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Class_14new

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

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1 PHYS809Class14Notes Separation of variables in cylindrical coordinates In cylindrical coordinates , , , r z θ the metric is 2 2 2 2 2 . ds dr r d dz θ = + + (14.1) Hence the scale factors are 1 2 3 1, , 1, h h r h = = = so that Laplace’s equation for the potential is 2 2 2 2 2 1 1 0. r r r r r z θ ∂Φ ∂ Φ ∂ Φ + + = (14.2) To separate the variables, let ( ) ( ) ( ) , R r Z z θ Φ = Θ 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 θ Θ + + = Θ (14.3) Assuming that θ is unrestricted, the angular dependence is , im e θ ± Θ = (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 Bessel’s 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 Legendre’s equation, we look for a series solution of form

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2 0 , n l l l R x a x = = (14.10) with 0 0. a Substitution into Bessel’s equation leads to ( ) 2 2 2 2 0 0 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 1 0. n m n m a - = + - = (14.12) The first equation gives , n m = ± and hence the second requires 1 0. a = Thus all odd terms in the series in (14.10) have coefficients equal to zero.
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Class_14new - PHYS809 Class 14 Notes Separation of...

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