Define the RC time constant and explain its role in charging a capacitor through a resistor.

Define the RC time constant and explain its role in charging a capacitor through a resistor.

• A. τ = R C; governs charging rate; Vc(t) = V_final (1 - e^{-t/τ}).
• B. τ = C / R; governs charging rate; Vc(t) = V_final e^{-t/τ}.
• C. τ = 1/(R C); governs discharging rate; Vc(t) = V_final (1 - e^{t/τ}).
• D. τ = RC^2; governs charging rate; Vc(t) = V_final (1 - t/τ).

Answer: (A)

The essential idea is that the RC time constant sets how fast a capacitor charges through a resistor. In a charging scenario, the capacitor voltage climbs gradually toward the supply voltage, following an exponential approach. The time constant is the product of the resistance and the capacitance, τ = R · C, and it defines the pace of that charging. The voltage across the capacitor is Vc(t) = V_final [1 − e^(−t/τ)], assuming the capacitor starts discharged and the final voltage is the supply. This means after one time constant, the capacitor reaches about 63% of the final voltage; after five time constants, it’s over 99%. The correct statement uses τ = RC and the proper charging form; the other forms either misstate τ, apply the wrong exponential behavior, or describe discharging rather than charging.

The essential idea is that the RC time constant sets how fast a capacitor charges through a resistor. In a charging scenario, the capacitor voltage climbs gradually toward the supply voltage, following an exponential approach. The time constant is the product of the resistance and the capacitance, τ = R · C, and it defines the pace of that charging. The voltage across the capacitor is Vc(t) = V_final [1 − e^(−t/τ)], assuming the capacitor starts discharged and the final voltage is the supply. This means after one time constant, the capacitor reaches about 63% of the final voltage; after five time constants, it’s over 99%. The correct statement uses τ = RC and the proper charging form; the other forms either misstate τ, apply the wrong exponential behavior, or describe discharging rather than charging.