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%?

Let's try another way. You can actually experiment yourself.

Take a 1000uF electrolytic cap and charge it up to +5V, then disconnect the
power supply. Put your voltmeter on the cap terminals and read +5V. Now take
a 10k resistor and put it across the terminals as well. The cap discharges
slowly. In the first moment the discharge current was 5V/10k = 0.5mA. With
this current the cap would be discharged in 10s, this is the time constant
"tau" = RC

But since the voltage is dropping also the discharge current drops. Now you
can use a stopwatch and read the voltage after 10s and you find it to be
1.84V, which is (1-0.624)5V. So this is where your 63% come from. Since we
are discharging, the value is 37% of the initial voltage. You can also note
down the values for 20s, 30s etc. until your meter has no more resolution
and find the corresponding values for multiple time constants.
BTW you do not need to do this experiment yourself but use a simulator or
solve the equations others have already written in their answers.

--
ciao Ban
Bordighera, Italy