Alan Horowitz wrote:

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%?

1-(1/e). Crack your textbook.

http://www.iop.org/EJ/S/UNREG/aS0aOQcupwxj4CNeiM5vaQ/article/-featured=jnl/0143-0807/23/1/304/ej2104.pdf
"Demonstration of the exponential decay law using beer froth"

Google
"exponential decay" 63 20,500 hits

Uncle Al gotta think of everything.

--
Uncle Al
http://www.mazepath.com/uncleal/
(Toxic URL! Unsafe for children and most mammals)
http://www.mazepath.com/uncleal/qz.pdf