire is very long compared to a + b, determine the emf between the ends of the short section. Assume v is (a) in the same direction as I, (b) in the opposite direction to I. FIGURE 29-45 Problem 35.

The generator of a car idling at 875-rpm produces 12.4 V. What w ill the output be at a rotation speed of 1550 rpm assuming nothing else changes?

A simple generator is used to generate a peak output voltage of 24.0 V. The square armature consists of windings that are 5.15 cm on a side and rotates in a field of 0.420 T at a rate of 60.0 rev/s. How many loops of wire should be wound on the square armature?

A simple generator has a 480-loop square coil 22.0 cm on a side. How fast must it turn in a 0.550-T field to produce a 120-V peak output?

Show that the rms output of an ac generator is Vrms = N A B a ) / \/ l where w = li r f .

A 250-loop circular armature coil with a diameter of 10.0 cm rotates at 120 rev/s in a uniform magnetic field of strength 0.45 T. What is the rms voltage output of the generator? What would you do to the rotation frequency in order to double the rms voltage output?

The back emf in a motor is 72 V when operating at 1200 rpm. What would be the back emf at 2500 rpm if the magnetic field is unchanged?

A motor has an armature resistance of 3.05 fi. I f it draws 7.20 A when running at fu ll speed and connected to a 120-V line, how large is the back emf?

What w ill be the current in the motor of Example 29-10 if the load causes it to run at half speed?

The back emf in a motor is 85 V when the motor is operating at 1100 rpm. How would you change the motors magnetic field if you wanted to reduce the back emf to 75 V

A dc generator is rated at 16 kW, 250 V, and 64 A when it rotates at 1000 rpm. The resistance of the armature windings is 0.400. (a) Calculate the no-load voltage at 1000 rpm (when there is no circuit hooked up to the generator). (b ) Calculate the full-load voltage (i.e. at 64 A ) when the generator is run at 750 rpm. Assume that the magnitude of the magnetic field remains constant.

A transformer has 620 turns in the primary coil and 85 in the secondary coil. What kind of transformer is this, and by what factor does it change the voltage? By what factor does it change the current?

Neon signs require 12 kV for their operation. To operate from a 240-V line, what must be the ratio of secondary to primary turns of the transformer? What would the voltage output be if the transformer were connected backward?

A model-train transformer plugs into 120-V ac and draws 0.35 A while supplying 7.5 A to the train, (a) What voltage is present across the tracks? (b ) Is the transformer step-up or step-down?

The output voltage of a 75-W transformer is 12 V, and the input current is 22 A. (a) Is this a step-up or a step-down transformer? (b ) By what factor is the voltage multiplied?

I f 65 MW of power at 45 kV (rms) arrives at a town from a generator via 3.0-fl transmission lines, calculate (a) the emf at the generator end of the lines, and (b) the fraction of the power generated that is wasted in the lines.

Assume a voltage source supplies an ac voltage of amplitude V0 between its output terminals. I f the output terminals are connected to an external circuit, and an ac current of amplitude / 0 flows out of the terminals, then the equivalent resistance of the external circuit is Req = Vq/ I 0. (a) I f a resistor R is connected directly to the output terminals, what is R eq? (b) I f a transformer with Np and N$ turns in its primary and secondary, respectively, is placed between the source and the resistor as shown in Fig. 29-46, what is Req? [Transformers can be used in ac circuits to alter the apparent resistance of circuit elements, such as loud speakers, in order to maximize transfer of power.]

Design a dc transmission line that can transmit 225 MW of electricity 185 km with only a 2.0% loss. The wires are to be made of aluminum and the voltage is 660 kV.

Suppose 85 kW is to be transmitted over two 0.100-ft lines. Estimate how much power is saved if the voltage is stepped up from 120 V to 1200 V and then down again, rather than simply transmitting at 120 V. Assume the transformers are each 99% efficient.

In a circular region, there is a uniform magnetic field B pointing into the page (Fig. 29-47). An xy coordinate system has its origin at the circular regions center. A free positive point charge +<2 = 1.0 juC is initially at rest at a position x = +10 cm on the x axis. If the magnitude of the magnetic field is now decreased at a rate of -0.10 T/s, what force (magnitude and direction) w ill act on +Q7FIGURE 29-47 Problem 54.

