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A capacitor's voltage cannot change instantaneously due to its inherent physical properties. When a DC source is abruptly applied to an RC circuit, the capacitor's voltage immediately after switching remains identical to its value before switching. This constraint is fundamental to capacitor behavior and requires solving a first-order differential equation to determine how the voltage changes over time.

A unit step function represents the abrupt application of a DC source voltage to an RC circuit. When the switch closes at time t=0, the input voltage jumps from zero to the source voltage instantaneously. This mathematical representation allows engineers to analyze the circuit's step response, which characterizes how the capacitor voltage and current evolve following this sudden change in input.

The complete response combines two parts: the circuit response for times before switching (t<0) and the step response for times after switching (t>0). By applying Kirchhoff's current law at t=0 and solving the resulting first-order differential equation through integration and exponentiation, you obtain the capacitor voltage for t>0. Combining this with the initial voltage gives the complete response showing exponential charging toward the source voltage.

After a DC source is applied, the capacitor voltage increases exponentially and gradually approaches the applied source voltage. This charging process is described mathematically by the step response derived from the first-order differential equation. The rate of voltage increase depends on the circuit's resistance and capacitance values, with larger RC products resulting in slower charging.

The current through the capacitor decreases exponentially with time after the DC source is applied. Initially, when the capacitor is uncharged, current is maximum. As the capacitor charges and its voltage approaches the source voltage, the current diminishes exponentially toward zero. This exponential decay reflects the changing voltage difference driving current through the circuit's resistance.

When a capacitor is initially uncharged, the complete response of the RC circuit is modified from the general case. The initial voltage across the capacitor is zero, simplifying the mathematical expression for the step response. This results in the capacitor voltage starting at zero and exponentially rising toward the source voltage, with the current beginning at its maximum value and decaying exponentially.

Kirchhoff's current law is applied at the moment the switch closes (t=0) to establish the relationship between voltage and current in the circuit. By applying this law and rearranging terms, a first-order differential equation is obtained that describes how current changes with time. Solving this equation through integration yields the step response, making Kirchhoff's law fundamental to understanding first-order circuits and their dynamic behavior.