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Imagine that one drilled a very small hole with smooth sides straight through the center of...

Imagine that one drilled a very small hole with smooth sides straight through the center of a charged sphere with total charge Q, radius R and uniform charge density p. A charge −? with mass ? is dropped into one end. Ignore effects from friction and gravity.

a. (10 points) The Coulomb force on an object of charge −? located inside the sphere a distance ? < ? from the center is due only to the charge that lies within the solid sphere of radius ?. Using any method you have learned thus far, determine the Coulomb force as a function of distance r from the center for points ? < ?? (effects from the small hole drilled through the center are negligible in this situation)

b. (10 points) Using the concept of force explain (in words) how this system is analogous to an object-spring system where ? = −??. What is ? for this system?

c. (10 points) Using the force from part (a) set up a differential equation of motion for this system. Use your equation to determine the period T of oscillations for the charge −?.

d. (10 points) Determine the time it takes for a -2.0 C charge to complete one full cycle if dropped from one end. Assume the radius of the sphere is R = 50,000 m and the total charge Q of the sphere is 2500 C.

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