The betatron, a device used to accelerate electrons to high energy, consists of a circular vacuum tube placed in a magnetic field (Fig. 29-48), into which electrons are injected. The electromagnet produces a field that (1) keeps the electrons in their circular orbit inside the tube, and (2) increases the speed of the electrons when B changes. (a) Explain how the electrons are accelerated. (See Fig. 29-48.) (b) In what direction are the electrons moving in Fig. 29-48 (give directions as if looking down from above)? (c) Should B increase or decrease to accelerate the electrons? (d) The magnetic field is actually 60 Hz ac; show that the electrons can be accelerated only during \ of a cycle (asos)- (During this time they make hundreds of thousands of revolutions and acquire very high energy.)FIGURE 29-48 Problems 55 and 56

Show that the electrons in a betatron, Problem 55 and Fig. 29-48, are accelerated at constant radius if the magnetic field B0 at the position of the electron orbit in the tube is equal to half the average value of the magnetic field (i?avg) over the area of the circular orbit at each moment: B0 = |i? avg. (This is the reason the pole faces have a rather odd shape, as indicated in Fig. 29-48.)

Find a formula for the net electric field in the moving rod of Problem 34 as a function of time for each case, (a) and (b).

Suppose you are looking at two current loops in the plane of the page as shown in Fig. 29-49. When the switch S is closed in the left-hand coil, (a) what is the direction of the induced current in the other loop? (b) What is the situation after a long time? (c) What is the direction of the induced current in the right-hand loop if that loop is quickly pulled horizontally to the right (S having been closed for a long time)?FIGURE 29-49 Problem 58.

A square loop 27.0 cm on a side has a resistance of 7.50 fi. It is initially in a 0.755-T magnetic field, with its plane perpendicular to B, but is removed from the field in 40.0 ms. Calculate the electric energy dissipated in this process

Power is generated at 24 kV at a generating plant located 85 km from a town that requires 65 MW of power at 12 kV. Two transmission lines from the plant to the town each have a resistance of 0.10 ft/km . What should the output voltage of the transformer at the generating plant be for an overall transmission efficiency of 98.5%, assuming a perfect transformer?

A circular loop of area 12 m2 encloses a magnetic field perpendicular to the plane of the loop; its magnitude is B(t) = (10T/s). The loop is connected to a 7.5-ft resistor

The primary windings of a transformer which has an 85% efficiency are connected to 110-V ac. The secondary windings are connected across a 2.4-ft, 75-W lightbulb. (a) Calculate the current through the primary windings of the transformer, (b) Calculate the ratio of the number of primary windings of the transformer to the number of secondary windings of the transformer.

A pair of power transmission lines each have a 0.80-ft resistance and carry 740 A over 9.0 km. If the rms input voltage is 42 kV, calculate (a) the voltage at the other end, (b) the power input, (c) power loss in the lines, and (d) the power output.

Show that the power loss in transmission lines, PL , is given by Pl = {Pj )2Rl/V 2, where PT is the power transmitted to the user, V is the delivered voltage, and RL is the resistance of the power lines.

A high-intensity desk lamp is rated at 35 W but requires only 12 V. It contains a transformer that converts 120-V household voltage, (a) Is the transformer step-up or stepdown? (b) What is the current in the secondary coil when the lamp is on? (c) What is the current in the primary coil? (d) What is the resistance of the bulb when on?

Two resistanceless rails rest 32 cm apart on a 6.0 ramp. They are joined at the bottom by a 0.60-ft resistor. A t the top a copper bar of mass 0.040 kg (ignore its resistance) is laid across the rails. The whole apparatus is immersed in a

A coil with 150 turns, a radius of 5.0 cm, and a resistance of 12 O surrounds a solenoid with 230 turns/cm and a radius of 4.5 cm; see Fig. 29-50. The current in the solenoid changes at a constant rate from 0 to 2.0 A in 0.10 s. Calculate the magnitude and direction of the induced current in the 150-turn coil.FIGURE 29-50 Problem 67.

A search coil for measuring B (also called a flip coil) is a small coil with N turns, each of cross-sectional area A. It is connected to a so-called ballistic galvanometer, which is a device to measure the total charge Q that passes through it in a short time. The flip coil is placed in the magnetic field to be measured with its face perpendicular to the field. It is then quickly rotated 180 about a diameter. Show that the total charge Q that flows in the induced current during this short flip time is proportional to the magnetic field B. In particular, show that B is given by B = QR 2NA where R is the total resistance of the circuit, including that of the coil and that of the ballistic galvanometer which measures the charge Q.

ring with a radius of 3.0 cm and a resistance of 0.025 f l is rotated about an axis through its diameter by 90 in a magnetic field of 0.23 T perpendicular to that axis. What is the largest number of electrons that would flow past a fixed point in the ring as this process is accomplished?

