`
`
`
`Jay Newman
`
`Momentum Dynamics Corporation
`Exhibit 1010
`Page 001
`
`Momentum Dynamics Corporation
`Exhibit 1010
`Page 001
`
`

`

`Jay Newman
`Union College
`Department of Physics and Astronomy
`Schenectady, NY 12308, USA
`
`ISBN: 978-0-387-77258-5
`DOI: 10.1007/978-0-387-77259-2
`
`e-ISBN: 978-0-387-77259-2
`
`Library of Congress Control Number: 2008929543
`
`© 2008 Springer Science+Business Media, LLC
`All rights reserved. This work may not be translated or copied in whole or in part without the written permission
`of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA),
`except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of
`informationstorage andretrieval, electronic adaptation, computer software, or by similar or dissimilar methodology
`now knownorhereafter developed is forbidden.
`The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not
`identified as such, is not to be taken as an expression of opinion as to whetheror not they are subject to proprietary
`rights.
`
`Printed on acid-free paper
`
`987654321
`
`springer.com
`
`Momentum Dynamics Corporation
`Exhibit 1010
`Page 002
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`Momentum Dynamics Corporation
`Exhibit 1010
`Page 002
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`

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`2. TORQUE AND FORCE ON A MAGNETIC DIPOLE
`
`At the end of the last section we considered the magnetic force on a straight current-
`carrying wire in a uniform magnetic field. Another important geometry of current
`flow, the current loop, is worthy of its own discussion. A current loop is a generic
`term for a simple circuit with a single closed loop, regardless of the exact trajectory
`of the current. Its importancelies not only in actual conducting wire circuits, but also
`in its use as a model for understanding the magnetic properties of matter through
`atomic electron current loops.
`loop, or in fact any current carrying
`In Section 4 we show that a current
`wire, generates its own characteristic magnetic field. Here we wish to examine the
`forces acting on a current
`loop placed in an external uniform magnetic field.
`Consider the rectangular current loop in Figure 17.11 lying in a region of uniform
`B field as shown. In this orientation, the two edges that are parallel to the magnetic
`field have no force acting on them, whereas the other two edges perpendicular to
`the B field each have a force on them given by Equation (17.6). Because the cur-
`rent direction is opposite in those two wire segments, the corresponding forces act
`in Opposite directions to create a couple (the torque due to equal and opposite
`forces) about the horizontal axis shown in the figure. There is no net force acting
`on the loop but the net torque acting will tend to produce a rotation of the loop
`as shown.
`
`torque
`Using the dimensions of the loop shown, we can calculate the net
`acting on the current
`loop about
`its central axis in the orientation shown in
`Figure 17.11 to be
`
`r= kB + eB = IewB = IAB,
`
`(17.7)
`
`where w/2 is the lever arm and A = €w is the area of the loop. If the loop is able to
`rotate, the couple will produce a rotation of the loop about the axis of rotation as
`shown. Equation (17.7) gives the maximum torque acting on the loop because, as can
`be seen in the side view shown in Figure 17.12, the lever arm distance changes with
`the orientation of the loop. With @ equal to the angle between the B field and the
`normalto the plane of the loop, the lever arm can be written as
`w
`= asin 8,
`2
`
`r
`
`so that in general the torque on a current loop in a uniform B field becomes a func-
`tion of the rotation angle
`
`T = wB sin 6,
`
`(17.8)
`
`where we have introduced the magnetic dipole moment ps = IA.
`The magnetic dipole momentis a vector quantity, just as is the electric dipole
`moment, and we chooseits direction to be perpendicular to the plane of the current
`loop. A simple second right-hand rule indicates which of the two directions
`perpendicular to the current loop plane is correct: if the fingers of your right hand
`are curled along the direction of current flow in a wire loop, your thumb will point
`in the proper direction of the magnetic dipole moment. Of course, if the current
`direction reverses so does the direction of the magnetic dipole moment,
`in
`accord with this right-hand rule. Note that if, instead of a single loop, we have a
`circuit with a tightly wound helical loop of N turns, we can replace this with N
`identical loops each having the same area and current so that the magnetic dipole
`moment of the circuit is 4 = N/A. Also note that Equation (17.8) is very similar to
`the equation for the torque on an electric dipole moment
`in an electric field
`(Equation (15.13))
`
`T = pEsin8@,
`
`TORQUE AND FORCE ON A MAGNETIC DIPOLE
`
`
`
`FIGURE 17.11 A current loop ina
`uniform magnetic field. The two
`forces shown are perpendicular to
`the plane of the paperas deter-
`mined by the right-handrule for
`Equation (17.6).
`
`
` |:=(w/2)sind
`
`FIGURE 17.12 Side view of a cur-
`rent loop in a uniform magnetic
`field. The normal to the loop makes
`an angle 0 with respect to the B
`field. The two forces that produce
`a net torque are shown with the
`moment arm r, as well as the
`magnetic dipole moment
`= IA
`along the normalto the loop. The
`net torque tendsto align the
`magnetic dipole moment with the
`magneticfield.
`
`Momentum Dynamics Corporation
`Exhibit 1010
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`Momentum Dynamics Corporation
`Exhibit 1010
`Page 003
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