Mid1sol - PHY 505 Classical Electrodynamics I Fall 2010...

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PHY 505 Classical Electrodynamics I Fall 2010 Midterm Exam 1 with solutions Problem M.1 (To be graded of 150 points). Calculate the spatial distributions of and E, created by a long, round cylinder of radius R , with the electric charge uniformly spread over its volume. Compare the result with that of for the uniformly charged sphere. Can you calculate the electrostatic energy of the system (per unit length)? If not, can you estimate the total energy of such cylinder of a finite length L >> R ? Solution : Due to the axial symmetry of the problem, we may write ( r ) = ( ), E ( r ) = n E ( ), where is the distance from the cylinder’s axis. Applying the Gauss law to a long cylinder of radius , coaxial with the charged cylinder, we get , for , / 1 , for , / 2 ) ( 2 0 R R R E  (*) where is the electric charge per unit length. Now integrating the relation E ( ) = - d ( )/ d which follows, for our symmetry, from the general expression for gradient in cylindrical coordinates, 1 we get  . for , / ln , for , 2 / 2 ) ( 2 1 2 2 0 R c R R c R Since the electrostatic potential has to be continuous at = R , the integration constants have to be related as 2 1 1 2 c c . Comparison of Eq. (*) with Eq. (1.19) and (1.22) of the lecture notes shows that while the field inside the cylinder changes similarly to that inside the uniformly charged sphere, outside the cylinder it changes much slower – as l/ rather than 1/ r 2 . Due the this change, integral (1.67) for the electrostatic
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Mid1sol - PHY 505 Classical Electrodynamics I Fall 2010...

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