Final answer:
The expression for the charge on a capacitor as a function of time, q(t), depends on whether it's an RC or AC circuit. For RC, it's q(t) = Qe^-t/T, while for AC, it's q(t) = CΥ0 sin(ωt). Current as a function of time in an AC circuit can be obtained by differentiating q(t) giving i_c(t) = Υ0Cω cos(ωt).
Step-by-step explanation:
To determine the expression for the charge on the capacitor as a function of time, q(t), we consider the different ways capacitors respond to circuits. For a discharging capacitor in an RC circuit, the charge decreases exponentially with time. The general expression for q(t) in this case is q(t) = Qe-t/T, where Q is the initial charge and T is the time constant, defined as T = RC. For an alternating current (AC) circuit, the charge varies sinusoidally as a function of time, given by q(t) = C Υ0 sin(ωt), with C representing the capacitance, Υ0 the peak voltage, ω the angular frequency, and t the time.
The time constant gives us an important point of reference in that when t = T, the charge or voltage has reached a certain fraction of its initial value. When charging, if we start at t = 0, this expression helps to describe the time course of the voltage across the capacitor reaching towards its maximum value, reflected by Q = CV, or q(t) = CΥ0(1 - e-t/T)
To find current as a function of time, we differentiate the charge with respect to time. In an AC circuit, the current can be found using ic(t) = Υ0Cω cos(ωt), where Υ0Cω is the amplitude of the current and cos(ωt) is the instantaneous value of the current relative to time.