What percent of full charge will a capacitor charge to in one RC time?

In a charging circuit involving a capacitor, the term "RC time" refers to the time constant, which is calculated as the product of resistance (R) and capacitance (C). This time constant (τ) represents the time it takes for the voltage across the capacitor to rise to approximately 63.2% of its maximum value (the full charge) during the charging process.

The mathematical relationship governing the charging of a capacitor can be expressed with the equation:

[ V(t) = V_{max} \left(1 - e^{-\frac{t}{RC}}\right) ]

Here, ( V(t) ) is the voltage across the capacitor at time ( t ), ( V_{max} ) is the maximum voltage the capacitor will eventually reach, and ( e ) is the base of the natural logarithm.

At one time constant (t = τ), substituting τ into this equation yields:

[ V(τ) = V_{max} \left(1 - e^{-1}\right) \approx V_{max} (1 - 0.3679) \approx V_{max} (0.6321) ]

This shows that the capacitor charges to about 63.2