Voltage Rise in a Resistor-Capacitor Circuit with an Applied DC Voltage

Consider the circuit below of a cell, switch, resistor and capacitor:

Let's assume that the circuit initially starts with the switch open, and that at a certain time t0 the switch is abruptly closed.

What potential difference will initially be present across the capacitor terminals? What will the potential difference be across the capacitor terminals after a long time has passed?

At time t0 when the switch is first closed we can assume that the capacitor has no charge, and so the potential difference across the capacitor terminals will be 0 Volts. The switch is assumed to be "perfect", in the sense that it can instantly close. The initial current will be (E/R) Amps at t0 but will fall as time progresses. In other words, the current is a maximum at t0, and falls from that point on. For the very briefest of moments (infinitessimally small), the current is (E/R).

MathML formulaFormula ImageStraight text formula
$v t = E 1 - e - 1 RC t$ v(t) = E(1-e^(-(1/(RC))t))

Where

v(t) = potential difference across capacitor at time t seconds after switch has been closed, Volts

E = Cell Voltage = Voltage that would be present across the capacitor terminals if the resistor was not present and the switch was closed, Volts