  It is well known that the form of grain size distribution strongly influences
the linear dielectric response of a dusty plasma. In previous results [IEEE
Trans. Plasma Sci. 29, 182 (2001)], it was shown that for a class of size
distributions, there is an equivalence to a Lorentzian distribution of
mono-sized particles. The electrostatic response to a slowly moving test
charge, using a second order approximation can then be found [Phys. Lett. A
305, 79 (2002)]. It is also well known [Phys. Plasmas 10, 3484 (2003)] that the
dynamical charging of grains in a dusty plasma enhances the shielding of a test
charge. It seems natural at this stage to seek the combined effects of grain
size distribution and grain charging dynamics to a test charge moving through
the dusty plasma. Here we consider the effects of both grain size distribution
and dynamical grain charging to a test charge moving slowly in a dusty plasma
by expressing the plasma dielectric response as a function of both grain size
distribution and grain charging dynamics. Both analytical as well as the
numerical results are presented. It is interesting to note that the previous
results can be retrieved by choosing appropriate values for different
parameters. This kind of study is relevant for both laboratory and space
plasmas.
