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exam1S - Physics 21 Fall 2004 Solution Hour Exam#1 The...

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Physics 21 Fall, 2004 Solution, Hour Exam #1 The graders for the problems were: 1 Pagan, 2 Lowe, 3 Shaffer, 4 Wentzel, 5 Hickman For questions about the grading, see the grader by Oct. 1. Problem 1. 1 4 2 2 I 1 I 3 I 2 1 V 8 V 9 V 1 2 (a) Write the loop and node equations needed to determine the currents I 1 , I 2 , and I 3 in the circuit shown. Indicate clearly the loop used to determine each loop equation. node: I 2 = I 1 + I 3 loop 1: 9 I 1 2 I 1 4 I 2 = 0 loop 2: 8 2 I 3 1 4 I 2 = 0 (b) Determine the currents by explicit solution of the equa- tions. You must show your work. Rearrange loop 1 and loop 2 equations: loop 1: 3 I 1 + 4 I 2 = 9 loop 2: 4 I 2 + 2 I 3 = 7 Use the node equation to eliminate I 3 = I 2 I 1 : 3 I 1 + 4 I 2 = 9 (1) 2 I 1 + 6 I 2 = 7 (2) Multiply (1) by 2 and (2) by 3; add and solve for I 2 . Substitute back for I 1 then I 3 . Results are I 1 = 1 . 0 A , I 2 = 1 . 5 A , I 3 = 0 . 5 A Problem 2. The circular ring of charge shown in the di- agram is in the xy plane centered at the origin and has a radius R . A total charge of Q is spread uniformly around the ring. R P (x=0,y=0,z) y z x Q θ (a) Find an expression for the linear charge density λ on the ring. λ = Total charge circumference = Q 2 πR (b) Give the components of the vector shown in the diagram from the point on the ring at angle θ (with respect to the x axis) to the point P . r r is the vector from the charge dQ to P : r r = R cos θ ˆ i R sin θ ˆ j + z ˆ k (c) Determine the electric field d E at the point P due to the element dQ of charge at the point θ on the ring.
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