Abstract
We determine the impedance of a cylindrical metal tube (resistor) of radius $a$, length $g$, and conductivity $\ensuremath{\sigma}$ attached at each end to perfect conductors of semi-infinite length. Our main interest is in the asymptotic behavior of the impedance at high frequency $(k\ensuremath{\gg}1/a)$. In the equilibrium regime, $k{a}^{2}\ensuremath{\ll}g$, the impedance per unit length is accurately described by the well-known result for an infinite length tube with conductivity $\ensuremath{\sigma}$. In the transient regime, $k{a}^{2}\ensuremath{\gg}g$, where the contribution of transition radiation arising from the discontinuity in conductivity is important, we derive an analytic expression for the impedance and compute the short-range wakefield. The analytic results are shown to agree with numerical evaluation of the impedance.
Highlights
We consider the longitudinal impedance of a cylindrical metal tube of radius a, length g, and conductivity attached at each end to perfect conductors of semi-infinite length (Fig. 1)
There are two regimes: (i) When the Rayleigh range ka2 corresponding to the tube radius is short compared to the resistor length g, the field pattern settles into an equilibrium in which the field is continually being eaten at the resistor while it is being replenished on axis by the deceleration of the beam
The impedance per unit length is well approximated by that of an infinite length tube with conductivity [2 –5]. (ii) When the Rayleigh range ka2 is short compared to g, equilibrium is not reached and the impedance per unit length differs from that of an infinite tube
Summary
We consider the longitudinal impedance of a cylindrical metal tube (resistor) of radius a, length g, and conductivity attached at each end to perfect conductors of semi-infinite length (Fig. 1). Expanding the field in a Fourier series in the axial coordinate z, the integral equation is rewritten as an infinite set of linear algebraic equations for the Fourier coefficients Truncating these equations by keeping only a limited number of Fourier coefficients, the equations are solved numerically. III we consider the impedance in the high frequency limit, ka 1, and derive a simpler integral equation for the electric field in the resistor which holds when ka g We solve this equation analytically, obtaining the impedance in the transient regime.
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