Final answer:
In a simple RC circuit, the voltage across a capacitor increases during charging according to V(t) = Vo(1 - e^-t/RC) and decreases during discharging as V(t) = Voe^-t/RC. The time constant τ (RC) plays a critical role in determining the rate at which the voltage approaches its final value.
Step-by-step explanation:
To calculate the voltage across the capacitor at time t in a simple RC circuit, you need to use the appropriate formula for either charging or discharging scenarios. During the charging process, the voltage across the capacitor increases according to the formula:
V(t) = Vo(1 - e-t/RC) (charging)
where Vo is the initial voltage (i.e., the electromotive force or emf), R is the resistance, C is the capacitance, and t is the time elapsed since the charging began. The function represents an exponential approach to the maximum voltage. By time t = RC, which is known as the time constant (τ or tau), the voltage will have reached about 63.2% of Vo.
On the other hand, if the capacitor is being discharged, the voltage decreases exponentially as follows:
V(t) = Voe-t/RC (discharging)
In this scenario, Vo represents the initial voltage across the capacitor at t = 0, and as time passes, the voltage falls towards zero. After each time constant τ, the voltage decreases to 36.8% of its previous value. Therefore, for a simple RC circuit consisting of one resistor and one capacitor, the answer is option b) V(t) = Vo(1 - e-t/RC) for charging and option a) V(t) = Voe-t/RC for discharging.