2008 Semifinal Exam Part B 9 Question B2 Consider a parallel plate capacitor with the plates vertical.The plates of the capacitor are rigidly supported in place.The distance between the plates is d.The plates have height h and area Ad2.Assume throughout this problem that the force of air resistance may be neglected;however,the force of gravity cannot be neglected.Neglect any edge effects as well as any magnetic effects. d/2 d a.A small metal ball with a mass M and a charge g is suspended from a string of length L that is tied to a rigid support.When the capacitor is not charged,the metal ball is located at the center of the capacitor-at a distance d/2 from both plates and at a height h/2 above the bottom edge of the plates. If instead a constant potential difference Vo is applied across the plates,the string will make an angle 0o to the vertical when the metal ball is in equilibrium. i.Determine 0o in terms of the given quantities and fundamental constants ii.The metal ball is then lifted until it makes an angle 6 to the vertical where 0 is only slightly greater than 0o.The metal ball is then released from rest.Show that the resulting motion is simple harmonic motion and find the period of the oscillations in terms of the given quantities and fundamental constants. iii.When the ball is at rest in the equilibrium position 0o,the string is cut.What is the maximum value for Vo so that the ball will not hit one of the plates before exiting?Express your answer in terms of the given quantities and fundamental constants. b.Suppose instead that the ball of mass M and charge g is released from rest at a point halfway between the plates at a time t=0.Now,an AC potential difference V(t)=Vo sinwt is also placed across the capacitor.The ball may hit one of the plates before it falls(under the influence of gravity)out of the region between the plates.If Vo is sufficiently large,this will only occur for some range of angular frequencies wmin <w<wmax.You may assume that wmin Vg/h and wmax >Vg/h.Making these assumptions,find expressions for wmin and wmax in terms of the given quantities and or fundamental constants. c.Assume that the region between the plates is not quite a vacuum,but instead humid air with a uniform resistivity p.Ignore any effects because of the motion of the ball,and assume that the humid air doesn't change the capacitance of the original system. i.Determine the resistance between the plates. ii.If the plates are originally charged to a constant potential source Vo,and then the potential is removed,how much time is required for the potential difference between the plates to decrease to a value of Vo/e,where Ine =1? iii.If the plates are instead connected to an AC potential source so that the potential difference across the plates is Vo sinwt,determine the amplitude lo of the alternating current through the potential source. Copyright C2008 American Association of Physics Teachers2008 Semifinal Exam Part B 9 Question B2 Consider a parallel plate capacitor with the plates vertical. The plates of the capacitor are rigidly supported in place. The distance between the plates is d. The plates have height h and area A ≫ d 2 . Assume throughout this problem that the force of air resistance may be neglected; however, the force of gravity cannot be neglected. Neglect any edge effects as well as any magnetic effects. h d/2 h/2 L h d Rigid Support String a. A small metal ball with a mass M and a charge q is suspended from a string of length L that is tied to a rigid support. When the capacitor is not charged, the metal ball is located at the center of the capacitor— at a distance d/2 from both plates and at a height h/2 above the bottom edge of the plates. If instead a constant potential difference V0 is applied across the plates, the string will make an angle θ0 to the vertical when the metal ball is in equilibrium. i. Determine θ0 in terms of the given quantities and fundamental constants. ii. The metal ball is then lifted until it makes an angle θ to the vertical where θ is only slightly greater than θ0. The metal ball is then released from rest. Show that the resulting motion is simple harmonic motion and find the period of the oscillations in terms of the given quantities and fundamental constants. iii. When the ball is at rest in the equilibrium position θ0, the string is cut. What is the maximum value for V0 so that the ball will not hit one of the plates before exiting? Express your answer in terms of the given quantities and fundamental constants. b. Suppose instead that the ball of mass M and charge q is released from rest at a point halfway between the plates at a time t = 0. Now, an AC potential difference V (t) = V0 sin ωt is also placed across the capacitor. The ball may hit one of the plates before it falls (under the influence of gravity) out of the region between the plates. If V0 is sufficiently large, this will only occur for some range of angular frequencies ωmin < ω < ωmax. You may assume that ωmin ≪ p g/h and ωmax ≫ p g/h. Making these assumptions, find expressions for ωmin and ωmax in terms of the given quantities and/or fundamental constants. c. Assume that the region between the plates is not quite a vacuum, but instead humid air with a uniform resistivity ρ. Ignore any effects because of the motion of the ball, and assume that the humid air doesn’t change the capacitance of the original system. i. Determine the resistance between the plates. ii. If the plates are originally charged to a constant potential source V0, and then the potential is removed, how much time is required for the potential difference between the plates to decrease to a value of V0/e, where ln e = 1? iii. If the plates are instead connected to an AC potential source so that the potential difference across the plates is V0 sin ωt, determine the amplitude I0 of the alternating current through the potential source. Copyright c 2008 American Association of Physics Teachers