HW04sol - Classical Electrodynamics I PHY 505 Fall 2010...

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1 C:\User\Teaching\505-506\F10\HW04sol.doc PHY 505 Classical Electrodynamics I Fall 2010 Homework 4 with solutions Problem 4.1 (to be graded of 20 points). Complete the cylinder problem started in class (see Fig. on the right), for the cases when voltage on the top lid equals: (i) V = V 0 J 1 ( 11 / R )sin , where 11 3.832 is the first root of function J 1 ( x ), and (ii) V = V 0 = const. For both cases, calculate the electric field in the centers of the lower and upper lids. Hint: For assignment (ii), an answer including series and/or integrals is satisfactory. Solution: In class, we have found the general solution to this problem:  R z n s n c R J z nm nm nm nm nm n sinh sin cos ) ( ) , , ( 01  , (*) where nm is the m- th root of the Bessel function J n ( x ). This solution already satisfies the boundary conditions on the sidewall and the bottom lid of the cylinder; hence coefficients c n and s n have to be found from the boundary condition on the top lid: ). , ( ) , , ( V L ( * * ) In assignment (i), this function has only one azimuthal harmonic proportional to sin , and only one radial harmonic, proportional to J 1 ( 11 / R ), so that only one of the coefficients c nm , s nm is not vanishing: 0 11 11 sinh V R L s , and Eq. (*) is reduced to a very simple analytical solution
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This note was uploaded on 09/10/2011 for the course PHY 505 taught by Professor Stephens,p during the Fall '08 term at SUNY Stony Brook.

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HW04sol - Classical Electrodynamics I PHY 505 Fall 2010...

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