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Assignments 2/15/11 10:29 AM Student View Summary View Diagnostics View Print View withEdit Assignment Answers Settings per Student MP_Assignment#2 [ Print ] Due: 8:00am on Monday, February 14, 2011 Note: You will receive no credit for late submissions. To learn more, read your instructor's Grading Policy Faraday's Law and Induced Emf Description: Discusses Faraday's law; presents a sequence of questions related to finding the induced emf under different circumstances. Learning Goal: To understand the terms in Faraday's law and to be able to identify the magnitude and direction of induced emf. Faraday's law states that induced emf is directly proportional to the time rate of change of magnetic flux. Mathematically, it can be written as , where is the emf induced in a closed loop, and is the rate of change of the magnetic flux through a surface bounded by the loop. For uniform magnetic fields the magnetic flux is given by , where is the angle between the magnetic field and the normal to the surface of area . To find the direction of the induced emf, one can use Lenz's law: The induced current's magnetic field opposes the change in the magnetic flux that induced the current. For example, if the magnetic flux through a loop increases, the induced magnetic field is directed opposite to the "parent" magnetic field, thus countering the increase in flux. If the flux decreases, the induced current's magnetic field has the same direction as the parent magnetic field, thus countering the decrease in flux. Recall that to relate the direction of the electric current and its magnetic field, you can use the right - hand rule: When the fingers on your right hand are curled in the direction of the current in a loop, your thumb gives the direction of the magnetic field generated by this current. In this problem, we will consider a rectangular loop of wire with sides uniform magnetic field and placed in a region where a exists (see the diagram). The resistance of the loop is . Initially, the field is perpendicular to the plane of the loop and is directed out of the page. The loop can rotate about either the vertical or horizontal axis, passing through the midpoints of the opposite sides, as shown. http://session.masteringphysics.com/myct/assignments Page 1 of 23 Assignments 2/15/11 10:29 AM Part A Which of the following changes would induce an electromotive force (emf) in the loop ? When you consider each option, assume that no other changes occur. Check all that apply. ANSWER: The magnitude of increases. The magnitude of decreases. The loop rotates about the vertical axis (vertical dotted line) shown in the diagram. The loop rotates about the horizontal axis (horizontal dotted line) shown in the diagram. The loop moves to the right while remaining in the plane of the page. The loop moves toward you, out of the page, while remaining parallel to itself. Part B Find the flux through the loop. Express your answer in terms of ANSWER: , , and . = Part C If the magnetic field steadily decreases from to zero during a time interval , what is the magnitude of the induced emf? Hint C.1 Find the change in magnetic flux What is the total change in magnetic flux Express your answer in terms of ANSWER: , , and during this time interval? . = http://session.masteringphysics.com/myct/assignments Page 2 of 23 Assignments 2/15/11 10:29 AM Express your answer in terms of , , , and . ANSWER: = Part D If the magnetic field steadily decreases from to zero during a time interval , what is the magnitude of the induced current ? Express your answer in terms of , , , , and the resistance of the wire. ANSWER: = Part E If the magnetic field steadily decreases from to zero during a time interval , what is the direction of the induced current ? ANSWER: clockwise counterclockwise The flux decreases, so the induced magnetic field must be in the same direction as the original (parent) magnetic field. Therefore, the induced magnetic field is out of the page. Using the right - hand rule, we deduce that the direction of the current is counterclockwise. Part F Which of the following changes would result in a clockwise emf in the loop ? When you consider each option, assume that no other changes occur. Check all that apply. ANSWER: The magnitude of increases. The magnitude of decreases. The loop rotates through 45 degrees about the vertical axis (vertical dotted line) shown in the diagram. http://session.masteringphysics.com/myct/assignments Page 3 of 23 Assignments 2/15/11 10:29 AM The loop rotates through 45 degrees about the horizontal axis (horizontal dotted line) shown in the diagram. The loop moves to the right while remaining in the plane of the page. The loop moves toward you, out of the page, while remaining parallel to itself. Clockwise emf implies that the induced magnetic field is directed into the page. Therefore, the magnetic flux of the original field must be increasing. Only the first option corresponds to increasing flux. Oscillations in an LC circuit. Description: Derive a differential equation for the current in an LC circuit, guess the solution, and answer conceptual questions about the solution. Learning Goal: To understand