The work considers the inverse source problem for the Laplacian in an insulated open upper half-plane \(\mathbb R^2_+ \subset\mathbb R^2\). To be more precise, the aim is to extract information on the mean-free and compactly supported source in the Poisson equation \( \Delta u =F\) in \(\mathbb R^2_+\), \( \frac{\partial u}{\partial x_2} =0\) on \(\Gamma\), \(\lim_{|x| \to \infty} |u(x)|=0\), from the value of the potential \(u\) on the boundary \(\Gamma= \partial\mathbb R^2_+\). The convex source support is, in essence, the smallest nonempty convex set that supports a source that produces the measured nontrivial data on the horizontal axis. In particular, it belongs to the convex hull of the support of any source that a compatible with the measurements. A previously introduced method for reconstructing the convex source support in bounded domains is modified to the case of the unbounded setting. The resulting numerical algorithm ia analysed both for the inverse source problem and for electrical impedance tomography with single part of boundary current and potential as the measurement data.