Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2016 / S19 / 1

Let $x_i=L/f_i$, with positive $f$ focusing in the first transverse plane. The doublet matrix is $M=Q_{2}D(L)Q_1$, with $Q_i=\left(\begin{array}{cc}1& 0\\ -1/f_i& 1\end{array}\right)$. Its lower-left entry is $C_x=-(x_1+x_2-x_1{x}_2)/L$. In the other plane both quadrupole signs reverse, giving $C_y=-(-x_1-x_2-x_1{x}_2)/L$. Thus positive effective focusing power in both planes requiresFor a parallel incident ray bundle, actual downstream focal distances are ${\ell }_x=L(1-x_1)/(x_1+x_2-x_1{x}_2)$ and ${\ell }_y=L(1+x_1)/(-x_1-x_2-x_1{x}_2)$. Requiring both foci downstream adds $|x_1|<1$. Positive effective power alone need not imply a real image after the final element. The original schematic below distinguishes these conditions.

The beam rigidity is $B\rho \simeq p/(0.2998\mathrm{GeV}/(c\mathrm{T}\mathrm{m}))\simeq 1.67\times 10^5$ T m. Equal opposing strengths have $f=B\rho /(G{L}_q)$, and weak separated-doublet net power is $1/F\simeq L/f^2$. Taking $F\sim L^{\ast }=50$ m gives the design familyThere is not a unique pair $(L_q,L)$ without a spacing/layout choice. A compact estimate $L\sim L_q$ gives about 18 m for each, showing immediately that neither is extremely small compared with 50 m and the advertised asymptotic approximation is only rough. For example $L=30$ m gives $L_q\sim 14$ m in the same estimate.

An equal-strength doublet also has exact parallel-input focal distances $f^2/L-f$ and $f^2/L+f$: it is astigmatic, so a single exact common 50 m focus cannot be imposed without specifying incoming optical functions or allowing unequal strengths. The family above is a scale estimate of net focusing, not a complete collider final-focus design.

A uniform nonrelativistic electron column has radial electric field $E_r=-n_{e}er/(2{ϵ}_0)$ inside its radius. A relativistic proton therefore has focusing strength $K=e^2{n}_e/(2{ϵ}_{0}pv)$ per unit length. Electron-current magnetic effects are small under the stated nonrelativistic assumption. To change the nominal doublet strength $1/L^{\ast }$ by $10^{-4}$ requiresOne illustrative choice is a one-metre lens with $n_e\simeq 1.1\times 10^{16}$ m${}^{-3}$; other density-length combinations work. The electron beam must cover the proton beam and have a sufficiently uniform profile. Its placement changes the detailed transfer-map correction; the estimate treats the required additional integrated focusing at the entrance and does not claim a uniquely specified electron-lens engineering design.