This is a very sound stupid question, but I'll go ahead anyway. Current is defined as the displacement of charge through a cross-section of a conductor per unit time. Okay. But how does relate to its math-definition, current = change of charge with respect to time, I=dQ/dt? I mean, for every charge leaving the cross-section of the conductor, an equivalent one will enter. This means that across the cross-section dQ/dt, the summed change of charge with respect to time, equals 0! I find I = Q/t, current = amount of charge passing the cross-section per unit time, a much more intuitive definition.. So can you guys help me make sense of this?
define a surface boundary. it doesn't have to be a closed surface, but it may be (and if it is, then continuity applies). electric current is defined to be the rate (w.r.t. time) that charge is crossing from one side of that surface boundary to the other. nothing else. if the surface is closed, then continuity requires that the negative of the rate of amount of charge contained inside the closed surface is equal to the current crossing that boundary from inside to out.
The water analogy for electricity isn't perfect but in effect you are saying.. "for every CC of water leaving the pipe an equivalent amount will enter" That's true. However electric current is equivalent to the flow rate of the water through the pipe not the change in volume of water in the pipe. I=dQ/dt is equivalent to measuring the instantaneous flow rate of the water. I=Q/t would be more akin to measuring the amount of water used over say a year and dividing acccordingly to give the average flow rate over that period.. or perhaps the total amout of water that has ever flowed through the pipe divided by the age of the pipe.
That word is where your problem lies. If you substitute the word 'flow' then your worry ceases to be.