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In an RC circuit, the cut-off frequency, denoted as \( \omega_c \), is an essential parameter. It defines the point at which the output signal's power is reduced to half its maximum value. The cut-off frequency can be mathematically expressed as \( \omega_c = \frac{1}{RC} \), where \( R \) is the resistance and \( C \) is the capacitance.
At this frequency, the circuit's response changes significantly, making it a crucial point of analysis in filters and signal processing applications. For example, in a low-pass filter, frequencies below \( \omega_c \) are allowed to pass, while those above are attenuated. This behavior is critical in shaping signal frequencies according to the circuit's needs.

Impedance is a measure of how much an RC circuit resists the flow of alternating current. It combines the effects of resistance and reactance into a single value. In our RC circuit, the impedance \( Z \) is calculated using the formula \( Z = \sqrt{R^2 + \left( \frac{1}{\omega C} \right)^2} \).
This formula shows impedance as a function of frequency \( \omega \), resistance \( R \), and capacitance \( C \). At the cut-off frequency \( \omega_c \), the reactance \( \frac{1}{\omega_c C} \) and resistance \( R \) combine such that the impedance reaches a special value \( R \sqrt{2} \). This value impacts how the current and voltage split between the resistor and capacitor, affecting their individual voltages.

The resistor voltage in an RC circuit, denoted as \( V_R \), is directly related to both the current through the resistor and its resistance value. It's described by the formula \( V_R = IR \), where \( I \) is the current.
At the cut-off frequency, the impedance affects the current, giving \( I = \frac{\mathcal{E}_0}{R \sqrt{2}} \). The voltage across the resistor then becomes \( V_R = I R = \frac{\mathcal{E}_0}{\sqrt{2}} \). This relationship shows how at the critical frequency, the resistor's voltage is reduced, exemplifying the effect of filtering in the RC circuit.

Capacitor voltage, denoted as \( V_C \), highlights the potential difference across the capacitor in an RC circuit. It's calculated using the equation \( V_C = \mathcal{E}_0 - V_R \), where \( \mathcal{E}_0 \) is the source voltage.
At the cut-off frequency, when \( V_R = \frac{\mathcal{E}_0}{\sqrt{2}} \), the remaining voltage across the capacitor also equals \( \frac{\mathcal{E}_0}{\sqrt{2}} \). This equivalence at \( \omega_c \) illustrates how the energy is equally shared between the resistor and the capacitor, thereby confirming that both contribute to the circuit's overall behavior at the cut-off point. Understanding this balance is key to mastering how RC circuits manage voltages across their components.