DUE October 31
Cavity Problem 1 (6 points)
Consider a pillbox cavity designed for frequency \(\omega\) and particle velocity \(\beta\) with peak surface electric field of \(E_0\).
- Calculate the radius \(R\).
- What are the Electric fields on the beam axis?
- What’s the maximum voltage this cavity can achieve?
- What’s the maximum voltage a beam could get?
- Assume constant \(\beta\); also, symmetric integrals are your friend
- What \(L\) optimizes this beam voltage?
- What’s the ratio of Maximum Beam Voltage/Maximum Cavity Voltage?
Cavity Problem 2 (4 points)
Now consider an equivalent of a pillbox cavity made with rectangular waveguide with sides \(a\), \(b\), and \(c\).
- Draw the fields of the accelerating mode; it should look pretty similar to a pillbox cavity.
- What mathematical form do these fields take?
- What’s the frequency of this cavity? Hint: Don’t overcomplicate it.
Cavity Problem 3 (12 points)
For a half-wave resonator:
- Calculate the Geometry Factor, given \(f\), \(a\), and \(b\).
- Calculate \(R/Q\) in terms of the \(TTF\) (transit time factor).
- (BONUS) Show that \(\frac{\beta\lambda}{2} = (b+a)\).
- Why does this make sense?
- Not trivial! Integral of \(sin(x)/x\) is not closed; either be more clever or demonstrate numerically?
- Assuming
c
above is proven, how would you go about optimizing \(b-a\)?