hwk_2_solutions

hwk_2_solutions - MasteringPhysics: Assignment Print View

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Unformatted text preview: MasteringPhysics: Assignment Print View http://sessi0n.masteringphysics.com/myct/assignmentPrint?assignmen... Assignment Display Mode [ Mimi PHYSICSSCFOB Etoci‘ric we E‘Ecios ti {Ease-2 at S‘ézfiéazt; or: atorzitay. {Emmet-st“ 3, $838 View Gradng Details Torque on a Dipole in a Uniform Field Description: Calculate the torque on and the potential energy of a dipole (two opposite charges) in a uniform electric field. I) Consider an electric dipole whose dipole moment (a vector pointing fromthe negitive charge to the positive charge) is oriented at angle {‘3 with respect to the y axis. There is an external electric field of magnitude (independent of the field produced by the dipole) pointing in the positive y direction The positive and negative ends of the ; dipole have charges q and m 1’}, respectively, and the two charges are a distance :5 apart. The dipole has a moment of inertia I about its center ofrmss. It will help you to imagine that the dipole is flee to rotate about a pivot through its center. Part A What is the net force thatthe dipole experiences due to the electric field? ' Part A What is the force on the positive charge? What is the force fiw that the charge «,1- q experiences due to the electric field in the y direction? ‘ Express if?” in terms ofthe given variables and the unit vectors x‘, ANSWER: gm 2 ’55:?) Express me in terms of the given variables and the unit vectors 2}, ANSWER: Part B What is em (F31, the magnitude of the torque that the electric field exerts about the center of mass of the dipole? Part 3.1 Find the torque on the positive charge What is the magnitude of the torque 7W about the center of mass of the dipole due to the force that the electric field exerts on the positive charge? Recall that the torque vector is given by 3’ v» if? x F, Where is the position vector from the center of the dipole to the applied force 1'31 . ’Part B.1.a Find the force on the positive charge i What is the force gm that the charge «t q experiences (he to the electric field in the y direction? Express your answerin terms of the given variables and the unit vectors a, 3}. ANSWER: 2% = Part B.1.b What is the moment arm? simple way to compute the magnitude of the torque about a pivot point is to multiply the magnitude of the force by the moment arm, which is the distance fi'omthe line of force to the chosen pivot point (and to apply a negative ' sign ifthe torque is clockwise). What is the moment arm {gm for the force on the positive charge in this problem? xpress your answer in terms of the given distances and angles. ANSWER: V {t z 3 Express the magnitude of the torque in terns of the given quantities. ANSWER: oi » 1 . yw = «Egmsgétf; 10f7 10/14/200811119AM MasteringPhysics: Assignment Print View http://sessionmasteringphysics.com/myct/assignmentPrint?assignmen... m Hint 8.2 Now considerthe negative charge Now find the magnitude torque on the negative charge. To find the magnitude of the net torque on the center of mass of the dipole, you need to consider whether the torques on the two charges act in the same or opposite directions. If both torques tend to rotate the dipole in the same direction, sum them to find the magnitude of the net torque, otherwise, subtract the smaller fiom the larger. Express the magnitude of the total torque in terms of the given quantities. ANSWER: . I _ enemy} {mtg qdfia’in {ifaemé Ea Part C fig Using the above result find the potential energy if” {{2} associated with the dipole‘s orientation in the field as a function of the angle 5? shown in the figure. Take the zero of the potential to occur when the dipole is at angle «f2; that is, {forge} a, é Hint CI The definition of potential energy The potential energy associated with a conservative force F is given by e w: ‘( my; M! firmer. m This definition ensures that Fix} mi?!) The potential energy function associated with a torque is found analogously, essentially by multiplying both sides of this equation by a distance (so force is convened to torque), and '0 then integrating with respect to angle rather than distance (since the distance has already been absorbed into the functions). Thus if the torque is given by the associated potential energy is we: we ~~ gaggrwa’, :Note that the lower bound of this integral is arbitrary Y 1T! Hint C.2 How to choose the constant of integration When you integrate to find 135*}, choose the constant of integration to satisfy the condition I’ ’ ANSWER: I ~ momma} m etfifi‘crxe t {from} ml’artD g Interms ofthe dipole moment I3, which ofthe following is anexpression for the torque on a dipole inthe field Hint D.l How to approach this part The general relation between the torque, dipole moment and electric field may be hard to derive. However, you can easily compare the magnitude and/or direction of the torque derived for the particular case considered above with the magnitudes and/or directions of the expressions given below for the same physical situation. i E E z where 2,? is the smaller angle between {3' and ‘3; The direction of 5 is given by the right-hand rule: stretch out the thumb of your right hand and curl the remaining fingers from {if to i? . Then your outstretched thumb gives the direction of i‘ , Hint 113 Formula for a » E; The formula for if > is gegems, 20f7 10/14/2008 11:19AM MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assigmnentPrint?assigmnen... : where it is the smaller angle between t? and g Hint DA More on torques The direction of the torque vector ?