Each time constant of an exponential curve is equal to what percentage of the maximum amount of charge?
The correct answer and explanation is :
Each time constant ((\tau)) of an exponential charging curve represents approximately 63.2% of the maximum charge.
Explanation:
In an RC circuit (resistor-capacitor circuit), when a capacitor charges through a resistor, the voltage and charge follow an exponential function:
[
Q(t) = Q_{\text{max}} \left(1 – e^{-t/\tau} \right)
]
where:
- ( Q(t) ) is the charge on the capacitor at time ( t ),
- ( Q_{\text{max}} ) is the maximum charge the capacitor can hold,
- ( \tau = RC ) is the time constant of the circuit,
- ( e ) is the mathematical constant ((\approx 2.718)).
At ( t = \tau ), we substitute into the equation:
[
Q(\tau) = Q_{\text{max}} \left(1 – e^{-1} \right)
]
Since ( e^{-1} \approx 0.3679 ), we get:
[
Q(\tau) = Q_{\text{max}} \times (1 – 0.3679) = Q_{\text{max}} \times 0.632
]
Thus, after one time constant, the capacitor has reached 63.2% of its maximum charge.
Interpretation:
- After 1 time constant ((\tau)): 63.2% of ( Q_{\text{max}} )
- After 2 time constants ((2\tau)): 86.5% of ( Q_{\text{max}} )
- After 3 time constants ((3\tau)): 95.0% of ( Q_{\text{max}} )
- After 5 time constants ((5\tau)): 99.3% of ( Q_{\text{max}} ) (essentially fully charged)
This concept is crucial in electronics for timing circuits, filters, and transient analysis. The same principle applies to an RL circuit where current follows a similar exponential pattern.
In summary, one time constant ((\tau)) corresponds to 63.2% of the maximum charge in an exponential charging process.