prelim2007

prelim2007 - Table of Contents squaresolid Complete...

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Unformatted text preview: Table of Contents squaresolid Complete squaresolid Partially Complete squaresolid Not Complete Preliminary Examination 2007 2 Problem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Problem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Problem 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Problem 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Problem 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Problem 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Problem 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Problem 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Problem 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Problem 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1 Preliminary Examination September , 2007 1. The space between two concentric spherical conducting shells is half-filled with a linear, isotropic dielectric material as shown in the figure. The shaded region contains the dielectric; the remainder is vacuum. The outer radius of the inner shell is a , the inner radius of the outer shell is b , and the dielectric has dielectric constant . Note: the boundary between the dielectric and the vacuum is a plane passing through the center of the sphere (normal to the page). Hence no radial vectors pass through this boundary. The inner sphere carries a charge Q and the outer sphere Q . (a) Find expressions of E , D , and P in the space between the spheres, both in the vacuum and dielectric regions. Be explicit about your uses of symmetry and boundary conditions for E and D in arriving at your results. (b) Is the free surface charge density on the inner conductor larger or smaller in the dielectric region compared with the vacuum region? Explain your answer. (c) Find the capacitance of the system. Solution Part C: This system is like two hemi-spherical capacitors in parallel, where the potential difference across the plates of each capacitor is the same. Denote the capacitor with the dielectric C 1 , and the capacitor without by C 2 . To find the capacitance C 1 , first slap a charge + q 1 on the inner surface, and q 1 on the outer surface. Determine E 1 between the plates. By symmetry, we can say that the electric field E 1 between two concentric hemispherical plates, where charge + q 1 is on the inner plate, is the same as the electric field between two concentric spherical plates where charge +2 q 1 is on the inner plate....
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prelim2007 - Table of Contents squaresolid Complete...

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