exam1sol(1).pdf - Exam I Grad problems are starred 1 Prove...

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Exam I - Grad problems are starred 1) Prove that 2 ( 1 r ) = - 4 πδ 3 ( r ), with the following steps consider the function G ( r, ) = K 1 ( r 2 + 2 ) 1 / 2 , with K a constant to be determined later, and evaluate the Laplacian function 2 G ( r, ) = D ( r, ). show that the D ( r, ) function approximates a Dirac delta everywhere, with the fol- lowing steps lim 0 D (0 , ) = , (1) lim 0 D ( r, ) = 0 , r 6 = 0 , (2) Z D ( r, ) d 3 ( r ) = constant. (3) From the integral, and using the known integral of the Dirac delta, evaluate the constant K . Solution: The Laplacian can be calculated in spherical coordinates to obtain D ( r, ) = - 3 2 ( r 2 + 2 ) 5 / 2 . The first limit goes like - 3 , while the second goes like 2 , so both limits are proven. The integral over all space can be done in spherical coordinates, with the solid angle integrating to 4 π . Z D ( r, ) d 3 ( r ) = ( - 4 π )(3 2 ) Z r 2 dr ( r 2 + 2 ) 5 / 2 . The indefinite integral in r is Z r 2 dr ( r 2 + 2 ) 5 / 2 = r 3 3 2 ( r 2 + 2 ) 3 / 2 , so that the integral result is - 4 π , and the constant K is 1. 2) Two semi-infinite grounded metal surfaces are in the ( z, y ) plane and ( z, x ) plane, with x, y > 0. A point charge + q is located at ( x 0 , y 0 , 0).
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  • Fall '15
  • Giovani Bonvicini

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