Using the Biot Savart Law: The Field of a Straight Wire

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2 Exective Editor: Nancy Whilton Project Manager: Katie Conley Development Editors: John Mrdzek, Matt Walker Editorial Assistant: arah Kabisch Development Manager: Cathy Mrphy Project Management Team Lead: Kristen Flathman enior Acqisitions Editor, Global Edition: Priyanka Ahja Project Editor, Global Edition: Amrita Naskar Manager, Media Prodction, Global Edition: Vikram Kmar enior Manfactring Controller, Prodction, Global Edition: Trdy Kimber Design Manager: Marilyn Perry Cover Designer: Lmina Datamatics Ltd. llstrators: Rolin Graphics Rights & Permissions Management: Timothy Nicholls Photo Researcher: tephen Merland, Jen immons Manfactring yer: Mara Zaldivar-Garcia Marketing Manager: Will Moore Cover Photo Credit: Georgii hipin/htterstock Acknowledgments of third party content appear on page 872, which constittes an extension of this copyright page. Pearson Edcation Limited Edinbrgh Gate Harlow Essex CM20 2JE England and Associated Companies throghot the world Visit s on the World Wide Web at: Pearson Edcation Limited 2016 The rights of Richard Wolfson to be identified as the athor of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act Athorized adaptation from the United tates edition, entitled Essential University Physics, Volme 2, 3rd edition, N , by Richard Wolfson, pblished by Pearson Edcation All rights reserved. No part of this pblication may be reprodced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, withot either the prior written permission of the pblisher or a license permitting restricted copying in the United Kingdom issed by the Copyright Licensing Agency Ltd, affron Hose, 6 10 Kirby treet, London EC 1N 8T. All trademarks sed herein are the property of their respective owners. The se of any trademark in this text does not vest in the athor or pblisher any trademark ownership rights in sch trademarks, nor does the se of sch trademarks imply any affiliation with or endorsement of this book by sch owners. PEARON, ALWAY LEARNNG and MasteringPhysics are exclsive trademarks in the U.. and/or other contries owned by Pearson Edcation, nc. or its affiliates. Unless otherwise indicated herein, any third-party trademarks that may appear in this work are the property of their respective owners and any references to third-party trademarks, logos or other trade dress are for demonstrative or descriptive prposes only. ch references are not intended to imply any sponsorship, endorsement, athorization, or promotion of Pearson s prodcts by the owners of sch marks, or any relationship between the owner and Pearson Edcation, nc. or its affiliates, athors, licensees or distribtors. N 10: N 13: ritish Library Cataloging-in-Pblication Data A cataloge record for this book is available from the ritish Library Typeset by Lmina Datamatics, nc. Printed and bond by RR Donnelley Kendallville in the United tates of America

3 558 Chapter 26 Magnetism: Force and Field dl and nr are perpendiclar. dl a rn x ame ; cos = a>r r P d Field contribtions perpendiclar to the axis cancel c where the integral redces to a simple form becase the distance x is the same for all points on the loop. The remaining integral is the sm of infinitesimal lengths arond the loop, or the loop circmference 2pa. o we have m 0 a 2 = 21x 2 + a 2 (26.9) 3/2 2 The direction of the field, as sggested in Fig b, is along the axis. Assess The field is strongest right at the loop center 1x = 02 becase here we re closest to the loop and so the contribtions from all segments of the loop are greatest. The field decreases as we move away from the loop. n general the field is a complicated fnction of distance, bt for large distances 1x W a2 it falls off as 1/x 3. That shold remind yo of the field we fond for an electric dipole in Chapter 20. We ll have more to say abot this dipole-like behavior in ection cleaving a net field along the axis. = d L Figre Finding the magnetic field on the axis of a crrent loop. Example 26.5 Using the iot avart Law: The Field of a traight Wire Find the magnetic field prodced by an infinitely long straight wire carrying steady crrent. nterpret This example, too, is abot the field prodced by a specified crrent distribtion. Develop Figre is or drawing of the wire on a coordinate system with the x-axis along the wire. ince the wire is infinite, the field magnitde mst be the same at all points eqidistant from the wire. We show one sch point P, a distance y from the wire. We also show an infinitesimal segment dl! of the wire and the nit vector rn toward the field point. Or plan is to calclate the field contribtions d from all sch crrent elements, and then integrate to find the field. Evalate oth dl! and rn lie in the plane of the page, so at P the vector dl! * rn in the iot avart law is ot of the page. This is tre for any segment of the wire. Therefore, we can sm the magnitdes of the contribtions d to find the magnitde of the net field, and we know its direction at P will be ot of the page. With rn a nit vector, dl! * rn = dl sin, where the triangle in Fig shows that sin = y/r = y/2x 2 + y 2. Then the iot avart law gives a field contribtion of magnitde d = m 0 dl! * rn r 2 = m 0 dl sin r 2 = m 0 y dl 1x 2 + y 2 2 3/2 ince the segment dl! lies along the x-axis, dl = dx. Also, y is a constant here, so the net field becomes = L d = m 0 y L - dx 1x 2 + y 2 2 3/2 where we chose the limits to inclde the entire infinite wire. The integral is a standard one, given in the integral tables of Appendix A; the reslt is Figre Calclating the magnetic field at P de to an infinite straight wire carrying crrent along the x-axis. = m 0 2py (26.10)