‘ can be detemtined from the vector equation ' where is? is a vector that points from the pivot to the point of action of the force it. m Part E In terms of the dipole moment 13, which of the following is an expression for the potential energy of a dipole in the field ff? Formula for t: x 2}” »Let€xeei§>tg,'llren ( 3 where (‘3 is the smaller angle between ('5' and g. The direction of at: is given by the right-hand rule: stretch out the thumb of yotn right hand and curl the remitting fingers from 13‘ to . Then your outstretched thumb gives the direction of E. Hint 15.2 Formula for a t the formula for a is a? gags (me, where $1 is the smaller angle between :3 and . A uniform electric field exists in the region between two oppositely charged parallel plates 1.70 cm apart. A proton is released fi'om rest at the surface of the positively charged plate and strikes the surface of the opposite plate in a WM“. \M\\\«mmmr«ww,w,i time interval 1.56><10'6 Part A Find the magnitude of the electric field. Hint A.l How to approach the problem Use the equation for the motion over time of a particle under constant acceleration, and use the acceleration you calculate to find the electric field needed to create that acceleration for the proton Hint A2 A relatiome between electric force and electric field Recall that a charged particle in an electric field will have an electric force on it equal to F: This means that the force will point in the same direction as if the particle is positively charged, and in the opposite direction as if the particle is negatively charged. m Pan A,3 Calculate the acceleration of the proton , : Calculate the acceleration (r of the proton. given the time taken to get fi'om one plate to the other and the separation of the plates. ANSWER: ‘21; TB Pan AA Calculate the force on the proton 3 of7 10/14/2008 11:19AM MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignmentPrint?assignmen... Calculate the force F on the proton needed to create the acceleration calculated in Part. A3. ANSWER: ; Part B Find the speed of the proton at the moment it strikes the negatively charged plate. Hint 3.1 How to approach the problem ANSWER: i I: I . mgr» Problem 20.79 Description: A magician friend is planning to use an electric field to levitate some srmll beads. They have a mass of meach, and he will give them q of charge. In order for the trick to work, the beads must remain stationary under the influence ofthe electric... A magician fiiend is planning to use an electric field to levitate sortie small beadsl They have a mass of 16 g each, and he will give them 3.0 ,1ng of charge. In order for the trick to work the beads must remain stationary under the influence ofthe electric field. Part A What should the strength of the electric field be? Express your answer using two significant figures. ANSWER. 5.: en Problem 20.35 Description: A proton moving to the right at v enters a region where a E electric field points to the lefi. (a) How far will the proton get before its speed reaches zero? (b) Describe its subsequent motion. 2 How far will the proton get before its speed reaches zero? Express your answer using two significant figures. ANSWER: i H 33m 5‘35? Part B Describe its subsequent motion ANSWER: Answer Key: 'The proton subsequently moves to the left, with the same constant acceleration in the field region, until it exits with the initial velocity reversed. Problem 20.58 F Description: The figure shows a device its inventor clains will separate isotopes of a particular element (Isotopes of the same element have nuclei with the same charge but different rmsses.) Atoms of the element are first stripped completely of their electrons.“ The figure shows a device its inventor clairm will separate isotopes of a particular element. (Isotopes of the same element have nuclei with the same charge but different rmsses.) Atoms of the element are first stripped completely of their electrons, then accelerated fi'om rest through an electric field chosen to give the desired isotope exactly the right speed to pass through the electrostatic analyzer (two curved metal plates establishing an electric field given by L alarm). Analyzer 4of7 10/14/200811:19AM MasteringPhysics: Assignment Print View http://session.masteringphysics.corn/myct/assigmnentPrint?assignmen.. i Part A Prove that the device won't work-that is, that it won't separate different isotopes. ANSWER: Answer Key: :When the device is in