Apply the variational method of Problem 8.9 to estimate the resonant frequency of the lowest TM mode in a “breadbox” cavity with perfectly conducting walls, of length d in the z direction, radius R for the curved quarter-circle “front” of the breadbox, and the “bottom” and “back” of the box defined by the plane segments (y = 0, 0 < x=””>< r)=”” and=”” (x=”0,” 0=””>< nf=””>< r),=”” respectively.=”” use=”” the=”” trial=”” function,=”” ez=”E0(p/R)v” (1=”” –=”” p/r)sin=”” 2n?=”” for=”” the=”” only=”” component=”” of=”” electric=”” field=”” present.=”” this=”” function=”” gives=”” vanishing=”” tangential=”” component=”” of=”” e=”” on=”” the=”” boundary=”” surfaces;=”” the=”” index=”” v=”” is=”” a=”” variational=”” parameter.=”” show=””>

Minimize with respect to v to find the best estimate of kR from the given trial function. Compare with the exact answer, kR = 5.13562, the first root ofJ2(x).