What percentage of source voltage is observed in a capacitor after one time constant?

The phenomenon described pertains to the charging process of a capacitor in an RC (resistor-capacitor) circuit, which is fundamental in understanding how capacitors work in electronic circuits. After one time constant, which is defined as the time it takes for the voltage across the capacitor to reach approximately 63.2% of the final steady-state voltage, the capacitor has a specific voltage characteristic.

The time constant (( \tau )) is calculated as the product of resistance (R) and capacitance (C), represented as ( \tau = R \times C ). When the circuit is powered, the voltage across the capacitor begins at zero and increases as it charges. The mathematical description of the charging voltage (V) at any time (t) is given by:

[ V(t) = V_s (1 - e^{-\frac{t}{\tau}}) ]

Where ( V_s ) is the source voltage and ( e ) is the base of the natural logarithm. When ( t = \tau ):

[ V(\tau) = V_s (1 - e^{-1}) ]

Since ( 1 - e^{-1} ) approximately equals 0.632 or 63.2%,