Magnetism. Clicker Questions. Question M1.01. Description: Introducing the directionality of the magnetic force on a charge.

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1 19 Magnetism Clicker Questions Question M1.01 Description: Introducing the directionality of the magnetic force on a charge. Question In a certain region of space there is a uniform magnetic field pointing in the positive z-direction (+z). In what direction should a negative point charge move to experience a force in the positive x-direction (+x)? 1. in the positive z-direction (+z). in the negative z-direction (z) 3. in the positive x-direction (+x) 4. in the negative x-direction (x) 5. in the positive y-direction (+y) 6. in the negative y-direction (y) 7. It can move in any direction. 8. It is impossible for the force to be in the +x-direction when the magnetic field is in the +z-direction. Commentary Purpose: To develop your understanding of the direction of the magnetic force on a moving point charge. Discussion: The direction of the magnetic force is always perpendicular to both the velocity of the point charge and the direction of the magnetic field. So, for example, if velocity v and magnetic field are in the xy plane, then the force is either in the +z or z-direction, because the z-axis is perpendicular to the xy plane. In this case, we want to have a force in the +x-direction and the magnetic field is in the +z-direction. Therefore, we need to have the velocity be somewhere in the yz plane without being along the z-axis. Many possible directions of motion lie in the xy plane, but only two are among the listed answers: (5) and (6). Which of those is correct? Should the velocity be in the +y-direction or the y-direction? According to the right-hand rule, the vector v is in the +x-direction when v is in the +y-direction and is in the +z-direction. ut the magnetic force is q(v ). Since q is negative, the force is in the +x-direction when v is in the y-direction and is in the +z-direction. Key Points: A magnetic force on a moving charge acts in a direction perpendicular to both the magnetic field and the charge s velocity. There are two directions perpendicular to any (non-parallel) magnetic field and velocity directions; the right-hand rule tells you which one is meant by the cross product. The magnetic force on a negative charge is in the opposite direction of a magnetic force on a positive charge. 171

2 17 Chapter 19 For Instructors Only Students may be less conversant with the right-hand rule than you think! (Also, some left-handed people inadvertently use their left hands.) Students choosing answer (5) are probably overlooking the sign of the charge. You should elicit the reasoning of students choosing the correct as well as incorrect answers, since some students commit two errors that cancel out. Students may improperly generalize from this result, thinking that only when the charge is moving along the y-axis does the force point along the x-axis. The minimal requirements (within the given constraints on force and magnetic field) are that the x component of velocity is zero and the y component of velocity is negative; there are no restrictions on the z component of velocity. This point can be raised during discussion, or used as the basis for a follow-up question. Question M1.0a Description: Exploring charged particle dynamics in magnetic fields. Question In each of the following situations, point charge q moves in a uniform magnetic field. The strength of the magnetic field is indicated by the density of field lines. In each situation, the initial speed v of the charge is the same. For which situation(s) will the charge q travel the longest distance in a certain time T? q v 1 v q 3 q v v q 4 5 q v & 3 7. & ,, 3 & ,, 3, 4 & Cannot be determined

3 Magnetism 173 Commentary Purpose: To develop your understanding of how a magnetic field affects a moving charge. Discussion: The magnetic force on a point charge is F = q(v ), where q is the amount of charge, v is the charge s velocity, and is the magnetic field strength at its location in space. The cross product means that the force is perpendicular to both the field and the velocity at all times. Since no other forces act on the point charge, the magnetic force is the net force, so the acceleration is also perpendicular to the velocity and magnetic field. Since the acceleration is always perpendicular to the velocity, the speed cannot change. Since the charge has the same speed in all five situations, it must travel the same distance in all of them. (Note, however, that its displacement will not be the same for all the situations.) Key Points: The magnetic force on a point charge q moving with velocity v in a magnetic field is q(v ). The magnetic force on a moving point charge is always perpendicular to its direction of motion and to the field. If the only force on a moving point charge is the magnetic force, its speed stays constant. For Instructors Only It is important to have students who pick one of the other choices verbalize their reasons. This will reveal the nature of their misunderstanding or confusion about the magnetic force (or perhaps about Newtonian motion). Some students might think that the magnetic field is much like the electric field, and therefore, the answer depends on the density of field lines or the angle between the direction of motion and the field. Students who choose answer (5) may be interpreting the question as asking about displacement rather than distance, and correctly reasoning that the charge traveling in a straight line will have the largest magnitude of displacement. Students choosing (10) might also be thinking about displacement, but not realizing they can reason without having a time and other values to calculate with. The next question in this set asks about the displacement. If you are using both questions, defer any discussion of displacement until then. One tactic is to present both questions in sequence without discussing or revealing the answer to the first, and then discussing them together. Students misinterpreting this question as about displacement will likely realize their mistake when they see the next. This is good: it sensitizes them to the importance of paying attention to detail and to the precise meaning words have in physics. Question M1.0b Description: Exploring charged particle dynamics in magnetic fields. Question In each of the following situations, point charge q moves in a uniform magnetic field. The strength of the magnetic field is indicated by the density of field lines. In each situation, the initial speed v of the charge is the same. For which situation(s) will the charge q have the largest displacement in a certain time T?

