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 z = E₀(p/R)^v (1 ? p/R)sin 2? 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 that

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 of J₂(x).

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(v + 2)(2v + 3)( + v + 4). v(2v + 1) kR? %3D