F10HW06 - 1 Problem 3.17 1.1 Equation 3.138 in Jackson r 2...

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Unformatted text preview: 1 Problem 3.17 1.1 Equation 3.138 in Jackson: r 2 G ( ~x; ~x ) = 4 ( ) ( ' ' ) ( z z ) Substituting in the de nition of the laplacian in cylindrical coordinates: 1 @ @ @G ( ~x; ~x ) @ + 1 2 @ 2 G ( ~x; ~x ) @' 2 + @ 2 G ( ~x; ~x ) @z 2 = 4 ( ) ( ' ' ) ( z z ) (1) Now, we want to show that the Green function can be expressed in the following form: G ( ~x; ~x ) = 4 L 1 X n =1 1 X m = 1 e im ( ' ' ) sin n z L sin n z L I m n L < K m n L > First, we're going to con rm that this solution satis es the above equation (that is, it's a valid Green function). Then, we're going to show that it is the Green function for the given setup. Plugging in the above Green function into equation (1): 4 L 1 X n =1 1 X m = 1 e im ( ' ' ) sin n z L sin n z L 1 @ @ @ @ I m n L < K m n L > (2) 1 2 m 2 + n 2 2 L 2 I m n L < K m n L > = 4 ( ) ( ' ' ) ( z z ) We will need the following identities of the Dirac delta function: ( ' ' ) = 1 X m = 1 e im ( ' ' ) (3) ( z z ) = 2 L 1 X n =1 sin n z L sin n z L (4) Hence, the right hand side of equation (2) becomes: 8 L ( ) 1 X n =1 1 X m = 1 e im ( ' ' ) sin n z L sin n z L 1 Thus, we have two summations equal to one another. Setting their coe cients equal to one another and canceling the exponential and sin terns from both sides of the equation: 4 L 1 @ @ @ @ I m n L < K m n L > 1 2 m 2 + n 2 2 L 2 I m n L < K m n L > = 8 L ( ) @ @ @ @ I m n L < K m n L > m 2 + n 2 2 2 L 2 I m n L < K m n L > = 2 ( ) Letting x = n L , this becomes the Bessel equation for x 6 = x . Clearly, I m ( x < ) K m ( x > ) is a solution to this equation. Hence, we have shown that the proposed Green function is indeed a valid Green function. To show that this is the Dirichlet Green function for the system at hand, we must show that is zero for ~x is on the surfaces: The exponential function has no zeroes for nite ' . sin n z L ¡ is zero at z = 0 ; L , as expected. Additionally, it is zero when z is an integer multiple of L , which is allowed since using the method of images would e ectively gives rise to an in nite number of surfaces at these places....
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F10HW06 - 1 Problem 3.17 1.1 Equation 3.138 in Jackson r 2...

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