the processes in a series circuit containing only an inductor and a capacitor. Consider the circuit shown in the figure. This circuit contains a capacitor of capacitance an inductor of inductance and . The resistance of all wires is considered negligible. Initially, the switch is open, and the capacitor has a charge . The switch is then closed, and the changes in the system are observed. It turns out that the equation describing the subsequent changes in charge, current, and voltage is very similar to that of simple harmonic motion, studied in mechanics. To obtain this equation, we will use the law of conservation of energy. Initially, the entire energy of the system is stored in the capacitor. When the circuit is closed, the capacitor begins to discharge through the inductor. As the charge of the capacitor decreases, so does its energy. On the other hand, as the current through the inductor increases, so does the energy stored in the inductor. Assuming no heat loss and no emission of electromagnetic waves, energy is conserved, and at any point in time, the sum of the energy stored in the capacitor and the energy stored in the inductor is a constant : , where is the charge on the capacitor and is the current through the inductor ( and are functions of time, of course). For this problem, take clockwise current to be positive. http://session.masteringphysics.com/myct/assignments Page 4 of 23 Assignments 2/15/11 10:29 AM Part A Using the expression for the total energy of this system, it is possible to show that after the switch is closed, , where is a constant. Find the value of the constant Hint A.1 Since . The derivative of the total energy is a constant, an expression for . Differentiate the expression for in the problem introduction to obtain . Hint A.1.1 A helpful derivative formula Recall that for any differentiable function , we have . This result is derived using the chain rule. ANSWER: = Hint A.2 The current in a series circuit In this one- loop circuit, the current through the capacitor is always equal to that through the inductor. Hint A.3 Charge and current Current is defined as the time derivative of charge, but watch the sign. With the conventions adopted here (clockwise current and a positive charge on the upper plate of the capacitor) . Therefore, . http://session.masteringphysics.com/myct/assignments Page 5 of 23 Assignments 2/15/11 10:29 AM Express your answer in terms of and . ANSWER: = This expression can also be obtained by substituting into the equation obtained from Kirchhoff's loop rule. It is always good to solve a problem in more than one way! Part B From mechanics, you may recall that when the acceleration of an object is proportional to its coordinate, , such motion is called simple harmonic motion , and the coordinate depends on time as , where , the argument of the harmonic function at , is called the phase constant . Find a similar expression for the charge correct value of Hint B.1 on the capacitor in this circuit. Do not forget to determine the based on the initial conditions described in the problem. The phase constant Initially, the charge of the capacitor is at a maximum; hence, the argument of the cosine function must be zero at the moment the switch is closed, . Hint B.2 Find the period of the oscillations What is the period of these oscillations ? Hint B.2.1 A relation between the period and angular frequency The period and the angular frequency are related by . Express your answer in terms of , , and other variables given in the introduction. ANSWER: = Express your answer in terms of , , and . Use the cosine function in your answer. ANSWER: http://session.masteringphysics.com/myct/assignments Page 6 of 23 Assignments 2/15/11 10:29 AM ANSWER: = Part C What is the current Hint C.1 in the circuit at time after the switch is closed ? Current as a derivative of the charge Note that . Use the equation you have found for Express your answer in terms of , , to find . , and other variables given in the introduction. ANSWER: = Part D Recall that the top plate of the capacitor is positively charged at . In what direction does the current in the circuit begin to flow immediately after the switch is closed ? ANSWER: There is no current because there cannot be any current through the capacitor. clockwise counterclockwise Part E Immediately after the switch is closed, what is the direction of the EMF in the inductor ? (Recall that the direction of the EMF refers to the direction of the back - current or the induced electric field in the inductor.) ANSWER: There is no EMF until the current reaches its maximum. clockwise counterclockwise In the remaining parts, assume that the period of oscillations is 8.0 milliseconds. Also, keep in mind that the top plate of the capacitor is positively charged at . http://session.masteringphysics.com/myct/assignments Page 7 of 23 Assignments 2/15/11 10:29 AM Part F At what time does the current reach its maximum value for the first time ? Hint F.1 How to approach the problem Try drawing a graph of the current as a function of time, in units of , where is the period of the current oscillation. Graph of Hint F.2 The figure shows a graph of current ANSWER: as a function of time . 