MRI is an non-invasive imaging technique used by many physicians to diagnose and treat diseases. The technique however is still very expensive and thus out of reach for developing countries. This has led to the goal to design a low-cost MRI system. The challenges that arise from this system make it necessary to design coils in a different way than conventional MRI. In this work the inverse boundary element method is used to create a coil design method for an arbitrary surface. This method is described and the mathematical framework is analyzed. A regularization method for the inverse problem has been designed in the form of a regularization matrix. This regularization matrix is constructed such that it can handle arbitrary surfaces. The regularization matrix is applied using Tikhonov regularization. To validate the design method a proof of concept radiofrequency coil for the low field MRI system at the LUMC has been realized. This coil is designed and has been used to image the human brain of an adult. The results from simulations beforehand are in agreement with the physically built coil showing that this method makes it possible to design and construct a physically feasible coil on an arbitrary surface.