Question #43238

A beam of electrons (rest mass $m_0$, charge $-e$) is accelerated from rest through a potential difference $V = \frac{m_0 c^2}{e}$, where $c$ is the speed of light. The beam then passes through an electron-optical system consisting of two identical thin magnetic lenses separated by a distance $d$. In electron optics, the optical power $P$ (reciprocal of focal length) of a thin magnetic lens is given by $P = \int_{-\infty}^{\infty} \left(\frac{\omega_c}{2v}\right)^2 dz$, where $v$ is the speed of the electron and $\omega_c$ is the local cyclotron frequency of the electron in the axial magnetic field $B_z(z)$. For each lens, the integral of the squared magnetic field is $\int_{-\infty}^{\infty} B_z^2 dz = \beta$. If the equivalent optical power of the two-lens system is zero (an afocal system), what is the required separation $d$?