# Focusing a beam of ions

A source of ions at point O produces a slightly diverging beam with half-angle of divergence $$\alpha_{0}\ll 1.$$ In order to collimate (re-focus) the beam, one can apply a uniform magnetic field along the z-axis (see the figure below). If all the ions have the same initial velocity and specific charge $$\frac{q}{m}$$, the beam focuses at point A, again at point 2A and so on. Assuming that $$\frac{q}{m}= 10^{5}~\mbox{C/kg}$$, $$B=10^{-3}~\mbox{T}$$, and $$v=100~\mbox{m/s}$$, find the distance $$\overline{\mbox{OA}}$$ in meters.

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