0.0 ms 2.0 ms 4.0 ms 6.0 ms 8.0 ms Part G At what moment does the EMF become zero for the first time ? Hint G.1 Relationship between the induced EMF and the current. The EMF is proportional to the rate of change of the current; the EMF becomes zero when Hint G.2 . Graph of http://session.masteringphysics.com/myct/assignments Page 8 of 23 Assignments 2/15/11 10:29 AM ANSWER: 0.0 2.0 4.0 6.0 8.0 Part H At what moment does the current reverse direction for the first time ? Hint H.1 How to approach the problem Try drawing a graph of the current as a function of time, in units of , where is the period of the current oscillation. Hint H.2 Graph of The figure shows a graph of current as a function of time. http://session.masteringphysics.com/myct/assignments Page 9 of 23 Assignments ANSWER: 2/15/11 10:29 AM 0.0 ms 2.0 ms 4.0 ms 6.0 ms 8.0 ms Part I At what moment does the EMF reverse direction for the first time ? Hint I.1 Relationship between the induced EMF and the current. The EMF is proportional to the rate of change of the current. Therefore, the EMF will change direction when Hint I.2 ANSWER: changes sign. Graph of 0.0 ms 2.0 ms 4.0 ms 6.0 ms 8.0 ms http://session.masteringphysics.com/myct/assignments Page 10 of 23 Assignments 2/15/11 10:29 AM Part J At what moment does the energy stored in the inductor reach its maximum for the first time ? Hint J.1 Formula for the energy in an inductor The energy in an inductor is given by , where Hint J.2 is the current flowing through the inductor. Graph of The figure shows a graph of current as a function of time. Hint J.3 Conservation of energy By conservation of energy, the energy in the inductor will be at a maximum when the energy in the capacitor is at a minimum. An examination of the function for the charge on the capacitor in Part B reveals that this occurs when the charge on the capicitor is zero. ANSWER: 0.0 ms 2.0 ms 4.0 ms 6.0 ms 8.0 ms Part K http://session.masteringphysics.com/myct/assignments Page 11 of 23 Assignments 2/15/11 10:29 AM At what time Hint K.1 do the capacitor and the inductor possess the same amount of energy for the first time ? Use conservation of energy Since the total energy is conserved, the energy of the capacitor (like that of the inductor) equals onehalf of the total energy at the time in question. Hint K.2 Energy and charge Since the energy of the capacitor is proportional to the square of its charge, you need to find a time when . Hint K.3 A useful trigonometric result Recall that . Express your answer in milliseconds. ANSWER: = Part L What is the direction of the current in the circuit 22.0 milliseconds after the switch is closed ? Hint L.1 Use the periodic nature of the process The current would have the same direction as it has 6.0 milliseconds after the switch is closed. (Can you see why ? ) Hint L.2 Graph of The figure shows a graph of the current as a function of time. http://session.masteringphysics.com/myct/assignments Page 12 of 23 Assignments ANSWER: 2/15/11 10:29 AM The current is zero. clockwise counterclockwise Part M What is the direction of the EMF in the circuit 42.0 milliseconds after the switch is closed ? Hint M.1 ANSWER: Graph of The EMF is zero clockwise counterclockwise Self - Inductance of a Solenoid Description: Walks through calculation of self - inductance for a single solenoid with some discussion at the end. Learning Goal: To learn about self - inductance from the example of a long solenoid. To explain self - inductance, it is helpful to consider the specific example of a long solenoid, as shown in the figure. This solenoid has only one winding, and so the EMF induced by its changing current appears across the solenoid itself. This contrasts with mutual inductance, where this voltage appears across a second coil wound on the same cylinder as the first. Assume that the solenoid has radius , length http://session.masteringphysics.com/myct/assignments Page 13 of 23 Assignments 2/15/11 10:29 AM along the z axis, and is wound with turns per unit length so that the total number of turns is equal to . Assume that the solenoid is much longer than its radius. As the current through the solenoid changes, the resulting magnetic flux through the solenoid will also change, and an electromotive force will be generated across the solenoid according to Faraday's law of induction: . Faraday's law implies the following relation between the self - induced EMF across the solenoid and the current passing through it: . The "direction of the EMF" is determined with respect to the direction of positive current flow, and represents the direction of the induced electric field in the inductor. This is also the direction in which the "back - current" that the inductor tries to generate will flow. Part A Suppose that the current in the solenoid is magnetic field due to this current ? Express your answer in terms of (such as ANSWER: . Within the solenoid, but far from its ends, what is the , quantities given in the introduction, and relevant constants ). = Note that this field is independent of the radial position (the distance from the axis of symmetry) as long as it is measured at a point well inside the solenoid. Part B What is the magnetic flux through a single turn of