4 174 Chapter 19 q v 1 v q 3 q v v q 4 5 q v & 3 7. & ,, 3, 4 & 5 9. None of the above 10. Cannot be determined Commentary Purpose: To develop your understanding of how a magnetic field affects a moving charge. Discussion: The magnetic force on a point charge is F = q(v ), where q is the amount of charge, v is the charge s velocity, and is the magnetic field strength at its location in space. The cross product means that the force depends on the angle between v and in a way that might seem unintuitive to you. Another way to write the magnetic force is to focus on its magnitude F = qvsin( θ ) = qv, where θ is the angle between the velocity and the magnetic field and v is the component of the velocity v perpendicular to the magnetic field. In situation (5), the velocity and magnetic field are parallel to each other, so v = 0, and therefore the acceleration is zero and the charge moves in a straight line. This is the only situation in which the charge moves in a straight line. Since all of the charges move the same distance (as discussed in the previous question) and the rest follow curved paths, the charge in (5) must have the largest displacement. Key Points: The magnetic force on a point charge q moving with velocity v in a magnetic field is q(v ). The magnitude of the magnetic force on a moving point charge can also be written F = qvsin( θ ) = qv. If a point charge s velocity is parallel or anti-parallel to the magnetic field, the magnetic force on it is zero. For Instructors Only The magnetic force is one of the few quantities students encounter that depends on the perpendicular, rather than parallel, component of a vector, so they may have difficulty learning to think about it. It is also common for students to focus on magnitudes rather than directions and think the magnetic force is simply qv, concluding that it is stronger when field lines are more dense. This may lead some students to

5 Magnetism 175 choose (3), (4), and (5). However, a larger force means that the circular path of q is smaller. Other students might pick (1) and (). In fact, it is impossible to determine which of the situations other than (5) has the largest displacement, because we d need to know the time interval T and other values to determine where along their circular or helical trajectories the particles are at the end. Additional Discussion Questions: 1. Which point charges move in circular paths? Describe the orientation of the path.. Of the charges moving in circular paths, order them from smallest to largest radius. 3. Which point charges move in helical paths? Describe the orientation of the path. 4. Of the charges moving in helical paths, order them from smallest to largest radius. 5. Compare the radii of the circular paths to the radii of the helical paths. Question M1.03 Description: Extending understanding of the Lorentz force law and link to magnetism to Newton s third law. Question A bar magnet moving with speed V passes below a stationary charge q. What can be said about the magnitude of the magnetic forces on the bar magnet (F b ) and on the charge q (F q ). v = 0 q N S V 1. F bar and F q are both zero.. F bar is zero and F q is not zero. 3. F bar is not zero and F q is zero. 4. F bar and F q are both non-zero. Commentary Purpose: To develop your understanding of the Lorentz force law, connecting it to Newton s third law. Discussion: We know that a charge moving through a magnetic field experiences a force. If the magnetic field moves past the charge, does the charge also experience a force? Yes. In the magnet s frame of reference, the charge is moving past it, and thus experiences a force. And since it experiences a force in one frame of reference, and both frames are inertial, it must experience a force in the other in the original frame, with a stationary charge.

6 176 Chapter 19 According to Newton s third law, if the charge experiences a force due to the bar, the bar must experience an equal-magnitude, opposite-direction force due to the charge. ut what is the physical mechanism by which the charge exerts a force on the bar? A moving charge is a current, and currents create magnetic fields. So, the bar magnet experiences a force due to another magnetic field. Since two magnets can attract or repel, we know that a magnetic field can exert a force on a magnet. Key Points: Newton s third law holds for magnetic forces. A moving charge is a current, which creates a magnetic field. A moving magnetic field exerts a force on a stationary charge, just as a stationary magnetic field exerts a force on a moving charge. For Instructors Only This is a good question for extending students understanding of magnetic forces into new territory, or for integrating their understanding of various magnetic-related forces. Students who realize the charge will experience a force might, through blind faith in Newton s second law, assert that F bar and F q are both non-zero without comprehending the mechanism by which the charge exerts a force on the bar (or vice-versa). The question serves as a context and motivation for discussing this. Critical students might wonder about our blithe statement that since it experiences a force in one frame of reference, it must experience a force in the other. Relativistically, the force may not be the same in both frames, but it must be nonzero. Question M1.04 Description: Introducing or developing understanding of superposition of magnetic fields. Question Two identical bar magnets are placed rigidly and anti-parallel to each other as shown. At what locations, if any, is the net magnetic field close to zero? A D N S D C D S N D A 1. A only. only 3. C only 4. D only 5. A and 6. A,, and C 7. C and D 8. None of the above.

