Charging a capacitor
When a current flows into a capacitor it does not charge instantly. If the capacitor is large and the resistance in the circuit is high so that the current is small, then it will take a significant time for the capacitor to become fully charged. The voltage (the potential difference) of the supply pushes the charge onto the capacitor, but as the charge builds up it repels further charge flowing in, so the current falls as the potential difference (voltage) rises. The change of current with time follows an exponential curve:
The current falls as the capacitor approaches a full charge but it may take a long time for the current to fall close to zero, that depends on the circuit.
When a capacitor is discharged it follows the same pattern but in reverse. As the capacitor begins to discharge the current is high and the rate of change of PD is large. The graph curves illustrate the fact that the changes are exponential.
There is a video on charge/discharge of a capacitor (second half of the video) below which you might find useful. The experimental set-up is like this:
The value of the resistance in the circuit is high, so when the switch is closed, the potential difference across the capacitor and the resistor is equal to the EMF of the battery. When the switch is released the capacitor will begin to discharge through the resistor.
The rate of discharge of the capacitor dQ/dt (which is the current) depends upon Q. The quantity of charge upon it, which in turn depends on the capacitance C
The constant of proportionality depends upon the value R , the resistance in the circuit. The product of C and R is called the time constant.
C x R = the time constant, which is the time taken to discharge 63% of the charge, 37% remaining. Hence the current falls to 37% of its value after this time.
In the graph above, which is drawn from experimental data, 37% of 30 is 11.1. The time taken for the current to fall to 11.1 micro ampere is about 53 seconds.
The time constant = C x R = 470 x10-6 x 105 = 47 seconds.
The calculation and the graph do not exactly agree, but the tolerance in value of the capacitor is 20% so the values are well within that range.