= Solution
A uniform nonrelativistic <electron> column has radial <electric field> $E_r=-n_ee r/(2\epsilon_0)$ inside its <radius>. A relativistic <proton> therefore has focusing strength $K=e^2n_e/(2\epsilon_0pv)$ per unit length. Electron-current magnetic effects are small under the stated nonrelativistic assumption. To change the nominal doublet strength $1/L^*$ by $10^{-4}$ requires
$$
\boxed{n_eL_e=\frac{2\epsilon_0pv}{e^2}\frac{10^{-4}}{L^*}\simeq1.1\times10^{16}\ \mathrm{m^{-2}}.}
$$
One illustrative choice is a one-metre lens with $n_e\simeq1.1\times10^{16}$ 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.
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