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Time-domain solutions of Maxwell's equations in homogeneous and isotropic media are paramount to studying transient or broadband phenomena. However, analytical solutions are generally unavailable for practical applications, while numerical solutions are computationally intensive and require significant memory. Semi-analytical solutions (e.g., series expansion), such as those provided by the current theoretical framework of the multipole expansion, can be discouraging for practical case studies. This paper shows how sophisticated mathematical tools standard in modern physics can be leveraged to find semi-analytical solutions for arbitrary localized time-varying current distributions thanks to the novel time-domain Cartesian multipole expansion. We present the theory, apply it to a concrete application involving the imaging of an intricate current distribution, verify our results with an existing analytical approach, and compare the proposed method to a finite-difference time-domain numerical simulation. Thanks to the concept of current "pixels" introduced in this paper, we derive time-domain semi-analytical solutions of Maxwell's equations for arbitrary planar geometries. [ABSTRACT FROM AUTHOR]