When a DC source is abruptly applied to an RC (Resistor-Capacitor) circuit, the voltage can be represented as a unit step function. The voltage across the capacitor, known as the step response, characterizes how the circuit reacts to this sudden change in input.

Due to the inherent properties of a capacitor, its voltage cannot change instantaneously. This means that immediately after the switch is closed, the capacitor's voltage remains the same as it was just before the switch was closed.

By applying Kirchhoff's current law at the moment the switch is closed (t=0), rearranging the terms, and rewriting the equation for times t>0, a first-order differential equation is obtained. This equation describes how the current through the circuit changes with time following the abrupt application of the DC source.

This differential equation is solved by integrating it, applying the limits, and taking the exponential on both sides. This yields the step response of the capacitor for times t>0. Combining this step response with the initial voltage across the capacitor (for t<0) gives the complete response of the RC circuit.

As time progresses, the voltage across the capacitor increases exponentially and approaches the applied source voltage. This process represents the charging of the capacitor.

If the capacitor is initially uncharged, the complete response of the circuit gets modified accordingly. From this modified response, the current through the capacitor is calculated. This current is observed to decrease exponentially with time, representing the gradual charging of the capacitor until it reaches the source voltage.

In conclusion, understanding the step response of an RC circuit provides valuable insights into how these circuits respond to sudden changes in input voltage. This knowledge is essential for designing and analyzing electronic circuits, particularly in applications such as signal processing, where capacitors are used extensively to filter or shape signals.