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hw2_p5406_s05

# hw2_p5406_s05 - E z anywhere between the plates Find the...

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HW 2 Phys5406 S05 due 2/3/05 1) In empty space the Green’s function is | r - r | - 1 . Integrate this over z’ to show that, if the volume charge density doesn’t depend on z’, the two dimensional Green’s function is, within a constant, g = - ln[( x - x ) 2 + ( y - y ) 2 ] . (1) Go to cylindrical coordinates and show that this can be written as g = - ln ρ 2 > - ln[1 + ( ρ < > ) 2 - 2( ρ < > ) cos( φ - φ )] , (2) = - ln ρ 2 > + 2 1 (1 /m )( ρ < > ) m cos m ( φ - φ ) , (3) where ρ <,> is the <, > of ρ and ρ . 2) Consider the case of the two infinite conducting plates studied in class. The lower plate with upper edge at z=0 is held at constant potential V 2 while the upper plate with lower edge at z=D is held at constant potential V 1 . If there is a uniform volume charge density ρ 0 between the plates, find Φ(
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Unformatted text preview: E ( z ) anywhere between the plates. Find the surface charge densities at z=0,D. 3) Consider the hollow conducting box studied in class. The inside surfaces at x=0,a, y=0,b, and z=0 are grounded. The inside surface at z=c is kept at constant potential V. Suppose the box is half ﬁlled, from z = 0 to z = c/2, with a dielectric of permitivity ± , what is the potential inside the box? Take a solution of Laplace’s equation inside and outside the dielectric and from the boundary conditions, including those at the dielectric interface, solve for all unknown coeﬃcients. One could solve for the Green’s function for this geometry. 1...
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