7 Magnetism 177 Commentary Purpose: To explore the superposition of magnetic fields. Discussion: Consider the magnetic field due to a single bar magnet. The magnetic field at any point due to the two magnets will be the sum of the magnetic fields due to each individual magnet ( superposition ). Notice that the magnets have opposite orientations, so at point C, the field lines due to one will be in the opposite direction of the field due to the other. Since the magnets are identical, C is the same distance from each magnet, and the lines are exactly anti-parallel at that point, the two fields will cancel exactly. At other points, however, the fields won t completely cancel. At points A, for example, the fields are antiparallel but not the same strength, so they ll only partially cancel. At points, fields have the same strength but aren t completely opposite in direction, so the net field won t be zero. And at D, the fields have neither the same strength nor opposing directions. Thus, (3) is the best answer. Key Points: Magnetic fields obey superposition: the total magnetic field from two sources is the vector sum of the magnetic fields due to each individual source, for every point in space. Two superposed fields will only cancel completely if they have the same magnitudes and exactly opposite directions. You can use the symmetries of a situation to deduce much about where fields will cancel. For Instructors Only For this question, finding out students reasons for their answer is more important than their actual answers. We recommend having students sketch the field lines, and describe how the strength of the field is related to the field line diagram. A demonstration with iron filings on an overhead projector would be illuminating. Question M1.06 Description: Developing understanding of electric and magnetic forces. Question A charged particle moves into a region containing both an electric field and a magnetic field. Which of the statements below is/are true? A. The particle cannot accelerate in the direction of.. The path of the particle must be a circle. C. Any change in the particle s kinetic energy is caused by the E field. 1. Only A. Only 3. Only C 4. oth A and 5. oth A and C 6. oth and C 7. All are true. 8. None are true.

8 178 Chapter 19 Commentary Purpose: To build your understanding of electric and magnetic forces. Discussion: Proving a statement true can be difficult; finding one counter-example that disproves a statement is often easier. Let s consider each statement, one at a time. Can the point charge accelerate in the direction of? That is, can the net force on the charge point in the direction of? Yes! If the velocity of the point charge is in the direction of, the magnetic force is zero. So, if the electric field is parallel to, then the electric force is parallel to also. Since the magnetic force is zero, the net force is in the direction of, as is the acceleration. Statement A must be false. Must the path of the point charge be in a circle? No! In the situation described above, the point charge would move in a straight line. Statement must be false. Can the magnetic force cause any change in kinetic energy? No! The direction of the magnetic force is always perpendicular to both the direction of motion and the direction of the magnetic field. Therefore, the magnetic force is always perpendicular to the displacement of the point charge, even if the charge moves along a curved path. Thus, the work done by the magnetic force is always zero, so the magnetic force cannot contribute to any changes in kinetic energy. Statement C is true. Key Points: An electric field acts on a moving charge by exerting a force in the direction of the field. A magnetic field acts on a moving charge by exerting a force perpendicular to both the field and the charge s velocity. A magnetic field can do no work on a moving charge. The behavior of a moving charge in a combination of electric and magnetic fields depends on the relative directions and strengths of the fields and on the initial velocity of the charge. For Instructors Only Students are most familiar with situations in which E and are perpendicular, and may assume that here without explicitly realizing it. This may cause some to believe that statement A is true. Statement should be easy for students to find a counter-example to. Discussing the proof of statement C provides a good opportunity to review the definition of work (and reiterate that force times distance is an inadequate definition). Question M1.07a Description: Introducing charged particle motion in electric and magnetic field combinations. Question A charge is released from rest in E and fields. oth fields point along the x-axis. Which of the following statements regarding the charge s motion are correct?

9 Magnetism 179 y z q x E 1. The charge will travel along a straight-line path.. The charge s speed will change as it travels. 3. The charge will travel in a helical path. 4. The charge will travel in a helical path of increasing pitch. 5. The charge will travel in a circle in the x y plane and only 7. and 4 only 8. None of the above Commentary Purpose: To check and refine your understanding of electric and magnetic forces on moving particles. Discussion: The full Lorentz force law is F = q E+ v. This implies that an electric field exerts a force on a charge in the direction of the field (for a positive charge) or in the opposite direction (for a negative charge). It also implies that a magnetic field exerts a force that points perpendicularly to both the field and the charge s velocity, and only if the charge s velocity has a component perpendicular to the magnetic field. In this situation, the electric field will cause the stationary charge to accelerate parallel to the magnetic field. Since the charge s velocity starts with no component perpendicular to the magnetic field, and never gains one, the magnetic force will remain zero. Thus, the particle will simply accelerate along the x-axis. Statements (1) and () are both valid, so answer (6) is best. Key Points: An electric field exerts a force on a charge parallel to the field, according to F = qe. A magnetic field exerts a force on a moving charge according to F = q v. If the charge has no velocity component perpendicular to the field, the force is zero. For Instructors Only This is the first of two related questions. Students who answer (3), (4), or (7) may be remembering the fact that charges in magnetic fields move along helical paths. In a sense, that is true here, for the limiting case of a zero-radius helix. A misunderstanding of the vector cross product or an inability to apply it reliably are common sources of error here. Students choosing answer (8) should be encouraged to describe the motion they expect. (One answer that arises occasionally is The charge first moves in a straight line until it gets some speed, and then.... )