4 26.6 Magnetic Dipoles 559 Assess This reslt for the magnetic field of a long crrent-carrying wire shold remind yo of or earlier finding for the electric field of a line charge; both decrease as the inverse of the distance from the line. t where the electric field of a line charge points radially otward, the magnetic field of a line crrent encircles the crrent, as shown in Fig Of corse, an infinite line crrent is impossible, bt or reslt is a good approximation to the fields of finite wires if we re close compared with the wire s length. Figre Magnetic field lines encircle a straight wire, with their direction given by the right-hand rle. The Magnetic Force etween Condctors n ection 26.4 we fond the force on a crrent-carrying wire in a magnetic field: F = l! *. Now yo ve seen that a straight wire prodces a magnetic field. That means crrent-carrying wires exert magnetic forces on each other. Figre shows the sitation for two parallel wires carrying crrents in the same direction. The wires are a distance d apart, so the field of wire 1 at the location of wire 2 follows from Eqation 26.10: 1 = m 0 1 /2pd. The field is perpendiclar to wire 2, so the force on a length l of wire 2 is F 2 = 2 l 1 = m l 2pd 1magnetic force between two wires2 (26.11) Figre shows that the direction of this force is toward wire 1, so the parallel crrents attract. Analyzing the force on wire 1 from wire 2 amonts to interchanging the sbscripts 1 and 2, giving an attractive force of the same magnitde. Reversing one of the crrents wold change the signs of both forces, showing that antiparallel crrents repel. The force between nearby condctors can be qite large, so engineers who design highstrength electromagnets mst provide enogh physical spport to withstand the magnetic force (Problem 85 considers this sitation). The hm yo often hear arond electrical eqipment comes from the mechanical vibration of nearby condctors in transformers and other devices, as they respond to the changing force associated with 60-Hz alternating crrent. 2 F l F 1 Figre The magnetic force between parallel crrents in the same direction is attractive. l 1 2 Got t? 26.5 A flexible wire is wond into a flat spiral as shown in the figre. (1) f a crrent flows in the direction shown, will the coil tighten or become looser? (2) Does yor answer depend on the crrent direction? Note: The crrent enters and leaves the coil throgh wires (not shown) at each end, perpendiclar to the page Magnetic Dipoles The crrent loop of Example 26.5 shows the essential characteristic of all steady-state crrents namely, a closed loop with crrent everywhere the same. Eqation 26.9 gives the field on the loop axis: = m 0 a 2 /21x 2 + a 2 2 3/2. For x W a we can ignore a 2 compared with x 2 in the denominator, giving m 0 a 2 /2x 3. Mltiply both sides by 2p to get 2m 0 A/x 3, where A is the loop area. Compare this reslt with the field on the axis of an electric dipole, Eqation 20.6b: oth show the inverse-cbe dependence of the dipole field, and both involve fndamental constants from the Colomb and iot avart

