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Unformatted text preview: SUNDAY MIDTERM 2 REVIEW PART I NATHAN MOORE Problems (1) Consider a uniformly charged sphere of radius R and charge Q > 0 centered at the origin. (a) What is the potential at the center of the sphere, V (0), assuming that the potential at infinity is zero? [Hint: line integrals are not needed] (b) We drill a very small hole to the center of the sphere along the positive x-axis and place a charge -e of mass m at the center. If we give the charge a velocity v in the x-direction, what is the minimum value of v needed for the the charge to escape the sphere? (c) What is E ( x,y,z ) at some point ( x,y,z )? (d) What is V ( x,y,z ) at some point ( x,y,z )? (e) What is the value of the potential on the surface of the sphere, V ( R )? (f) What is U of the sphere? (g) How much work must be done in order to squash the sphere into a uniformly charged sphere of half the radius and charge Q? (2) Consider a parallel plate capacitor consisting of two rectangular plates of area A , separated by a distance d which is much less than the length and width of the plates. The bottom plate lies in the x-y plane with charge- Q and the top plane lies in the plane z = d with charge Q . There is a dielectric filling the space between the plates of dielectric constant K ( z ) = 1 + z d . (a) What is the capacitance of the system? (b) How much work does it take to remove the dielectric slab?...
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