10 180 Chapter 19 Question M1.07b Description: Introducing charged particle motion in electric and magnetic field combinations. Question A charge has an initial velocity parallel to the y-axis in E and fields. oth fields point along the x-axis. Which of the following statements regarding the charge s motion are correct? y z q v x E 1. The charge will travel along a straight-line path.. The charge s speed will change as it travels. 3. The charge will travel in a helical path. 4. The charge will travel in a helical path of increasing pitch. 5. The charge will travel in a circle in the x y plane and only 7. and 4 only 8. None of the above Commentary Purpose: To check and refine your understanding of electric and magnetic forces on moving particles. Discussion: The full Lorentz force law is F= q E+ v. This implies that an electric field exerts a force on a charge in the direction of the field (for a positive charge) or in the opposite direction (for a negative charge). It also implies that a magnetic field exerts a force that points perpendicularly to both the field and the charge s velocity, and only if the charge s velocity has a component perpendicular to the magnetic field. The charge begins with a velocity in the y-direction, perpendicular to the magnetic field. Therefore, the cross-product rule says that the magnetic force on it will be in the z-direction. (If the charge is negative, the force will be in the +z-direction.) Since the magnetic force is perpendicular to the charge s velocity, it does no work, and causes the velocity vector to change direction but not magnitude. If there were no electric field, the magnetic force would continue to bend the particle s path without changing its speed, causing it to move in a circle in the yz plane. At the same time, the electric field exerts a constant force on the charge in the +x-direction, parallel to the electric field. (If the charge is negative, the force will be in the x-direction.) This will cause it to experience a constant acceleration in the x-direction. ecause of the cross product, the magnitude and direction of the magnetic force do not depend on any x velocity the particle might have. So, the particle moves in a circle in the yz plane while simultaneously accelerating in the x-direction. This results in the particle following a helical (spiral) trajectory of increasing pitch (distance between turns): answer (4). If the charge is positive, the helix will proceed to the right; if negative, it will proceed to the left.

11 Magnetism 181 Key Points: An electric field exerts a force on a charge parallel to the field, according to F = qe. A magnetic field exerts a force on a moving charge according to F = q v. If the charge has no velocity component perpendicular to the field, the force is zero. When magnetic and electric fields are parallel, their effects on a charge s motions are independent: the electric field influences motion parallel to the fields, and the magnetic field influences motion in the plane perpendicular to them. For Instructors Only This is the second of two related questions. Students who know that the particle will travel in a helical path often forget or don t realize that the pitch will change due to the electric field s influence, and consequently will answer (3). Students are likely to struggle with the vector cross product and the direction the magnetic force will point as the particle s trajectory curves. Question M1.08 Description: Verbalizing and picturing magnetic field lines. Question Consider a long thin straight wire with a current I. Which of the following statements about the magnetic field lines is true? A. Field lines are parallel to the wire.. Field lines are perpendicular to the wire. C. Field lines are directed radially away from the wire. D. Field lines are circles centered on any point on the wire. 1. A only. only 3. C only 4. D only 5. A and C only 6. and D only 7. and C only 8. None of them is true.

12 18 Chapter 19 Commentary Purpose: To develop your ability to describe and picture magnetic field lines. Discussion: The magnetic field lines due to current flow through a wire forms circles centered on the wire, perpendicular to the wire. One form of the right-hand rule for magnetic fields says that if you point your right thumb along a wire in the direction current flows, and curl your fingers, they indicate the direction that these circles of magnetic field lines point. This result can be derived mathematically from the iot-savart law, but it s worth memorizing so you can use it to reason about current and magnetic field problems. If the field lines were parallel to the wire, they would not obey the direction indicated by the cross product in the iot-savart law. If they pointed radially outward, they would have to end on the wire, and magnetic field lines never begin or end; they always form loops. Key Points: The magnetic field lines generated by a current in a straight wire form circular loops around the wire, perpendicular to it and with a direction given by the right-hand rule. Magnetic field lines always form closed loops, and never begin or end. For Instructors Only Students may have seen diagrams of the line geometry, and even be able to reproduce those diagrams, but be unable to describe them in words. This question provides an opportunity to practice relating graphical to verbal representations. Question M1.09 Description: Introducing and understanding the force a magnetic field exerts on a current-carrying wire. Question A very long wire lies in a plane with a short wire segment. The long wire carries current I, while the short wire of length L carries current i. The two wires are parallel to each other. Which of the following statements are true? I i d A. The direction of the magnetic force exerted by the long wire on the short wire is directed away from the long wire.. The magnitude of the force on the short wire is μ0iil πd. C. The long wire experiences a force of exactly the same magnitude as the force experienced by the short wire. L