A flashlight can be made that is powered by the induced current from a magnet moving through a coil of wire. The coil and magnet are inside a plastic tube that can be shaken causing the magnet to move back and forth through the coil. Assume the magnet has a maximum field strength of 0.05 T. Make reasonable assumptions and specify the size of the coil and the number o f turns necessary to ligh t a standard 1-watt, 3-V flashlight bulb.

A small electric car overcomes a 250-N friction force when traveling 35 km/h. The electric motor is powered by ten 12-V batteries connected in series and is coupled directly to the wheels whose diameters are 58 cm. The 270 armature coils are rectangular, 12 cm by 15 cm, and rotate in a 0.60-T magnetic field, (a) How much current does the motor draw to produce the required torque? (b ) What is the back emf? (c) How much power is dissipated in the coils? (d) What percent of the input power is used to drive the car?

What is the energy dissipated as a function of time in a circular loop of 18 turns of wire having a radius of 10.0 cm and a resistance of 2.0 XI if the plane of the loop is perpendicular to a magnetic field given by

A thin metal rod of length rotates with angular velocity (o about an axis through one end (Fig. 29-51). The rotation axis is perpendicular to the rod and is parallel to a uniform magnetic field B. Determine the emf developed between the ends of the rod. FIGURE 29-51 Problem 73

The magnetic field of a shunt-wound dc motor is produced by field coils placed in parallel with the armature coils. Suppose that the field coils have a resistance of 36.0 O and the armature coils 3.00 Cl. The back emf at fu ll speed is 105 V when the motor is connected to 115 V dc. (a) Draw the equivalent circuit for the situations when the motor is just starting and when it is running fu ll speed, (b) What is the total current drawn by the motor at start up? (c) What is the total current drawn when the motor runs at fu ll speed?

Apply Faradays law, in the form of Eq. 29-8, to show that the static electric field between the plates of a parallel-plate capacitor cannot drop abruptly to zero at the edges, but must, in fact, fringe. Use the path shown dashed in Fig. 29-52.FIGURE 29-52 Problem 75.

A circular metal disk of radius R rotates with angular velocity co about an axis through its center perpendicular to its face. The disk rotates in a uniform magnetic field B whose direction is parallel to the rotation axis. Determine the emf induced between the center and the edges.

What is the magnitude and direction of the electric field at each point in the rotating disk of Problem 76?

A circular-shaped circuit of radius r, containing a resistance R and capacitance C, is situated with its plane perpendicular to a spatially uniform magnetic field B directed into the page (Fig. 29-53). Starting at time t = 0, the voltage difference V\>a = Va across the capacitor plates is observed to increase with time t according to = K)(l - e~t/T), where V0 and t are positive constants. Determine dB /d t, the rate at which the magnetic field magnitude changes with time. Is B becoming larger or smaller as time increases?FIGURE 29-53

In a certain region of space near Earths surface, a uniform horizontal magnetic field of magnitude B exists above a level defined to be y = 0. Below y = 0, the field abruptly becomes zero (Fig. 29-54). A vertical square wire loop has resistivity p, mass density pm, diameter d, and side length t It is initia lly at rest with its lower horizontal side at y = 0 and is then allowed to fall under gravity, with its plane perpendicular to the direction of the magnetic field. (a) While the loop is s till partially immersed in the magnetic field (as it falls into the zero-field region), determine the magnetic drag force that acts on it at the moment when its speed is v. (b) Assume that the loop achieves a terminal velocity vT before its upper horizontal side exits the field. Determine a formula for v^. (c) I f the loop is made of copper and B = 0.80 T, find v j.FIGURE 29-54 Problem 79.

In an experiment, a coil was mounted on a low-friction cart that moved through the magnetic field B of a permanent magnet. The speed of the cart v and the induced voltage V were simultaneously measured, as the cart moved through the magnetic field, using a computer-interfaced motion sensor and a voltmeter. The Table below shows the collected data: Speed, v (m/s) 0.367 0.379 0.465 0.623 0.630 Induced voltage, V (V) 0.128 0.135 0.164 0.221 0.222 (a) Make a graph of the induced voltage, V, vs. the speed, v. Determine a best-fit linear equation for the data. Theoretically, the relationship between V and v is given by V = B N lv where N is the number of turns of the coil, B is the magnetic field, and is the average of the inside and outside widths of the coil. In the experiment, B = 0.126 T, N = 50, and = 0.0561 m. (b) Find the % error between the slope of the experimental graph and the theoretical value for the slope, (c) For each of the measured speeds v, determine the theoretical value of V and find the % error of each.