the solenoid ? Express your answer in terms of the magnetic field , quantities given in the introduction, and any needed constants. ANSWER: = Part C http://session.masteringphysics.com/myct/assignments Page 14 of 23 Assignments 2/15/11 10:29 AM Suppose that the current varies with time, so that . Find the electromotive force induced across the entire solenoid due to the change in current through the entire solenoid. Hint C.1 Find the flux in terms of the current In Part B you found the flux through a single turn of the solenoid. Now find the flux through the entire solenoid. Express your answer in terms of constants such as . ANSWER: Hint C.2 , other quantities given in the introduction, and various = Find the EMF for the entire solenoid You now have an expression for the magnetic flux that passes through the solenoid. From this, you should be able to derive an expression for the EMF in the solenoid. Suppose the total magnetic flux through the solenoid is . What is the electromotive force generated in the solenoid by the changing flux ? Express your answer in terms of and its derivative, and other variables given in the introduction. ANSWER: = Hint C.3 Putting it together You have now worked out three things: the magnetic field from ; the flux from this field; the EMF for the entire solenoid. Put them together and you have the answer! Express your answer in terms of , , , and . ANSWER: = Part D http://session.masteringphysics.com/myct/assignments Page 15 of 23 Assignments 2/15/11 10:29 AM The self - inductance is related to the self - induced EMF for a long solenoid. (Hint: The self - inductance by the equation will always be a positive quantity.) Express the self - inductance in terms of the number of turns per length and . Find , the physical dimensions , and relevant constants. ANSWER: = This definition of the inductance is identical to another definition you may have encountered: , where is the magnetic flux due to a current in the inductor. To see the correspondence you should differentiate both sides of this equation with respect to time and use Faraday's law, i.e., . Now consider an inductor as a circuit element. Since we are now treating the inductor as a circuit element, we must discuss the voltage across it, not the EMF inside it. The important point is that the inductor is assumed to have no resistance. This means that the net electric field inside it must be zero when it is connected in a circuit. Otherwise, the current in it will become infinite. This means that the induced electric field deposits charges on and around the inductor in such a way as to produce a nearly equal and opposite electric field such that fields produced by charges (like ! Kirchhoff's loop law defines voltages only in terms of ), not those produced by changing magnetic fields (like ). So if we wish to continue to use Kirchhoff's loop law, we must continue to use this definition consistently. That is, we must define the voltage alone (note that the integral is from A to B rather than from B to A, hence the positive sign). So finally, , where we have used of and the definition . Part E Which of the following statements is true about the inductor in the figure in the problem introduction, where is the current through the wire? Hint E.1 A fundamental inductance formula The self - inductance is related to the voltage across the inductor through the equation . Note that unlike the EMF, it does not have a minus sign. ANSWER: If is positive, the voltage at end A will necessarily be greater than that at end B. http://session.masteringphysics.com/myct/assignments Page 16 of 23 Assignments 2/15/11 10:29 AM If is positive, the voltage at end A will necessarily be greater than that at end B. If is positive, the voltage at end A will necessarily be less than that at end B. If is positive, the voltage at end A will necessarily be less than that at end B. Part F Now consider the effect that applying an additional voltage to the inductor will have on the current already flowing through it (imagine that the voltage is applied to end A, while end B is grounded). Which one of the following statements is true ? Hint F.1 A fundamental inductance formula The self - inductance is related to the voltage across the inductor through the equation . Note that unlike the EMF, it does not have a minus sign. However, when applying Kirchhoff's rules and traversing the inductor in the direction of current flow, there will be a term , just as traversing a resistor gives a term . ANSWER: If is positive, then will necessarily be positive and will be will necessarily be negative and will be negative. If is positive, then negative. If is positive, then could be positive or negative, while will necessarily be negative. If is positive, then will necessarily be positive and will be positive. If is positive, then could be positive or negative while will necessarily be positive. If is positive, then will necessarily be negative and will be positive. Note that when you apply Kirchhoff's rules and traverse the inductor in the direction of current flow, you are interested in , just as traversing a resistor gives a term . In sum: when an inductor is in a circuit and the current is changing, the changing