13 Magnetism A only. only 3. C only 4. A and 5. A and C 6. and C 7. A,, and C 8. None of them are true Commentary Purpose: To explore and develop familiarity with the force a magnetic field exerts on a current-carrying wire. Discussion: To answer this question, we need to know the magnetic field created by the long wire, and the force the short wire experiences because of that field. We will assume that we can treat the very long wire as infinitely long. (Why else would we be told it is very long and not given a variable for its length?) The magnetic field created by an infinitely long, straight wire carrying current I is μ0i πd a distance d away from the wire. This result is worth remembering. It can be derived via the iot-savart law (complicated) or via Ampere s law and a symmetry argument (simpler). The magnetic field lines form circles surrounding the wire, with a direction given by the right-hand rule, so in the figure the field will point into the page at every point along the short wire. The force exerted on a segment of wire by a magnetic field (due to some source other than the segment itself) is F = i L, where i is the current in the segment, L is the length of the segment, and the direction of the vector L is the direction the current is flowing along the segment. (If is not the same at every point along the segment for example, if the segment were perpendicular to the very long wire in this question you must use that equation separately for every infinitesimal piece of the wire segment, and integrate.) Putting our expression for the magnetic field due to the long wire into this force equation, we find that statement is true. How about the direction of the force? If the magnetic field due to the long wire points into the page everywhere along the short segment, the right-hand rule indicates that the force exerted will be towards the long wire. So, statement A is false. Statement C must be true, because Newton s third law is valid for magnetic forces as well as all other forces. Furthermore, according to that law, the direction of the force on the long wire must be towards the short wire: equal magnitudes, opposite directions. So, the two wires attract each other. The best answer to this question is therefore (6). Key Points: The magnetic field a distance d away from an infinitely long wire carrying a current I is μ0i πd, circulating around the wire according to the right-hand rule. A segment of wire of length L carrying current i, located in an externally generated magnetic field, experiences a magnetic force F = i L where the direction of L points along the wire in the direction of current flow. Two parallel current-carrying wires will attract each other if their currents flow in the same directions. This is called the pinch effect. (If the currents flow in opposite directions, the wires will repel.)

14 184 Chapter 19 For Instructors Only This is a good question for students just encountering the force law for magnetic fields acting upon currentcarrying wires. Students claiming statement A is true (answers 1, 4, or 7) might be unable to apply the right-hand rule correctly or reliably. They might also have an intuitive reason for believing the short wire is repelled (for example, by analogy with like charges repelling), so getting them to explain their reasoning is important. For students claiming statement is false (answers 1, 3, 5, and 8), your next diagnostic step should be to ask what they think the magnitude of the force is. Those that think it is correct except for the factor of p might be accidentally remembering the magnetic field at the center of a circle of current, rather than for an infinite line. One can reason about many of the factors in the answer: for example, the force should get stronger if either current is increased, so i and I must be in the numerator. A longer short segment provides more current to be acted upon by the field, so L should be in the numerator as well. The effect should get weaker if the wires are farther apart, so some power of d must appear in the denominator. And μ 0 appears in pretty much any magnetic field expression. In fact, using dimensional analysis (i.e., considering units), one can figure out what the force magnitude must be to within a multiplicative constant. This exercise can be valuable for students to engage in or at least see you talk through: it helps them learn how to check their answers for reasonability, as well as believe that the mathematics in physics ought to be interpreted. Confusion about variables is also likely, since this question uses d where r or R is often seen, and has both i and I. If such confusion occurs, we recommend taking the opportunity to make a strong point about the importance of understanding what the variables in definitions, laws, and other equations mean, of being able to use them comfortably with any choice of variables, and of being defensive about the fact that many letters are used with different meanings. To help students develop this facility, we recommend that you make a general effort to be inconsistent in your notation (between problems only, not within them, of course) and often at odds with the textbook s conventions. Students claiming statement C is false (1,, 4, or 8) should be engaged in a discussion to determine whether they really think Newton s third law is violated (or just forgot about it), and if so, why. This could indicate a serious misunderstanding. Question M.01 Description: Reasoning with the iot-savart law and superposition of magnetic fields. Question In all cases the wire shown carries a current I. For which situation is the magnitude of the magnetic field maximum at the point P?

