How many time constants does it take for the voltage across the capacitor to reach the value of the source voltage?

The voltage across a capacitor in an RC (resistor-capacitor) circuit approaches the source voltage asymptotically as time progresses. The concept of time constants is crucial in understanding this behavior. One time constant, which is defined as the time it takes for the voltage to rise to about 63.2% of the source voltage, occurs when the circuit is switched on.

After one time constant, the voltage is roughly 63.2% of the maximum value. After two time constants, the voltage reaches approximately 86.5% of the source voltage. By the time three time constants have passed, the voltage across the capacitor approaches about 95% of the source voltage.

Continuing this pattern, after five time constants the voltage will have reached about 99.3% of the source voltage. This is generally considered to be sufficiently close to the source voltage for practical purposes, as the voltage is very near to the source voltage at this point. Therefore, after five time constants, the voltage across the capacitor is effectively seen as having reached the supply voltage.

Recognizing this behavior is important in the timing and functional aspects of electrical systems involving capacitors. Thus, five time constants is the answer that best captures how long