Charging a capacitor:

In the diagram below, all of the resistance in the circuit is added together and shown as a single value R.

With S₁ closed and S₂ open, the capacitor will charge up.

Note that Kirchhoff's second law always applies, that is:

                   E  =  V_R  +  V_C

The charging sequence is as follows:

• On closing S₁, no current has yet flowed, the capacitor plates have no charge on them and hence, there is no voltage across it.
• The whole of the applied voltage is developed across the resistor:

                   V_R  =  E

• The initial charging current is equal to the current through the resistor: 
• 
  As C charges, the potential difference across it (V_C) increases, opposing the applied voltage (E) so that the charging current is progressively reduced.
• Finally the capacitor is fully charged (V_C = E) and current ceases (consequently V_R = O).
• This sequence is shown graphically below.
    

The curves are called 'exponential' curves and it can be seen that the slopes dVc/dt and dl/dt are progressively decreasing as time increases.