15 Magnetism 185 I 1 P R πr R I 3 R I P R/ 4 R/ R I P Commentary Purpose: To reason with the iot-savart law and develop your intuition for the magnetic fields currents generate. Discussion: First, let s review some basic results that are very useful for thinking about magnetic fields due to steady currents. A distance R from an infinitely long straight wire carrying a current I, the magnetic field induced by the current is μ0i πr. Similarly, the magnetic field at the center of a current-carrying circular wire of radius R is μ 0 I R. oth of these can be derived from the iot-savart law. (The field for an infinite wire can also be derived from Ampere s law.) The field at the center of a half-circle of current has a magnitude of one-half the value for a whole circle of the same radius. This is because the field contribution is in the same direction and has the same magnitude for each infinitesimal piece of the wire, so the contributions add up without any components that cancel. If you have half as many infinitesimal segments, you get half as much total field strength. So, each half-circle contributes μ I πr. 0 4 The magnetic field due to a finite line segment of current is zero for points directly in line with (ahead of or behind) the current, because the angular part of the cross product in the iot-savart law involves the sine of the angle between the current s direction of flow and the displacement vector from the current location to the point in question. Now, we can use these results to reason about the relative strengths of the magnetic field due to the current arrangements shown. Case 4 must also produce a stronger field than case 1, because both involve two semicircular currents, but for case 3 one of the semicircles has a smaller radius and therefore creates a stronger field at P. The other half is the same for both cases, and the small straight segments in case 4 flow directly towards or away from P and don t create any field there at all.

16 186 Chapter 19 Case 4 must have a larger field at P than case 3, because both involve similar geometry, but for case 3 the two semicircular currents flow in opposite directions around P, so their contributions will partially cancel. For case 4, the field due to the two semicircular segments are both in the same direction. Case 4 must also be larger than case. This can be reasoned two ways. First, the field at the center of a circular current of radius R is larger (by a factor of p ) than the field a distance R away from an infinite wire having the same current. Since case has only a finite segment of wire, the field it creates will be smaller than for an infinite wire and, therefore, less than case 1, which has already been shown to be less than case 4. The second way is more straightforward. The length of the wire segment in case 1 is the same as for case, but all of the infinitesimal elements are the same distance away from point P. Since the contribution to the field from each element drops off as 1/r, the field at P due to 1 must be larger than in case. Thus, we know case 4 must have the largest magnetic field at P. Key Points: We can compare the magnetic fields due to many different current arrangements without actually calculating them. Knowing expressions for the magnetic fields created by various standard current arrangements (infinite lines, circular loops, etc.) is helpful. The iot-savart law lets you reason qualitatively about magnetic fields and the variables they depend on. For Instructors Only Students should be encouraged (or admonished) to reason to the answer, rather than determining expressions for the field magnitude for all four cases. This can be accomplished by limiting the time students have to decide upon their answers, and when they complain that they haven t had enough time telling them they should be reasoning qualitatively, not calculating. Some general points that you should try to help students appreciate through this question include: that magnetic field strength falls off as 1/R; that a circular loop around a point creates a stronger field than an infinite line at the same distance; and that a point inside a closed current path generally experiences a stronger field than a point outside a similar path. A good follow-up question, perhaps as an informal rhetorical question for students to ponder as they leave class, asks students to well-order the four cases according to increasing magnetic field strength at P. Question M.0 Description: Reasoning with the iot-savart law and superposition of magnetic fields. Question Order the following situations according to the magnitude of the magnetic field at the point P. Order from highest to lowest.

17 Magnetism ACD. ADC 3. DAC 4. CAD 5. DAC 6. None of the above Commentary Purpose: To reason with the iot-savart law and develop your intuition for the magnetic fields currents generate. Discussion: From the iot-savart law, we can find that the magnetic field at the center of a circular current loop of radius R is μ 0 I R, and the magnetic field a distance R away from an infinitely long, straight current is μ I πr, where I is the current in both cases. 0 Similarly, the iot-savart law lets us reason that the magnetic field due to half of a circular current loop will be one-half the value for a complete circle, and the field due to a semi-infinite line (half an infinite line, starting at the point on the infinite line closest to P) will be one-half the value for an infinite line. The other piece of information we ll need to reason about the current arrangements in this problem is that the magnetic field lines created by a current make circular loops around the current. If you point the thumb of your right hand along the wire in the direction current flows, and curl your fingers around the wire, your fingers will point the direction of the magnetic field. Once we know these facts, we can reason about the current arrangements shown in the question. Case A consists of one infinite line (equivalent to two semi-infinite lines) and one circular loop (equivalent to two semicircles), with current flowing such that the magnetic fields due to the two oppose each other. We ll notate this as C L, meaning the field strength at P is due to two semicircles minus two opposing semi-infinite lines. Case consists of two semi-infinite lines and one semicircle, such that the fields from all three are in the same direction and add without cancellation. Thus, the field for case is 1C + L. Similarly, the field at P is C + 4L for case C. For case D, it is 1.5C + L. So, we can rank the cases in order of decreasing field strength as: C (C + 4L) > D (1.5C + L) > (1C + L) > A (C L).

