We analyze the use of non-fitting meshes for simulating the propagation of electromagnetic waves inside the Earth. In order to take into account subcell conductivity variations, we propose a simple " exact integration technique " that avoids the use of parameter homogenization and employs standard edge finite element basis functions. For our geophysical applications, we consider a 3D Maxwell's system with piecewise constant conductivity and globally constant permittivity and permeability. The model is analysed and discretized using both the E and H-formulations. Our main contribution is to develop a sharp error estimate for both the electric and magnetic fields. In the presence of singularities, our estimate shows that the magnetic field approximation is converging faster than the electric field approximation. As a result, we conclude that error estimates available in the literature for Nédélec's elements with fitting meshes are sharp with respect to the electric field error, but provide pessimistic convergence rates for the magnetic field in our geophysical applications. Another surprising consequence of our analysis is that non-fitting meshes deliver the same convergence rate than fitting meshes to approximate the magnetic field. Our theoretical results are numerically illustrated via 2D experiments. For the analyzed cases, the accuracy loss due to the use of non-fitting meshes is limited, even for high conductivity contrasts.