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HW10 - MasteringPhysics 3:36 PM Assignment Display Mode...

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5/10/09 3:36 PM MasteringPhysics Page 1 of 13 http://session.masteringphysics.com/myct Assignment Display Mode: View Printable Answers phy260S09 HW10 Due at 11:00pm on Tuesday, April 28, 2009 View Grading Details Charged Ring Description: Find the electric field from a uniformly charged ring (qualitative and quantitative parts) at points along its axis. Then use this to find the frequency of small oscillations of a oppositely charged object placed on the axis. Consider a uniformly charged ring in the xy plane, centered at the origin. The ring has radius and positive charge distributed evenly along its circumference. Part A What is the direction of the electric field at any point on the z axis? Hint A.1 How to approach the problem Approach 1 In what direction is the field due to a point on the ring? Add to this the field from a point on the opposite side of the ring. In what direction is the net field? What if you did this for every pair of points on opposite sides of the ring? Approach 2 Consider a general electric field at a point on the z axis, i.e., one that has a z component as well as a component in the xy plane. Now imagine that you make a copy of the ring and rotate this copy about its axis. As a result of the rotation, the component of the electric field in the xy plane will rotate also. Now you ask a friend to look at both rings. Your friend wouldn't be able to tell them apart, because the ring that is rotated looks just like the one that isn't. However, they have the component of the electric field in the xy plane pointing in different directions! This apparent contradiction can be resolved if this component of the field has a particular value. What is this value? Does a similar argument hold for the z component of the field? ANSWER: parallel to the x axis parallel to the y axis parallel to the z axis in a circle parallel to the xy plane Part B What is the magnitude of the electric field along the positive z axis? Hint B.1 Formula for the electric field You can always use Coulomb's law, , to find the electric field (the Coulomb force per unit charge) due to a point charge. Given the force, the electric field say at due to is . In the situation below, you should use Coulomb's law to find the contribution to the electric field at the point from a piece of charge on the ring at a distance away. Then, you can integrate over the ring to find the value of . Consider an infinitesimal piece of the ring with charge . Use Coulomb's law to write the magnitude of the infinitesimal at a point on the positive z axis due to the charge shown in the figure. [ Print ]
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5/10/09 3:36 PM MasteringPhysics Page 2 of 13 http://session.masteringphysics.com/myct Use in your answer, where . You may also use some or all of the variables , , and . ANSWER: = Hint B.2 Simplifying with symmetry By symmetry, the net field must point along the z axis, away from the ring, because the horizontal component of each contribution of magnitude is exactly canceled by the horizontal component of a similar contribution of magnitude from the other side of the ring. Therefore, all we care about is the
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