We present and discuss the properties of the coherent electromagnetic fields of a very short, ultra-relativistic bunch in a rectangular vacuum chamber inside a bending magnet. The analysis is based on the results of a direct numerical solution of Maxwell's equations together with Newton's equations. We use a new dispersionfree time-domain algorithm which employs a more efficient use of finite element mesh techniques and hence produces self-consistent and stable solutions for very short bunches. We investigate the fine structure of the CSR fields. We also discuss coherent edge radiation. We present a clear picture of the field using the electric field lines constructed from the numerical solutions. This method should be useful in the study of existing and future concepts of particle accelerators and ultrafast coherent light sources, where high peak currents and very short bunches are envisioned. Show Partial Abstract