magnetic field in the inductor produces an electric field. This field opposes the change in current, but at the same time deposits charge, producing yet another electric field. The net effect of these electric fields is that the current changes, but not abruptly. The "direction of the EMF" refers to the direction of the first, induced, electric field. http://session.masteringphysics.com/myct/assignments Page 17 of 23 Assignments 2/15/11 10:29 AM Tactics Box 34.1 Using Lenz's Law Description: Knight Tactics Box 34.1 Using Lenz's Law is illustrated. Learning Goal: To practice Tactics Box 34.1 Using Lenz's Law. Lenzs law is a useful rule for determining the direction of the induced current in a loop. Specifically, it says that there is an induced current in a closed conducting loop if and only if the magnetic flux through the loop is changing. The direction of the induced current is such that the induced magnetic field opposes the change in the flux. The following tactics box summarizes the essential steps in using Lenz's law. TACTICS BOX 34.1 Using Lenz's law Determine the direction of the applied magnetic field. The field must pass through the loop. Determine how the flux is changing. Is it increasing, decreasing, or staying the same? Determine the direction of an induced magnetic field that will oppose the change in the flux: Increasing flux: The induced magnetic field points opposite the applied magnetic field. Decreasing flux: The induced magnetic field points in the same direction as the applied magnetic field. Steady flux: There is no induced magnetic field. Determine the direction of the induced current. Use the right - hand rule to determine the current direction in the loop that generates the induced magnetic field you found in Step 3. Follow the steps above to solve the following problem: A wire carries a current in the direction indicated in the figure. A loop is located next to the wire. If the current in the wire increases, is there a clockwise current around the loop, a counterclockwise current, or no current ? Part A Determine the direction of the magnetic field of the current - carrying wire. Hint A.1 The right- hand rule for fields Place your right thumb in the direction of the current. Curl your fingers around the wire to indicate a circle. Your fingers point in the direction of the magnetic field lines around the wire. http://session.masteringphysics.com/myct/assignments Page 18 of 23 Assignments 2/15/11 10:29 AM ANSWER: upward downward into the plane of the screen out of the plane of the screen Part B As the current in the wire increases, the magnetic field around the wire increases in magnitude without changing its direction. Does this affect the flux through the loop, and if so, how? ANSWER: As the magnetic field of the wire increases, the flux through the loop increases as well. As the magnetic field of the wire increases, the flux through the loop decreases. As the magnetic field of the wire increases, the flux through the loop remains steady. Now, follow Step 3 in the tactics box above and find the direction of the induced magnetic field. Part C Based on the information found in the previous parts, complete the statement below. Hint C.1 The right- hand rule for fields Place your right thumb in the direction of the current. Curl your fingers around the wire to indicate a circle. Your fingers point in the direction of the magnetic field lines around the wire. ANSWER: There is a counterclockwise current induced in the loop. Self - Inductance of a Coaxial Cable Description: Find the self - inductance per unit length of a coaxial cable. A coaxial cable consists of alternating coaxial cylinders of conducting and insulating material. Coaxial cabling is the primary type of cabling used by the cable television industry and is also widely used for computer networks such as Ethernet, on account of its superior ability to transmit large volumes of electrical signal with minimum distortion. Like all other kinds of cables, however, coaxial cables also have some self - inductance that has undesirable effects, such as producing some distortion and heating. Part A http://session.masteringphysics.com/myct/assignments Page 19 of 23 Assignments 2/15/11 10:29 AM Consider a long coaxial cable made of two coaxial cylindrical conductors that carry equal currents opposite directions (see figure). The inner cylinder is a small solid conductor of radius in . The outer cylinder is a thin walled conductor of outer radius , electrically insulated from the inner conductor. Calculate the self - inductance per unit length cable and of this coaxial cable. ( is the inductance of part of the is the length of that part.) Due to what is known as the "skin effect", the current flows down the (outer) surface of the inner conducting cylinder and back along the outer surface of the outer conducting cylinder. However, you may ignore the thickness of the outer cylinder. Hint A.1 How to approach the problem The self - inductance of the cable can be found if both the current and the magnetic flux through the cable are known. Using Ampre's law, you can calculate the magnetic field due to the cable; you will find that there is a magnetic field in the region between