18 188 Chapter 19 In order to place the last case (case A), we had to determine that 1C + L > C L. We can show this by noting that L = C/p, substituting this into the inequality, and simplifying: 1C + L > C L 4 > p QED. Key Points: We can compare the magnetic fields due to many different current arrangements without actually calculating them. Knowing expressions for the magnetic fields created by various standard current arrangements (infinite lines, circular loops, etc.) is helpful, as is knowing the field values for standard fractions of these arrangements. The iot-savart law lets you reason qualitatively about magnetic fields and the variables they depend on. For Instructors Only Students should be strongly pushed to reason qualitatively about the ranking of the cases, rather than deriving expressions for each. (Some quantitative work is necessary to place case A, but that is best done in terms of an inequality.) This problem is about reasoning with magnetic field generation involving qualitative features of the iot-savart law, symmetry, and superposition not about calculating fields for various current loop shapes. You might want to open a discussion of what portions of these standard current arrangements (loops and infinite lines) are easy to deduce the field for, and what require detailed calculation. For example, since all points on a circular current loop contribute the same amount to the net magnetic field at the center, we can find the field due to any fraction of a circle by taking that fraction of the field due to an entire circle. For lines, however, only infinite and semi-infinite lines (beginning at the point of closest approach to P) are simple to determine; any other fragment requires integrating the iot-savart law. Students often have difficulty determining what current arrangements Ampere s law can or cannot be productively applied to. Although Ampere s law is not particularly useful for answering this question (except as a way to find the field due to an infinite line), students may be wondering about it, so a discussion may be warranted. This question works well with Question M.01: the reasoning is similar, but the application is different enough that one may be used to introduce and develop the ideas and the other to reveal (to students as much as to the instructor) whether students grasp and can apply them. Question M.03a Description: Developing ability to apply the iot-savart law, and recognize where it (rather than Ampere s law) is required. Question The diagram shows a circular wire loop of radius R carrying current I. What is the direction of the magnetic field,, at the center of the loop?

19 Magnetism 189 Top View farthest from you I closest to you R I 1. Left. Right 3. Up 4. Down 5. None of the above Commentary Purpose: To develop your understanding of the iot-savart law. Discussion: According to the iot-savart law, an infinitesimal segment of current I with length ds creates 3 an infinitesimal magnetic field d= ( μ0 4 π) I ds r r at a point if r is the displacement vector from the current element to the point in question. For a current-carrying wire, one can find the total magnetic field by adding up (integrating) the contributions from all of the infinitely many infinitesimal current segments. Note the cross product: this means that the magnetic field created will have a direction that is perpendicular to both the current flow and the displacement vector r. This means the magnetic field from that segment must form circular loops around the segment. The direction can be determined from the right-hand rule: place your thumb pointing along the current in the direction of positive current flow, and curl your fingers partially, and your fingers will show you the direction in which the magnetic field points as it circles around the segment. You can use this to determine that for any infinitesimal segment of the circular current path shown, the infinitesimal magnetic field at the origin will point upward (out of the page in the right-hand diagram). If you add up all the contributions from the entire loop, therefore, the total magnetic field at the center must point upward as well. Answer (3). Key Points: The iot-savart law describes the magnetic field due to current flow. The magnetic field due to a current loop is the sum of the magnetic field contributions from each infinitesimal segment of the loop. Current flowing through a wire causes magnetic field lines to circle around the wire, with a direction given by the point the thumb, curl the fingers version of the right-hand rule. For Instructors Only This is a good question for wading into the vector calculus of the iot-savart law. Students frequently remain confused about the direction of r, whether it points from the source point to the field point or vice-versa and subsequently have difficulty reconciling the various forms of the right-hand rule. For students who already know that the magnetic field lines due to a straight wire forms circles around the wire, and can find the direction, the question is simpler and can be approached by a common-sense superposition argument. (The question is even easier for those familiar with magnetic dipoles.)

20 190 Chapter 19 This question is a good lead-in to the next one, which asks students to determine the magnitude of the field. While that may seem straightforward and trivial, it remains difficult for students who are not yet comfortable with vectors and calculus. Question M.03b Description: Developing ability to apply the iot-savart law, and recognize where it (rather than Ampere s law) is required. Question The diagram shows a circular wire loop of radius R carrying current I. What is the magnitude of the magnetic field,, at the center of the loop? Top View farthest from you I closest to you R I μ 0 I/4p R 3. μ 0 I/p R 4. μ 0 I/4R 5. μ 0 I/R 6. None of the above. Commentary Purpose: To develop your understanding of the iot-savart law and help you distinguish it from Ampere s law. Discussion: According to the iot-savart law, an infinitesimal segment of current I with length ds creates 3 an infinitesimal magnetic field d= ( μ0 4 π) I ds r r at a point if r is the displacement vector from the current element to the point in question. For a current-carrying wire, one can find the total magnetic field by adding up (integrating) the contributions from all of the infinitely many infinitesimal current segments. All infinitesimal segments around the circular current loop in this problem create magnetic fields pointing in the up direction (as discussed in the previous question), so we can find the magnitude of d due to one segment and integrate to find the total magnitude. The magnitude d we get when we put our known values and angles into the iot-savart law is μ0 4π same relative angles). Since the value of d we found is the contribution per length ds of the loop, and the total length of the loop is p R, the total field at the origin must be πrd= μ 0 I R: answer (5). Ids R. All segments make exactly the same contribution to the field at the origin (same distance, Key Points: The iot-savart law describes the magnetic field due to infinitesimal current element. The magnetic field due to a current loop is the sum of the magnetic field contributions from each infinitesimal segment of the loop.

