CourseNotes.5 - 3.5 Problems Problem 22.25 In figure 2 two...

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Unformatted text preview: 3.5 Problems Problem 22.25 In figure 2, two curved plastic rods, one of charge +q and the other of charge −q , form a circle of radius R = 8.50 cm in an xy plane. The x axis passes through both of the connecting points, and the charge is distributed uniformly on both rods. If q = 15.0 pC , what are the a) magnitude and b) dircction (relative to the positive direction of the x axis) of the electric field E produced at P , the center of the circle? Figure 2: Problem 22.25 Solution Part a) As with all of these problems, we begin with the differential form of the electric field. dE = 1 dq 4π 0 r 2 We can relate the differential element of charge dq to the differential element of length along the hoop via the linear charge density. The charge density will be the same for the top and the bottom q portion of the ring with opposite signs λ = ± πR . We can use this to describe the differential element of charge in terms of the differential element of length dθ. dq = λ R dθ = ± q dθ π We must be careful here. Our goal is to sum up all of the contributions from all of the differential elements of charge to the field at the center of the ring. The electric field is a vector quantity however, and so we can not simply sum it up as if it was a scaler. The symmetry of the problem allows us to greatly simplify the problem though. If we consider an element of charge dq located at a position θ (where θ is chosen to be 0 at the positive x axis) then it will have a corresponding piece of charge at π − θ which will cancel out the x portion of the field. Hence, only the y component of the field will contribute for each differential piece of charge. The y component of the field is a scaler quantity and hence can be integrated in the normal way. dEy = 1 dq q sin θ = ± 2 sin θ dθ 4π 0 R2 4π 0 R2 Integrating gives the total electric field q 2π π sin θ dθ − sin θ dθ 4π 2 0 R 2 0 π q π 2π = − cos θ|0 − − cos θ|π 4π 2 0 R 2 q =2 = 2.02 N C π 0 R2 E= 5 ...
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This note was uploaded on 12/05/2011 for the course PHY 2049 taught by Professor Any during the Spring '08 term at University of Florida.

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