hw5_p5406_s07

# hw5_p5406_s07 - HW 5 Phys5406 S07 due 1 For the 1D Green...

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HW 5 Phys5406 S07 due 3/1/07 1) For the 1D Green function problem solved in class add the following volume charge den- sities: a) σδ ( z - D/ 2), b) ρ 0 , and c) ρ 0 z/D . For each solve for the E between the plates and accurately sketch E vrs z. For part a) solve via Gauss’s law and convince yourselves that you get the same solution as with the 1D Green function. 2)Use the solution found in class for the potential inside a conducting box of sides (a,b,c) in the (x,y,z) directions when all sides are grounded except for the side at z = c that is maintained at potential Φ 0 . We will use the solution to get the Green’s function G D for all problems with this geometry. a) To see the connection between 2 G D ( r , r ) = - 4 πδ ( r - r ) and a sine series, expand a 1D delta function in the range (0,a) as below, δ ( x - x ) = j [ A j sin jπx/a sin jπx /a ] , (1) and find the coefficients A j . This is called the completeness relation for the sine series, indicating any function f(x), x in the range (0,a), can be expressed as, m B m sin mπx/a. (2) Find B m .

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