Geometric efficiency for a circular detector and a linear source of arbitrary orientation and position

A new axisymmetric radiation vector potential which is singular along its entire axis of symmetry is derived for a spherically symmetric point radiation source. This potential and a previously given non-singular point source potential are integrated to give radiation vector potentials for a straight linear source of constant strength. Analytical solutions are given for the geometric efficiency G of a line source and a circular disk detector when the line source is parallel to the detector axis. The analytical solution is also given for the case where the line source is parallel to the disk surface, such that the source axis and the detector axis intersect. All other cases are given as simple one-dimensional trigonometric integrals. Numerical results for G are given for all cases considered, and results given previously by Pommé for a line source parallel to the detector plane have been verified. The methods and vector potentials presented here can be adapted for calculations with many different geometries, and many results are applicable in other fields such as electromagnetism, gravity and fluid mechanics.