hw6_p5406_s03 - HW 6 Phys5406 S03 due 3/27/03 1) The object...

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Unformatted text preview: HW 6 Phys5406 S03 due 3/27/03 1) The object is to find the Green’s function inside a cylindrical volume using, in part, the solution to the cylindrical problem of JDJ. This is a cylinder of radius a and length L. All surfaces are grounded except that at z=L and it is kept at potential V(ρ, φ). The solution is in JDJ and was worked out in class. a) Write the solution for the potential in terms of an integral involving the derivative of GD . This should indicate what terms are required in GD aside from constants you will determine below. b) Recall the result for sinh γz /sinh γL expanded in a sine series and that for the sine series expansion of δ (z − z ). c) Show that δ (φ − φ ) = AΣm eim(φ−φ ) and find A. d) Show that δ ( ρ− ρ ) ρ e) Noting that 2 = Σn Bn Jm ( χmn ρ )Jm ( χmn ρ ) and find Bn . a a GD = −4πδ (r − r ), put these pieces together and obtain GD . 2) Solve the problem when the surfaces at z=0 and z=L are at constant potential V and the round surface of the cylinder is grounded. You can use either the Green’s function or directly from LaPlace’s equation. For practise, the adventurous can show that both give the same result. 3) A spherical volume of radius a has a volume charge density ρ = Qx/a4 . Find the potential inside and outside of the sphere. Hint use spherical harmonics and the spherical addition theorem. 1 ...
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