Transverse electromagnetic wave (TEM mode) with an angular frequency ω and a power P is propagating in the z-direction along a coaxial transmission line (P is the time-averaged power flow, to be rigorous). The coaxial line is consisting of two concentric circular cylinders of copper. See Fig. 2 a) and b) for the definitions of an inner radius a and an outer radius b. In between the two cylinders, the line is filled with a uniform lossless dielectric (μ, ε). See the definitions of a radius r and an azimuthal angle ϕ at the figures.

question 1

Sketch both the electric and magnetic field lines in the Fig. 2 c) and d) at t = 0.

question 2

Sketch both the electric and magnetic field lines in the Fig. 2 e) and f) at ωt = 90º.

question 3

This cavity can be used for particle acceleration, if one machines apertures along y axis as shown in Fig. 2 g) Consider particles moving very close to the speed of light, and suppose all the particles are in a bunch of delta function (that is, a bunch length is 0). Find the maximum acceleration energy as a function of E0, keeping it in mind that the particles are accelerated twice at y < 0 side and y >0 side, respectively, as shown in Fig. 2 h). Here, assume the value of a smaller than λ/8, and that of b fixed to λ, where λ is a wavelength.