Consider a piece of conductor of cross sectional area #A#.
Now, if #n# be the concentration of charge carriers (electrons or Cooper pairs or electron-holes or whatever it may be). Then, in #dt# time, the number of charge carriers that pass through the cross section is,
#dN = nv_dAdt# where #v_d# is the drift speed.
This holds because #v_d*dt# is length of conductor whose free charges pass through #A# in time #dt# and hence the associated volume is #dV = v_ddt*A# whose charges constitute the current in #dt# time through area #A#.
Then charge passing through the area is,
#dQ = nqv_dAdt# where #q# is charge on each charge carrier.
Therefore the current,
#I = (dQ)/(dt) = nqv_dA#
