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Correct Answer: τ = 1 s
the time constant, denoted as \(\tau\), of an rc (resistor-capacitor) circuit is a crucial parameter that determines how quickly the capacitor charges up to approximately 63.2% of the supply voltage or discharges to about 36.8% of its initial charge. the time constant \(\tau\) is calculated using the formula:
\[
\tau = r \times c
\]
where \( r \) is the resistance in ohms (\(\omega\)) and \( c \) is the capacitance in farads (f).
in the given problem, the resistance \( r \) is specified as 10 k\(\omega\) (or 10,000 \(\omega\)) and the capacitance \( c \) is 100 \(\mu\)f (or 100 x 10^{-6} f). plugging these values into the formula yields:
\[
\tau = 10,000 \, \omega \times 100 \times 10^{-6} \, f = 1 \, \text{second}
\]
thus, the time constant \(\tau\) of this rc circuit is 1 second. this means that it takes about 1 second for the capacitor in this circuit to charge up to about 63.2% of the source voltage when charging, or to discharge to about 36.8% of its initial voltage when discharging. this characteristic time constant helps in understanding the behavior of the circuit in response to changes in voltage, influencing how it performs in various applications such as filtering, timing, and coupling in electronic circuits.
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