"Alan Horowitz" wrote in message
om...
when a current just starts flowing into a RL or RC circuit, how does
the voltage "know" that it should be increasing exactly 63% during
each time-constant period?

And whence the number 63%?

The voltage knows nothing about how it's "supposed" to behave. It just does
its thing without a care in the world.
The thing it does though, will always result in exactly the same voltage
shape, because with a fixed R and C and supply voltage it can do no other.
As the C voltage grows, the voltage across the R must drop. If the R voltage
drops then the charging current must drop. If the charging current drops,
then the C voltage must rise at a slower rate, ... and so on and so on ...
Everything slows down more and more as time goes on. A bit of thought and
you'll notice that the C can never actually charge exactly to the supply
voltage.
As this RL RC (dis)charging process must always result in this particular
shape or curve and this quite 'natural' curve turns up across all branches
of science, engineering and finance, it wasn't long before the
mathematicians found they could usefully model, or describe the curve
accurately, using an equation based on the 2.718 "e" value used for working
out 'natural' logarithms.
Hence the maths numbers and formulae that are taught are a good descriptive
model or analogue of what's happening in the circuit but have nothing to do
with the actual circuit workings.
Be wary when relying purely on maths models. They confer 'expertise' into
how something works, without offering 'understanding' of how something
works. The difference can be crucial.
regards
john