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spring_05_exam1_sol_phy2049

# spring_05_exam1_sol_phy2049 - PHY2049 Spring 2005 Prof...

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PHY2049 Spring 2005 1 Prof. Darin Acosta Prof. Paul Avery Feb. 7, 2005 PHY2049, Spring 2005 Exam 1 Solutions 1. A central particle of charge 3 q is surrounded by two circular rings (radius R and 2 R , respectively) of charged particles, as shown in the figure. What are the magnitude and direction of the net electrostatic force on the central particle due to the other particles? Most charges are distributed symmetrically around the circle, so that the net force on the central particle is zero. The only unbalanced forces lie along the y axis from 3 charges: ( )( ) ( )( ) ( )( ) 2 2 2 2 2 2 2 2 2 2 2 3 2 3 4 3 6 4 4 6 3 9 9 up 2 2 y y q q q q q q F k k k R R R q q q q F k k k k R R R R + + + = − + = + = 2. Charge is uniformly distributed around a ring of radius 10cm and the electric field E is measured along the ring's central axis (perpendicular to the plane of the ring. At what distance (in cm) from the ring's center is the magnitude of E maximized? The electric field from a continuous charge distribution is solved by integration. Keep in mind that by symmetry, only the field pointing along the axis perpendicular to the ring plane, let’s call it the z axis, is non-zero. ( ) ( ) 2 2 2 2 3/2 2 2 3/2 2 2 cos cos 2 z z z dq z dE R z R z ds z dE R z R z E R z θ θ λ π λ = = + + = + = + To find where the field is a maximum, take a derivative: +4 q +6 q 2 q 2 q 2 q 2 q +2 q 7 q 7 q + q + q −3 q

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PHY2049 Spring 2005 2 ( ) ( ) ( ) 3/2 5/ 2 2 2 2 2 2 2 2 3 0 2 2 0 3 7.1 cm 2 z dE R z z R z z dz R z z R z = = + + + = + = = 3. Three concentric conducting shells have radii
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spring_05_exam1_sol_phy2049 - PHY2049 Spring 2005 Prof...

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