21 Magnetism 191 If all the segments create field contributions with the same direction and magnitude at some point, we don t actually have to solve an integral to find the total field. It is important to recognize when you can find the magnetic field due to steady currents using Ampere s law and when you must use the iot-savart law. For Instructors Only How best to explain and discuss this question depends on your students mathematical sophistication and experience with the iot-savart law. It provides a good situation for working quantitatively with the law while avoiding ugly integrals. A general discussion of why Ampere s law is not useful here may be in order. Students often have a difficult time recognizing when they can use Ampere s law and when they must use the iot-savart law. The magnetic field due to a long wire is often found using the iot-savart law to demonstrate that the result is the same as when Ampere s law is used. The significance of this demonstration is frequently lost on many students. Even when they can assert that symmetry is required to use Ampere s law, they remain uncertain what the symmetry statement applies to. Answer (3) might indicate that students are recalling the magnetic field due to a straight infinite wire and not incorrectly applying the iot-savart law. QUICK QUIZZES 1. (b). The force that a magnetic field exerts on a charged particle moving through it is given by F = qvsinθ = qv, where is the component of the field perpendicular to the particle s velocity. Since the particle moves in a straight line, the magnetic force (and hence, since qv 0) must be zero.. (c). The magnetic force exerted by a magnetic field on a charge is proportional to the charge s velocity relative to the field. If the charge is stationary, as in this situation, there is no magnetic force. 3. (c). The torque that a planar current loop will experience when it is in a magnetic field is given by τ = IA sin θ. Note that this torque depends on the strength of the field, the current in the coil, the area enclosed by the coil, and the orientation of the plane of the coil relative to the direction of the field. However, it does not depend on the shape of the loop. 4. (a). The magnetic force acting on the particle is always perpendicular to the velocity of the particle, and hence to the displacement the particle is undergoing. Under these conditions, the force does no work on the particle and the particle s kinetic energy remains constant. 5. (a) and (c). The magnitude of the force per unit length between two parallel current carrying wires is F l = ( μ0i1i) ( πd). The magnitude of this force can be doubled by doubling the magnitude of the current in either wire. It can also be doubled by decreasing the distance between them, d, by half. Thus, both choices (a) and (c) are correct. 6. (b). The two forces are an action-reaction pair. They act on different wires and have equal magnitudes but opposite directions.

22 19 Chapter 19 ANSWERS TO MULTIPLE CHOICE QUESTIONS 1. The electron moves in a horizontal plane in a direction of 35 N or E, which is the same as 55 E of N. Since the magnetic field at this location is horizontal and directed due north, the angle between the direction of the electron s velocity and the direction of the magnetic field is 55. The magnitude of the magnetic force experienced by the electron is then F = q vsin θ = C m s T sin N ( )( ) = The right-hand rule number 1 predicts a force directed upward, away from the Earth s surface for a positively charged particle moving in the direction of the electron. However, the negatively charged electron will experience a force in the opposite direction, downward toward the Earth s surface. Thus, the correct choice is (d).. If the magnitude of the magnetic force on the wire equals the weight of the wire, then Ilsinθ = w, or = w Ilsinθ. The magnitude of the magnetic field is a minimum when θ = 90 and sin θ = 1. Thus, min w = = Il N 0.10 A 050. m = 00. T and (a) is the correct answer for this question. 3. The z-axis is perpendicular to the plane of the loop, and the angle between the direction of this normal line and the direction of the magnetic field is θ = Thus, the magnitude of the torque experienced by this coil containing N = 10 turns is τ IAN sin θ T 0. A 00. m 030. m ( 10) sin = N m = = meaning that (c) is the correct choice. 4. A charged particle moving perpendicular to a magnetic field experiences a centripetal force of magnitude F = m v c r = q v and follows a circular path of radius r = mv q. The speed of this proton must be 19 3 qr ( C )( T )( m) 3 v = = = ms 7 m kg and choice (e) is the correct answer. 5. The magnitude of the magnetic field at distance r from a long straight wire carrying current I is = μ0i πr. Thus, for the described situation, 7 ( 4π 10 TmA / )( 1A) 7 = = 1 10 T π m making (d) the correct response. 6. The force per unit length between this pair of wires is 7 F I I l = μ d = 4π 10 T m A 3A 0 1 π π m and (d) is the best choice for this question. 7 6 = 9 10 N 1 10 N 7. The magnitude of the magnetic field inside the specified solenoid is N = ni = I 7 μ0 μ0 = 4π 10 T m A l which is choice (e) m (. ) = A T