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The impedance of a capacitor in the s-domain is given by the formula \( Z(s) = \frac{1}{sC} \). In this expression, \( s \) represents a complex frequency variable used in Laplace transforms, where \( C \) is the capacitance in farads. The use of the s-domain is particularly useful in control systems and engineering applications because it allows for the analysis of circuits in a frequency domain where systems can be represented in terms of their dynamics.

When we derive this impedance, we start from the time-domain relationship for a capacitor, where the current \( I \) through the capacitor is equal to the derivative of the voltage \( V \) across it multiplied by the capacitance \( C \):

\[ I(t) = C \frac{dV(t)}{dt}. \]

Applying the Laplace transform to this equation, we switch from the time domain to the s-domain. The Laplace transform of the time derivative introduces the term \( s \) into our expressions. This transformation results in:

\[ I(s) = sC V(s), \]

which can be rearranged to show that the voltage across the capacitor is related to the current through its impedance:

\[ V(s)