q1s - Physics 2212 K Summer 2009 Quiz#1 Solutions...

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Physics 2212 K Quiz #1 Solutions Summer 2009 ² 0 Permittivity constant m p Mass of a proton μ 0 Permeability constant m e Mass of an electron e Fundamental charge c Speed of light K Coulomb constant g Acceleration due to gravity Unless otherwise directed, friction, drag, and gravity should be neglected, and all batteries and wires are ideal. Any integrals in free-response problems must be evaluated. I . (20 points) Two identical particles, each with mass m and positive charge q , are attached by ideal cords to the same point on an infinite uniform horizontal plane of positive charge, as illustrated. The plane causes a uniform electric field with magnitude E 0 . When the particles are at equilibrium, there is an angle θ between their cords. What is the length, L , of the cords? Express your answer in terms of parameters defined in the problem, and physical or mathematical constants. ( On Earth, do NOT neglect gravity! ) . . . . . . . . . . . . . . Use Newton’s Second Law. Sketch a Free-Body Diagram. The forces on one of the particles are a weight mg , a tension T , an electric force due to the charged plane qE 0 , and an electric force due to the other particle Kq 2 /r 2 . Note that r = 2 L sin θ 2 . Choose a coordinate system. Write out Newton’s Second Law in terms of magnitudes for each dimension. X F x = Kq 2 r 2 - T x = ma x = 0 T sin θ 2 = Kq 2 4 L 2 sin 2 θ 2 and X F y = qE 0 - mg - T y = ma y = 0 T cos θ 2 = qE 0 - mg There are two unknowns, L and T . Since L is the answer to the question, eliminate T . One way is to divide the expression found from the x component of the forces by that found from the y component. T sin θ 2 T cos θ 2 = sin θ 2 cos θ 2 = Kq 2 4 L 2 sin 2 θ 2 qE 0 - mg Solve for L . L 2 = Kq 2 cos θ 2 4 ( qE 0 - mg ) sin 3 θ 2 L = s Kq 2 cos θ 2 4 ( qE 0 - mg ) sin 3 θ 2 Quiz #1 Solutions Page 1 of 4
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1. (10 points) An electron orbits a fixed uniformly charged sphere. The orbital radius is R , and the speed of the electron is v e . With what speed, v p , will a proton orbit that sphere at the same radius?
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