Figure 3: Spatiotemporal evolution of 
and ratio between the magnitude of typical Coulomb and gravitational forces, Rcg.
and ratio between the magnitude of typical Coulomb and gravitational forces, Rcg.From: Self-charging of identical grains in the absence of an external field
This ratio is given in cgs units12 by 
, where 
is the mean of the distance to each grain’s nearest neighbor, 
is the mean grain mass, and g is gravity. (a) Spatiotemporal evolution of bed charges using χe = 0.6 and η = 1. Inset shows the same plot over longer time, (to 14 sec.). (b) Time evolution of 
and Rcg (blue: averaged over the entire bed, green: averaged over the layers 14–16), using χe = 0.6 and η = 1. Note that although Rcg is only 20% when averaged over the entire bed, in the fastest charging region, around height = 15, Rcg = 1. (c) Spatiotemporal evolution of 
using χe = η = 1: again inset shows same plot over longer time, (to 14 sec.), highlighting periodic oscillations in charge. Notice that for χe = η = 1, the entire bed becomes highly charged. (d) Here, Rcg averaged over the entire bed exceeds one, at which point the charge starts to plummet. Data in (a) and (c) obtained by dividing the bed into one-mean-grain-diameter slices and calculating the sum of the absolute values of charges in each slice.
