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 \(E_0\).

  1. Calculate the radius \(R\).
  2. What are the Electric fields on the beam axis?
  3. What’s the maximum voltage this cavity can achieve?
  4. What’s the maximum voltage a beam could get?
    • Assume constant \(\beta\); also, symmetric integrals are your friend
  5. What \(L\) optimizes this beam voltage?
  6. 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 \(c\).

  1. Draw the fields of the accelerating mode; it should look pretty similar to a pillbox cavity.
  2. What mathematical form do these fields take?
  3. What’s the frequency of this cavity? Hint: Don’t overcomplicate it.

Cavity Problem 3 (12 points)

For a half-wave resonator:

  1. Calculate the Geometry Factor, given \(f\), \(a\), and \(b\).
  2. Calculate \(R/Q\) in terms of the \(TTF\) (transit time factor).
  3. (BONUS) Show that \(\frac{\beta\lambda}{2} = (b+a)\).
    1. Why does this make sense?
    2. Not trivial! Integral of \(sin(x)/x\) is not closed; either be more clever or demonstrate numerically?
  4. Assuming c above is proven, how would you go about optimizing \(b-a\)?