Concepts & Calculations

Note: Each of these problems consists of Concept Questions followed by a related quantitative Problem. The Concept Questions involve little or no mathematics. They focus on the concepts with which the problems deal. Recognizing the concepts is the essential initial step in any problem-solving techinque.

| 75 | Concept Questions Two circuits contain an emf produced by a moving metal rod, like that in Figure 22-4b. The speed of the rod is the same in each circuit, but the bulb in circuit 1 has one-half the resistance of the bulb in circuit 2. The circuits are otherwise identical. Is (a) the motional emf and (b) the current in circuit 1 greater than, the same as, or less than, that in circuit 2? (c) If the speed of the rod in circuit 1 were doubled, how would the power delivered to the light bulb compare to that in circuit 2? Provide a reason for each of your answers.
Problem The resistance of the light bulb in circuit 1 is , and that in circuit 2 is . Determine (a) the ratio of the emfs and (b) the ratio I1/I2 of the currents. (c) If the speed of the rod in circuit 1 is twice that in circuit 2, what is the ratio P1/P2 of the powers? Check to see that your answers are consistent with your answers to the Concept Questions.

| 76 | Concept Questions A magnetic field passes through a stationary wire loop, and its magnitude changes in time according to the graph in the drawing. The direction of the field remains constant, however. There are three equal time intervals indicated in the graph: 0–3.0 s, 3.0–6.0 s, and 6.0–9.0 s. (a) Is the induced emf equal to zero during any of the intervals? (b) During which interval is the magnitude of the induced emf the largest? (c) If the direction of the current induced during (the first interval is clockwise, what is the direction during the third interval? In all cases, provide a reason for your answer.

Problem The loop consists of 50 turns of wire and has an area of 0.15 m2. The magnetic field is oriented parallel to the normal to the loop. For purposes of this problem, this means that in Equation 22.2. (a) For each interval, determine the induced emf. (b) The wire has a resistance of . Determine the induced current for the first and third intervals. Make sure your answers are consistent with your answers to the Concept Questions.

| 77 | Concept Questions The drawing shows a straight wire carrying a current I. Above the wire is a rectangular loop that contains a resistor R. (a) Does the magnetic field produced by the current I penetrate the loop and generate a magnetic flux? (b) When is there an induced current in the loop, if the current I is constant or if it is decreasing in time? (c) When there is an induced magnetic field produced by the loop, does it always have a direction that is opposite to the direction of the magnetic field produced by the current I? Provide a reason for each answer.

Problem If the current I is decreasing in time, what is the direction of the induced current through the resistor R—left to right or right to left? Give your reasoning.

| 78 | Concept Questions A flat coil of wire has an area A, N turns, and a resistance R. It is situated in a magnetic field such that the normal to the coil is parallel to the magnetic field. The coil is then rotated through an angle of 90°, so that the normal becomes perpendicular to the magnetic field. (a) Why is an emf induced in the coil? (b) What determines the amount of induced current in the coil? (c) How is the amount of charge that flows related to the induced current I and the time interval during which the coil rotates?
Problem The coil has an area of 1.5 × 10−3 m2, 50 turns, and a resistance of . During the time when it is rotating, a charge of 8.5 × 10−5 C flows in the coil. What is the magnitude of the magnetic field?

| 79 | Concept Questions A constant current I exists in a solenoid whose inductance is L. The current is then reduced to zero in a certain amount of time. (a) If the wire from which the solenoid is made has no resistance, is there a voltage across the solenoid during the time when the current is constant? (b) If the wire from which the solenoid is made has no resistance, is there an emf across the solenoid during the time that the current is being reduced to zero? (c) Does the solenoid store electrical energy when the current is constant? If so, express this energy in terms of the current and the inductance. (d) When the current is reduced from its constant value to zero, what is the rate at which energy is removed from the solenoid? Express your answer in terms of the initial current, the inductance, and the time during which the current goes to zero.
Problem A solenoid has an inductance of L = 3.1 H and carries a current of I = 15 A. (a) If the current goes from 15 to 0 A in a time of 75 ms, what is the emf induced in the solenoid? (b) How much electrical energy is stored in the solenoid? (c) At what rate must the electrical energy be removed from the solenoid when the current is reduced to zero in 75 ms?

| 80 | Concept Questions The rechargeable batteries for a laptop computer need a much smaller voltage than what a wall socket provides. Therefore, a transformer is plugged into the wall socket and produces the necessary voltage for charging the batteries. (a) Is the transformer a step-up or a step-down transformer? (b) Is the current that goes through the batteries greater than, equal to, or smaller than the current coming from the wall socket? (c) If the transformer has a negligible resistance, is the electric power delivered to the batteries greater than, equal to, or less than the power coming from the wall socket? In all cases, provide a reason for your answer.
Problem The batteries of a laptop computer are rated at 9.0 V, and a current of 225 mA is used to charge them. The wall socket provides a voltage of 120 V. (a) Determine the turns ratio of the transformer, (b) What is the current coming from the wall socket? (c) Find the power delivered by the wall socket and the power sent to the batteries. Be sure your answers are consistent with your answers to the Concept Questions.

| 81 | Concept Questions The drawing shows a coil of copper wire that consists of two semicircles joined by straight sections of wire. In part a the coil is lying flat on a horizontal surface. The dashed line also lies in the plane of the horizontal surface. Starting from the orientation in part a, the smaller semicircle rotates at an angular frequency about the dashed line, until its plane becomes perpendicular to the horizontal surface, as shown in part b. A uniform magnetic field is constant in time and is directed upward, perpendicular to the horizontal surface. The field completely fills the region occupied by the coil in either part of the drawing. (a) In which part of the drawing, if either, does a greater magnetic flux pass through the coil? Account for your answer. (b) As the shape of the coil changes from that in part a of the drawing to that in part b, does an induced current flow in the coil, and, if so, in which direction does it flow? Give your reasoning. To describe the flow, imagine that you are above the coil looking down at it. (c) How is the period T of the rotational motion related to the angular frequency , and in terms of the period, what is the shortest time interval that elapses between parts a and b of the drawing?

Problem The magnitude of the magnetic field is 0.35 T. The resistance of the coil is , and the smaller semicircle has a radius of 0.20 m. The angular frequency at which the small semicircle rotates is 1.5 rad/s. Determine the average current, if any, induced in the coil as the coil changes shape from that in part a of the drawing to that in part b.

| 82 | Concept Questions In a television set the power needed to operate the picture tube comes from the secondary of a transformer. The primary of the transformer is connected to a wall receptacle. (a) How is the power delivered by the receptacle to the primary related to the power delivered by the secondary to the picture tube? Give your answer in the form of an equation, and explain what assumptions are implied when this equation is used. (b) How is the turns ratio of the transformer related to the currents in the primary and the secondary? (c) How is the turns ratio of the transformer related to the voltage across the primary and the voltage across the secondary? (d) Express the turns ratio Ns/Np of the transformer in terms of the power P used by the picture tube, the voltage Vp across the primary, and the current Is in the secondary.
Problem The primary of the transformer is connected to a 120-V receptacle. The picture tube of a television set uses 91 W, and there is 5.5 mA of current in the secondary coil of the transformer to which the tube is connected. Find the turns ratio Ns/Np of the transformer.

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