Some problems in electro dynamics have been studied in the light of electromagnetic field theory and computational techniques. One is that of the retrograde motion of the electric arc – an old and intriguing problem of physics. A more complete model based on electrodynamics has been proposed for study of this problem. In this model, the plasma column is treated as a flexible conductor. The associated 3-dimensional boundary-value problem was solved by the boundary element method (BEM). It has been shown that the retrograde motion can indeed be understood in terms of electrodynamics and a number of the reported features of this phenomenon have been explained by the proposed model. The others are concerned with a recent controversy in electrodynamics. It has been discussed in the thesis that Lorentz and Ampere force laws are equivalent in the sense of magnetostatics provided that the latter is appropriately used. The interactions of electromagnetic forces on the systems of the electromagnetic jet propulsion and the electromagnetic impulse pendulum have been examined. It turns out that the observations are consistent with the laws of electromagnetic energy and momentum. Two aspects of the water-arc explosion problem have been studied. One is the electromagnetic (EM) aspect which is calculated by the method of separation of the variables, the other is the magnetohydrodyuamic (MHD) aspect which is solved by the finite difference method (FDM). The numerical results of the two aspects show that neither the pure EM model nor the MHD model for the incompressible fluid could explain the explosive phenomenon; this strongly suggests that the shock wave produced by the under-water arc must account for the phenomenon. On the wire fragmentation, a preliminary study has shown that this phenomenon could be caused by the electromagnetically driven stress waves and a theoretical model has been proposed for further investigation. This is described in an Appendix to the thesis.