Explicit Relation Between Volume and Lower Bound for Q for Small Dipole Topologies

A rigorous relation is derived between any local change of the volume of an electrically small radiating device and the lower bound of its radiation Q factor. The relation concerns the actual volume and not the circumscribing volume. This means that also (incremental) changes in volume can be studied where the circumscribing volume remains the same. The relation clearly proves that any arbitrary increase in volume decreases the Q, and any arbitrary reduction in volume increases the Q. When directly applied to volumes embedded within a sphere, it is almost trivial to rigorously prove the well-known fact that the full sphere provides the absolute minimal Q. The communication ends with a simple analytical proof of the limit as introduced by Thal for dipole type spherical TM modes. To the knowledge of the author, these explicit relations between Q factor and occupied volume have not been described in literature yet.