HW02sol - Classical Electrodynamics I PHY 505 Fall 2010...

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1 C:\User\Teaching\505-506\F10\HW02sol.doc PHY 505 Classical Electrodynamics I Fall 2010 Homework 2 with solutions Problem 2.1 (to be graded of 10 points). Use the Gauss law to calculate the mutual capacitance of the following 2-electrode systems, with the same cross- section (see Fig. on the right): (i) a conducting sphere inside a concentric spherical cavity in another conductor, and (ii) a conducting cylinder inside a coaxial cavity in another conductor. (In this case, we speak about capacitance per unit length.) Compare the results with those obtained in class using the Laplace equation solution. Solutions: (i) Applying the Gauss Law to a sphere of radius r in the range a < r < b , for the electric field E = E(r) n r , we get , 4 ) ( 2 0 r Q r E  where Q is the charge of the inner conductor. Integrating this result, we find that voltage V ( b ) - ( a ) between the electrodes is , 1 1 4 4 ) ( 0 2 0  b a Q r dr Q dr r E V b a b a so that the mutual capacitance a b ab b a V Q C m 0 1 0 4 1 1 4 , in agreement with Eq. (2.56) of the lecture notes. (ii) An absolutely similar calculation, but applied to a cylinder, with charge Q / L per unit length, gives ) / ln( 2 , 2 ) ( 0 0 a b L C r r E m , in agreement with Eq. (2.49). Problem 2.2 (10 points). Following the class discussion of two weakly coupled conducting spheres, find an approximate expression for the mutual capacitance (per unit length) between two thin, parallel wires, each with a round cross-section, but its own diameter. Both diameters, D 1 and D 2 , are much smaller than the distance d between the wires. Compare the result with that for two spheres, and interpret the difference. a b a 0
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2 C:\User\Teaching\505-506\F10\HW02sol.doc Solution : This problem is close to that solved in class for two small spheres, but we should be more
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HW02sol - Classical Electrodynamics I PHY 505 Fall 2010...

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