 Solution

Substituting the electron Lorentz factor in the supplied energy-loss formula givesWithout replenishment, the electron's momentum decreases. The reference dipoles then bend it more tightly, producing a displaced orbit. In linear beam optics, $x=D_x\delta$ and $y=D_y\delta$, where $\delta =\Delta p/p\simeq -U_0/E\simeq -1.11\times 10^{-3}$ per uncompensated turn. In an ideal planar ring the horizontal dispersion is nonzero and the orbit shifts horizontally, inward where $D_x>0$. There is no corresponding vertical closed-orbit shift when $D_y=0$. Real coupling or vertical dispersion can change that idealization.

With current $I=0.500$ A, the number of circulating particles is $N=I{T}_{\mathrm{rev}}/e$, so the radiated power isThe $2\mu \mathrm{s}$ revolution time cancels when using beam current. Radio-frequency cavities restore the lost energy longitudinally, with synchronous gain $e{V}_{\mathrm{RF}}\sin {\phi }_s=U_0$ in the sine-phase convention. Their RF power system must supply at least this beam power, in addition to its own losses. Radiation damping and quantum excitation determine beam emittances; they should not be confused with the dispersive closed-orbit shift from a momentum change.