the two conductors. Thus, calculate the magnetic flux per unit length through the region between the two cylinders. Since the current in the cable is given, now you should have enough information to calculate the self - inductance of the cable. Hint A.2 Find the magnetic field inside the inner cylinder Consider the inner conducting cylinder. It carries a current magnitude of the magnetic field at a distance along its outer surface. What is the from the axis of the cylinder, in the region inside the cylinder? Hint A.2.1 Ampre's law Ampre's law states that the line integral of the magnetic field calculated around any closed path is given by , where is the permeability of free space, and is the net current through the area enclosed by the path. The best loop to use is a circle centered on the cylinders' axis, because the magnetic field will be tangent to the loop at all points, making the dot product easier to compute. Express your answer in terms of some or all of the variables , , and , the permeability of free space. ANSWER: http://session.masteringphysics.com/myct/assignments Page 20 of 23 Assignments 2/15/11 10:29 AM ANSWER: = Hint A.3 Find the magnetic field between the cylinders Now consider both conducting cylinders, each carring equal currents magnitude of the magnetic field at a distance in opposite directions. Find the from the axis of the cable in the region between the two cylinders. Hint A.3.1 Ampre's law Ampre's law states that the line integral of the magnetic field calculated around any closed path is given by , where is the permeability of free space, and is the net current through the area enclosed by the path. The best loop to use is a circle centered on the cylinders' axis, because the magnetic field will be tangent to the loop at all points, making the dot product easier to compute. Express your answer in terms of some or all of the variables , , and , the permeability of free space. ANSWER: = Between the conducting cylinders, the magnetic field is identical to one that would be produced by a wire carrying current located along the cylinders' axis. Hint A.4 Find the magnetic field outside the cable Finally, find the magnitude of the magnetic field at a distance region outside the cable. Again, assume that a current from the axis of the cable in the flows along the surface of the inner cylinder and flows back on the surface of the outer cylinder. Hint A.4.1 Ampre's law Ampre's law states that the line integral of the magnetic field calculated around any closed path is given by , where is the permeability of free space, and is the net current through the area enclosed by the path. The best loop to use is a circle centered on the cylinders' axis, because the magnetic field will be tangent to the loop at all points, making the dot product easier to compute. http://session.masteringphysics.com/myct/assignments Page 21 of 23 Assignments 2/15/11 10:29 AM Express your answer in terms of some or all of the variables , , and , the permeability of free space. ANSWER: = The fields due to the opposite currents cancel each other out. Hint A.5 Find the magnetic flux in the cable Find the magnetic flux per unit length inner cylinder and in the cable due to the currents . Let be the radius of the that of the outer cylinder. Hint A.5.1 How to approach the problem The magnetic field of the coaxial cable exists only in the region between the two cylinders. Therefore, you need to calculate the magnetic flux per unit length through this portion of the cable. To do that, think about the direction of the magnetic field (using the right - hand rule) and what area element to use. Hint A.5.2 How to calculate the magnetic flux Given a finite area in a magnetic field , the magnetic flux through the area is defined by the following integral calculated over the area, , where is an infinitesimal area element. The magnetic field due to the cable exists only in the region between the cylinders and it is perpendicular to any plane containing the cable axis. Therefore, to calculate the magnetic flux it is most convenient to choose an area element of length parallel to the cable axis, and of width lying in the plane containing the axis. The figure below shows such an area element with the cross section of the cable parallel to the cable axis depicted. Such an area element is perpendicular to the magnetic field, making the dot product easy. http://session.masteringphysics.com/myct/assignments Page 22 of 23 Assignments 2/15/11 10:29 AM Express your answer in terms of some or all the variables , , , and , the permeability of free space. ANSWER: = Hint A.6 Formula for self - inductance The self inductance of a circuit that carries a current is given by , where is the magnetic flux through the closed loop of the circuit. Express your answer in terms of some or all the variables , , , and , the permeability of free space. ANSWER: = The capacitance per unit length , where of such a coaxial cable is . So the product is the speed of light in vacuum! As you might have guessed, this is not a coincidence, but a result that is quite generally true for such systems. Score Summary: Your score on this assignment is 0%. 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