Opt1'sol

# Opt1'sol - PHY 506 Classical Electrodynamics II Spring 2011...

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1 C:\User\Teaching\505-506\S11\Opt1'sol.doc PHY 506 Classical Electrodynamics II Spring 2011 Optional Problems Set 1 with solutions Problem O.1 . Modify the results of Homework Problem 2.1 (ii) for a superconductor microstrip line, taking into account the London depth of magnetic field penetration into both the strip and the ground plane. Solution : In this case, the expression for L has to be replaced with the result of PHY 505 Homework Problem 11.2:  L 0 2 d w L , while the capacitance C 0 per unit length remains the same as in Problem 2.1, because the electric field penetrates into conductors (including superconductors) by a much smaller length – see Sec. 2.1 of the lecture notes. As a result, the TEM wave propagation speed in such a line, 2 / 1 L 2 / 1 L 2 / 1 2 / 1 0 0 / 2 1 1 2 1 ) ( 1 d v d d C L v'  , is lower that that ( v ) of plane waves. 1 Note that in this case the “general” relation (7.109) is not satisfied, because it hinges on the similarity of boundary conditions for the magnetic and electric fields. Problem O.2 . For a metallic coaxial cable with the circular cross-section (Fig. 7.19 of the lecture notes), find the lowest non-TEM mode and calculate its cutoff frequency. Solution : The analysis repeats that of the hollow circular waveguide (Fig. 7.22a) up to the Bessel equation (7.136). However, in the coaxial cable, point = 0 is inaccessible for the field, and hence, instead of Eq. (7.137), we have to look for the radial part R ( ) in the form of a linear superposition of the Bessel functions of the fist and second kind:    ,... 2 , 1 , 0 , cos ) ( ) ( 0 2 1 n n k Y c k J c f nm n nm n nm , where eigenvalues k nm of the transversal wave number k t have to be chosen to satisfy boundary conditions at = a and = b , and give nonuniform longitudinal field, f nm const. For the E modes, the boundary condition Eq. (7.120) yields the following system of two equations: 0 ) ( ) ( , 0 ) ( ) ( 2 1 2 1 b k Y c b k J c a k Y c a k J c nm n nm n nm n nm n , These two linear, homogeneous equations for constants c 1,2 are compatible if ) ( ) ( ) ( ) ( a k Y b k J b k Y a k J nm n nm n nm n nm n , where integer index m = 1, 2, … numbers the roots of this characteristic equation (for each fixed angular quantum number n ). Introducing the dimensionless variable nm k nm a , we can rewrite the equation as 1 This difference is especially large ( v’ / v ~ 0.1) in the so-called long (or “ distributed ”) Josephson junctions with d ~ 1 nm and L ~ 100 nm, enabling very interesting effects of interactions between the slow TEM waves and magnetic-field-induced “waves” of tunneling supercurrent – see, e.g., Section 6.4 of the monograph by M. Tinkham, recommended in Sec. 6.3.

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2 C:\User\Teaching\505-506\S11\Opt1'sol.doc  nm n nm n nm n nm n Y a b J a b Y J . (*) Table on the right shows results for combination ( b / a – 1) nm = k nm ( b a ), obtained by numerical solution 2 of this equation for a few values of ratio b / a , and for a few lowest numbers n and m .
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## This note was uploaded on 09/10/2011 for the course PHY 506 taught by Professor Stephens,p during the Spring '08 term at SUNY Stony Brook.

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Opt1'sol - PHY 506 Classical Electrodynamics II Spring 2011...

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