5 560 Chapter 26 Magnetism: Force and Field Direction of Figre Finding the direction of a crrent loop s magnetic dipole moment. Far away, the fields look similar c E + Figre The electric field of an electric dipole and the magnetic field of a crrent loop. Far from their sorces, both have the shape and the 1/r 3 dependence of the dipole field. m cbt close in, they re different. laws that relate fields and their sorces. Where the electric-field expression contains the electric dipole moment p, the prodct of charge and separation, the magnetic-field expression contains A, the prodct of the loop crrent and loop area. We identify A as the magnitde, m, of the crrent loop s magnetic dipole moment. Then the on-axis magnetic dipole field becomes = m 0 m 1on@axis field, magnetic dipole2 (26.12) 2p 3 x The magnetic dipole moment is a vector whose direction follows from the right-hand rle shown in Fig f we describe the loop by a vector of magnitde A whose direction is perpendiclar to the loop as shown in Fig , then we can write the magnetic dipole moment as m! = A. Practical crrent loops often have mltiple trns; since each carries the same crrent, an N-trn loop has effective crrent N, so its dipole moment becomes m! = NA a magnetic dipole moment, N@trn crrent loop b (26.13) Althogh we ve fond the magnetic field for a crrent loop only on the loop axis, a more elaborate calclation shows that the magnetic field anywhere far from the loop has exactly the same configration as the electric field far from an electric dipole. And althogh we developed the magnetic dipole moment for a circlar loop, Eqation in fact gives the dipole moment of any closed loop of crrent. We conclde that any crrent loop constittes a magnetic dipole, and that far from the loop, its field will be that of a dipole. Electric and magnetic dipoles are analogos: oth have the same field configration and mathematical form far from their sorces (Fig ), and both are characterized by their respective dipole moments. t their fields aren t the same. One is an electric field, its origin in static electric charge; the other is a magnetic field, its origin in moving electric charge specifically, charge moving in a closed loop. And the similarity in field configrations holds only at large distances; as Fig shows, the fields near electric and magnetic dipoles are very different, reflecting the different strctres that give rise to each. Crrent loops are biqitos, and so are dipole magnetic fields. Mltiple trns of crrent-carrying wire prodce the strong magnetic fields of electromagnets, and spercondcting loops provide the fields in MR scanners. At the atomic level, orbiting and spinning electrons constitte miniatre magnetic dipoles. Even planets and stars have magnetic dipole fields. Application Magnetic Fields of Earth and n Many astrophysical objects have magnetic fields reslting from the interaction of condcting flids with the objects rotation. Earth s field arises in its liqid-iron oter core, where convective flows work with Earth s rotation to prodce electric crrents. The figre shows that Earth s field approximates that of a dipole; the magnitde of the dipole moment is approximately m = 8.0 * A # m 2. The direction of the dipole moment vector differs from that of Earth s rotation axis, which acconts for the difference between magnetic and tre north. Earth s field reverses roghly every million years, and geologists track seafloor spreading from the reslting magnetization in rocks. Farther ot, Earth s magnetic field traps high-energy particles and ths protects s from dangeros radiation. Yo can see from the figre that magnetic field lines concentrate toward the polar regions, which is why energetic particles tend to enter Earth s atmosphere near the poles, making the arora a high-latitde phenomenon (recall Example 26.3). The n s gaseos natre makes its magnetic field mch more dynamic, and magnetism is the dominant force in its hot, electrically condcting atmosphere. The n s field reverses approximately every 11 years, coinciding with the rise and fall of snspots regions of intense magnetic field that are often sorces of violent otbrsts. Magnetic axis m Rotation axis 11 N

6 26.6 Magnetic Dipoles 561 Dipoles and Monopoles Atoms, molecles, and radio antennas are among the many strctres that behave as electric dipoles. n all these, separation of positive and negative electric charge gives rise to the dipole. Magnetism is different. No one has ever fond an isolated magnetic north or soth pole analogos to an electric charge. Electromagnetic theory doesn t rle ot sch magnetic monopoles, and indeed some theories sggest that monopoles might have formed in the ig ang. t they ve never been fond. All magnetic fields we ve ever seen come from moving electric charge. As yo ll see in ection 26.7, that incldes the fields of permanent magnets. ecase steady crrents form closed loops, the simplest magnetic entity is the dipole. Electric field lines generally begin or end on electric charges. t there aren t any magnetic charges magnetic monopoles. Magnetic field lines don t begin or end, bt form closed loops encircling the moving electric charges that are their sorces. n Chapter 21 we developed Gass s law to qantify the statement that the nmber of electric field lines emerging from any closed srface depends only on the charge enclosed. ecase there s no magnetic charge, the net nmber of magnetic field lines and therefore the magnetic flx A # da emerging from any closed srface is always zero. Ths Gass s law for magnetism is C # da = 0 1Gass>s law for magnetism2 (26.14) Like Gass s law for electricity, Eqation is one of the for fndamental laws that govern all electromagnetic phenomena in the niverse. We ll meet the remaining two laws shortly. Althogh Gass s law for magnetism has zero on its right side, it s not devoid of content; rather, it says that all magnetic fields are configred so that their field lines have no beginnings or endings. F top Got t? 26.6 The figre shows two fields. Which cold be a magnetic field? -F side a m F side Forces on top and bottom cancel. -F top b m points along yor right thmb when yo crl yor fingers in the crrent s direction. The Torqe on a Magnetic Dipole n ection 20.5 we fond that an electric dipole p! in a niform electric field E experiences a torqe t! = p! * E ; in a nonniform field there s a net force as well. The same is tre for a magnetic dipole in a magnetic field, as yo can see by considering the rectanglar crrent loop in a niform field shown in Fig a. Crrent flowing along the top and bottom of the loop reslts in pward and downward forces of eqal magnitde, and neither a net force nor a net torqe is associated with these forces. Crrents flowing along the vertical sides also reslt in eqal bt opposite forces. However, as Fig b shows, these forces reslt in a net torqe abot a vertical axis throgh the center of the loop. The vertical sides have length a and the crrents are perpendiclar to the horizontal magnetic field, so the force on each has magnitde F side = a. The vertical sides are half the loop width b from the axis, so the torqe de to each is t side = 1 2 bf side sin = 1 2 ba sin. Torqes on the two sides are in the same direction (ot of the page in Fig b), so the net torqe is t = ab sin = A sin, with A the loop area. We ve already identified A as the magnitde of the loop s magnetic dipole moment m! and, given the direction of m! as Forces on sides also cancel to give zero net force, bt prodce a net torqe. + -F side down Figre A rectanglar crrent loop in a niform magnetic field. Top view of the loop, showing that magnetic forces on the vertical sides reslt in a net torqe. b 2 m F side p

