Final answer:
To determine the charge as a function of time in a decaying circuit, you must solve a calculus-based differential equation using the given resistance, capacitance, and exponentially decaying voltage function from the battery.
Step-by-step explanation:
To find the charge as a function of time in a series circuit consisting of a resistor, a capacitor, and a battery with a voltage that decays exponentially, we use the equation q(t) = Qe-t/τ where Q is the maximum charge the capacitor could hold if the battery were at a steady state value, and τ is the time constant (RC) of the circuit. For an exponentially decaying battery voltage function e = emf e-5t, the initial current and charge in the circuit are zero, since the capacitor starts uncharged. Integrating the current over time gives the charge on the capacitor as a function of time.
Given that the initial charge is 0 (q(0) = 0) and the battery voltage decays according to e = 200e-5t, to find the charge q(t), you must solve a differential equation that includes the battery's voltage function, the resistance R, and the capacitance C. The solution to this differential equation typically involves an integral which is unique to the time-dependent nature of the battery voltage. This question requires knowledge of calculus as applied to electric circuits.
Additionally, to find the initial charge Q that the capacitor would have with a steady battery, you can set the voltage across the capacitor equal to the battery's emf and solve for Q = C × emf.