University of Illinois, Urbana ChampaignvPHYS 212vDQ2C_solution

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Discussion Question 2C P212, Week 2 Forces Between Point Charges 1 Q Q h d q, m Consider two charges Q placed at fixed positions a horizontal distance d apart, as shown in the figure. A third... particle of mass m and unknown charge q is then placed a vertical distance h above the fixed charges. The position of this third particle is arranged so that it is suspended in mid-air above the other two! This equilibrium condition occurs because the electrical and gravitational forces on the charge precisely cancel each other. (a) Calculate the charge q that is required to balance the third particle in mid-air. Express your answer in terms of the given parameters plus physical constants. 2 2 2 2 3 / 2 2 2 2 0 ( / 2) ( / 2) ( / 2) 2 kQq h mg d h d h mg q d h KQh × − = + + = + ⎡ ⎤ ⎣ ⎦ (b) Does your answer make sense? Find three limiting cases where you know what the answer should be and make sure your expression for q gives the right result. . Hint: The best algebraic check is probably d → 0 . Does the sign of q make sense? Want q and Q to have same sign to insure repulsion. As h→0 : q→∞ 9 As d →0: q= 2 2 (2 ) 2 k Q q mgh mg q h KQ = → = 9 As m→0 : q→0 (c) Equilibrium situations like the one you have analyzed can be described as either stable or unstable. Stable equilibrium means that the system will return to its equilibrium configuration when disturbed by a small amount (like a swinging pendulum). Unstable equilibrium describes situations like a ball balanced on top of a hill – the slightest disturbance will drastically affect the system. Think about your suspended charge problem for a moment ... what sort of equilibrium condition is present? Take your best guess before you turn to the next page. No restoring force in ± z (in-out of page) thus unstable equilibrium [Show More]

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