HW10sol - 1 C:\User\Teaching\505-506\F10\HW10sol.doc PHY...

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Unformatted text preview: 1 C:\User\Teaching\505-506\F10\HW10sol.doc PHY 505 Classical Electrodynamics I Fall 2010 Homework 10 with solution Problem 10.1 (to be graded of 20 points). AC current of frequency is being flown through a long conductor with a round cross-section of radius R which is generally comparable with the skin depth ( ). In the quasistationary approximation, find the current density distribution across the wire. Analyze the limits R << ( ) and R >> ( ) . Solution : The solution may be found directly from Eq. (6.14) of the lecture notes for vector B, which for the symmetry of this problem (see Fig. on the right) may be presented as n B ( , t ). However, the expression for the Laplace operator of such a vector in polar coordinates is relatively complex. 1 Instead, let us stop at half-way of the derivation of that equation - see the second form of Eq. (6.13), j B 1 t , and take curl of both sides. According to the Maxwell equation j = H = ( B / ), the left-hand side equals to j / t , while the right yields 2 j / (since in the quasistationary limit, j = 0). Hence the current density obeys the differential equation similar to that = 0)....
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HW10sol - 1 C:\User\Teaching\505-506\F10\HW10sol.doc PHY...

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