operation, an isotope, of nuclear charge q and mass m, is accelerated from rest to a speed v, in a distance d, by the field E#l, where v”2=2*al*d=2*(q*El/m)*d. This will be the proper speed to pass through the analyzer if a2=q*E2/m=vA2/r=2*(q*El /m)*d/r, or E2=2*El*d/r. This condition depends on the fields and the geometry, but not on q/m, so different isotopes cannot be separated. Gauss's Law Description Two multiple~choice questions about how to use Gauss's law. Learning Goal: To understand the meaning of the variables in Gauss‘s law, and the conditions under which the law is applicable. Gauss's law is usually written twfifissmm; where $85 a it?” Cg; v In?) is the permittivity ofvacuurn. How should the integral in Gauss's law be evaluated? ANSWER: I around the perimeter of a closed 100p _ over the surface bounded by a closed loop over a closed surface is the area of the surface element. The direction of is normal to the surface element, pointing out of the » represents an infinitesimal surface element. The nagnimde of (in enclosed volume. s i i E In Gauss's law, to what does emf; refer? ANSWER: the net charge inside the closed surface the charge residing on insulators inside the closed surface all the charge in the physical system any charge inside the closed surface that is arranged symmetrically § The major of Gauss‘s law is to etermme an electric field whenthe charge distribution both inside‘anduoutside the Gaussian surface, symmetric. Of electric field can alwaysube fouml by adding up (or \ 'ntegrating) the contributions of all the charge in the problem In highly symmetric situations, however, Gauss‘s law is much simpler computationally than dealing with all such contributions, and it provides better physical might, too. s, Calculating Electric Flux through a Disk Description: Calculate the electric flux through a disk orientated at an angle with respect to a unifonn electric field. 9 Suppose a disk with area A is placed in a uniform electric field ofrmgnitude E. The disk is oriented so that the vector nomtal to its surface, it, makes an angle 1? with the electric field, as shown in the figure. i 50f7 10/14/200811:19AM MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assigmnentPrint?assigmnen... g The electric flux <33; througha surface is given by 4);; 2e o where in? is an infinitesimal element ofarea onthe surface. m Hint A.2 Simplifying the integrand _ Note that the formula for electric flux can be expressed in the following way: to... [signifieth Since we are dealing with a constant electric field, which therefore does not vary across the surface of the disk, we can take 3; o 3‘? outside of the integral. This gives a much simpler expression~ m Part A.3 Evaluate the scalar product ind . y“; interms one: and e. ANSWER: g . is; = £99} :: Express your answer in terms of IS, xi, and ii ANSWER: ; oi; = fitméfim Problem 21.44 Description: (a) What is the flux through the hemispherical open surface of radius R shown in the figure ? The uniform field has magnitude E. Part A ‘ What is the flux through the hemispherical open surface of radius it? shown in the figure ? The uniform field has magifitude E. V VHintAJ Don't use a messy integral! Give your answerin terms of I? and I13. ANSWER= E ca: «52215:: ‘Problem21.48 ’ V ' V ' E Description: A solid sphere 2.0 cm in radius carries a uniform volume charge density. The electric field 1.0 cm from the spheres center has magnitude 39 kN/C. (a) At what other distance fi’om the center does the field have this magnitude? (b) What is the net... A solid sphere 2.0 m; in radius carries a uniform volume charge density. The electric field 1.0 em from the sphere's center has magnitude 39 kaC. Part A At what other distance from the center does the field have this magnitude? ‘ Express your answer using two significant figures. ANSWER: : lg 65 = cm 2.8 ‘ Part B What is the net charge onthe sphere? i E. Express your answer using two significant figures. 2 i i- ANSWER: ' .5; = 33;, “(f 60f7 10/14/2008 11:19AM MasteringPhysics: Assignment Print View htlp://session.masteringphysics.com/myct/assigtttnentPrint?assignmen... Problem 21 70 i Description: A friend is working on a biology experin and needs to create an electric field of magnitude E at a distance of 10 cm fiomthe central portion of a large nonconducting square plate a on each side. She needs to know ’ how much charge to put on E S k A friend is working on a biology experiment and needs to create an electric field of magnitude 380 at a distance of 10 can: from the Central portion of a large nonconducting square plate 4.9 m on each side. She needs to know how much charge to put on the plate. Part A 3 What do you tell her? Express your answer using two significant figures. ANSWER: gagfiaigwfix‘afifi IQ! = i‘ 1.6x10‘7 Summary 2 of 107ilems complete (10% avg. score“ - 20 of 13 points ; 7of7 10/14/200811:19AM ...
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hwk_2_solutions - MasteringPhysics: Assignment Print View

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