Class_14new - 1 PHYS809 Class 14 Notes Separation of...

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Unformatted text preview: 1 PHYS809 Class 14 Notes Separation of variables in cylindrical coordinates In cylindrical coordinates , , , r z the metric is 2 2 2 2 2 . ds dr r d dz = + + (14.1) Hence the scale factors are 1 2 3 1, , 1, h h r h = = = so that Laplaces equation for the potential is 2 2 2 2 2 1 1 0. r r r r r z + + = (14.2) To separate the variables, let ( ) ( ) ( ) , R r Z z = so that 2 2 2 2 2 1 1 1 1 1 0. d dR d d Z r R r dr dr r d Z dz + + = (14.3) Assuming that is unrestricted, the angular dependence is , im e = (14.4) where m is an integer. Equation (14.3) is then 2 2 2 2 1 1 1 . d dR m d Z r R r dr dr r Z dz - = - (14.5) The right hand side can be equal to a positive or negative constant. We will first consider the case in which the constant is negative. In particular, we take 2 2 2 , d Z k Z dz = (14.6) which has solutions of form . kz Z e = (14.7) This form of solution is appropriate for problems in which z goes to infinity. The radial equation is now 2 2 2 1 0. d dR m r k R r dr dr r +- = (14.8) This is Bessels equation. By multiplying by 2 r to get ( ) 2 2 2 0, d dR r r k r m R dr dr +- = (14.9) we see that the solution is a function of x = kr . As for Legendres equation, we look for a series solution of form 2 , n l l l R x a x = = (14.10) with 0. a Substitution into Bessels equation leads to ( ) 2 2 2 2 2 . l n l n l n l l l l l l l n m a x a x a x + + + +- = = = +- = - = - (14.11) Since the lowest order term on the right hand side has degree n + 2 , the coefficients of the x n and x n +1 terms must both be identically zero. Hence ( ) 2 2 2 2 1 0 and...
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This note was uploaded on 12/02/2011 for the course PHYS 809 taught by Professor Macdonald during the Fall '11 term at University of Delaware.

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Class_14new - 1 PHYS809 Class 14 Notes Separation of...

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