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Class_9new - PHYS809 Class 9 Notes Separation of variables...

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1 PHYS809Class9Notes Separation of variables in polar coordinates To illustrate the method of separation of variables, consider the two dimensional problem in which two semi-infinite conducting planes at potential V meet along their edges at an angle α . The geometry suggests that polar coordinates will be useful for solving Laplace’s equation for the potential between the two planes. In this coordinate system, 2 2 2 1 1 0. r r r r r φ φ θ + = (1.1) To separate the variables, we assume that the potential is the product of a function of r and a function of θ , ( ) ( ) ( ) , . r R r φ θ θ = Θ (1.2) On dividing by the potential, Laplace’s equation leads to 2 2 1 . r d dR d r R dr dr d θ Θ = - Θ (1.3) Each side of the equation is a function of a single variable. This is possible only if the two sides are equal to the same constant. For reasons that will become apparent, we will set this constant to be ν 2 , where ν is non-negative. The equations for R and Θ are now 2 , d dR r r R dr dr ν = (1.4) and 2 2 2 , d d ν θ Θ = - Θ (1.5) which have solutions , 0, ln , 0, ar br R a b r ν ν ν ν - + = + = (1.6) and sin cos , 0, , 0, c d c d νθ νθ ν θ ν + Θ = + = (1.7) where a , b , c and d are constants. In general, the solution is a linear superposition of terms of form r θ
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2 ( ) ( ) ( )( ) sin cos , 0, ln , 0.
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