7 562 Chapter 26 Magnetism: Force and Field shown in Figs and 26.23b, we can incorporate the directionality and the factor sin into a cross prodct: t! = m! * 1torqe on a magnetic dipole2 (26.15) analogos to the torqe on an electric dipole. The magnetic torqe of Eqation cases magnetic dipoles crrent loops to align with their dipole moment vectors along the magnetic field. t takes work to rotate a dipole ot of alignment with the field, and in analogy with Eqation the associated potential energy is U = -m! # (26.16) n a nonniform field, a dipole also experiences a net force. That s why the nonniform field near the poles of a bar magnet attracts magnetic materials that, as we ll see in the next section, contain magnetic dipoles. The torqe on a magnetic dipole is important in many technologies, inclding electric motors and MR imaging. ome satellites se the torqe prodced by Earth s magnetic field to orient themselves in space; with electricity generated from solar panels powering crrent loops, there s no fel to rn ot. Application Electric Motors Commtator + attery N rshes Rotating loop Electric motors are so mch a part of or lives that we hardly think of them. Yet refrigerators, disk drives, sbway trains, vacm cleaners, power tools, food processors, fans, washing machines, water pmps, hybrid cars, and most indstrial machinery wold be impossible withot electric motors. At the heart of every electric motor is a crrent loop in a magnetic field. t instead of a steady crrent, the loop carries a crrent that reverses to keep the loop always spinning. n direct-crrent (DC) motors, this is achieved throgh the electrical contacts that provide crrent to the loop. The figre shows how crrent flows to the loop throgh a pair of stationary brshes that contact rotating condctors called the commtator. The crrent loop rotates to align with the field, bt jst as it does so, the brshes cross the gaps in the commtator and reverse the loop s crrent and therefore its dipole moment vector. Now the loop swings another 180 to its new desired position, bt again the commtator reverses the crrent and so the loop rotates continosly. A rigid shaft spinning with the coils delivers mechanical energy. Ths the motor is a device that converts electrical energy to mechanical energy; the magnetic field is an intermediary in this energy conversion. Example 26.6 Torqe on a Crrent Loop: Designing a Hybrid-Car Motor Toyota s Pris gas electric hybrid car ses a 60-kW electric motor that develops a maximm torqe of 207 N # m. ppose yo want to prodce this torqe in a motor like the one in the preceding Application, consisting of a 700-trn rectanglar coil measring 30 cm by 20 cm in a niform field of 50 mt. How mch crrent does the motor need? nterpret This problem is abot an electric motor, which according to the Application is essentially a crrent loop in a magnetic field. We re given the torqe and asked for the crrent. Develop Eqation 26.15, t! = m! *, determines the torqe on a crrent loop. Figre is a sketch of the loop at the point of maximm torqe, t max = m, which occrs when sin = 1. To solve for the crrent, we need the magnetic dipole moment from Eqation 26.13, m = NA. Then t max = NA. Magnetic dipole moment is ot of page, so m #, giving maximm torqe. 700-trn coil Figre Loop for the motor of Example 26.6, shown in the position of maximm torqe.

Introductory Statistics Neil A. Weiss Ninth Edition

Introductory Statistics Neil A. Weiss Ninth Edition Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world Visit us on the World Wide Web at: