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Unformatted text preview: 1 PHYS809 Class 9 Notes 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 . The geometry suggests that polar coordinates will be useful for solving Laplaces equation for the potential between the two planes. In this coordinate system, 2 2 2 1 1 0. r r r r r + = (1.1) To separate the variables, we assume that the potential is the product of a function of r and a function of , ( ) ( ) ( ) , . r R r = (1.2) On dividing by the potential, Laplaces equation leads to 2 2 1 . r d dR d r R dr dr d = - (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 2 , where is non-negative. The equations for R and are now 2 , d dR r r R dr dr = (1.4) and 2 2 2 , d d = - (1.5) which have solutions , 0, ln , 0, ar br R a b r - + = + = (1.6) and sin cos , 0, , 0, c d c d + = + = (1.7) where a , b , c and d are constants....
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- Fall '11