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Definition of current

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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?

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